Research ArticleAn Iterative Regularization Method for Identifying the SourceTerm in a Second Order Differential Equation
Fairouz Zouyed and Sebti Djemoui
Applied Math Lab University Badji Mokhtar Annaba PO Box 12 23000 Annaba Algeria
Correspondence should be addressed to Fairouz Zouyed fzouyedgmailcom
Received 28 May 2015 Accepted 21 September 2015
Academic Editor Peter Dabnichki
Copyright copy 2015 F Zouyed and S Djemoui This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper discusses the inverse problem of determining an unknown source in a second order differential equation frommeasuredfinal data This problem is ill-posed that is the solution (if it exists) does not depend continuously on the data In order to solvethe considered problem an iterative method is proposed Using this method a regularized solution is constructed and an a priorierror estimate between the exact solution and its regularized approximation is obtained Moreover numerical results are presentedto illustrate the accuracy and efficiency of this method
1 Introduction
Let119867 be a separableHilbert space with the inner product (sdot sdot)and the norm sdot Consider the problem of finding the sourceterm 119891 isin 119867 in the following system
with the additional data119906 (119879) = 119892 (2)
where 119860 119863(119860) sub 119867 rarr 119867 is a positive self-adjoint linearoperator with a compact resolvent we denote by 120590(119860) thespectrum of the operator 119860
The problem (1) is an abstract version of the system
which arises in the mathematical study of structural dampedvibrations of string or a beam [1ndash3] Also this problem can beconsidered as a biparabolic problem in the abstract settingFor physical motivation we cite the biparabolic model pro-posed in [4] for more adequate mathematical description ofheat and diffusion processes than the classical heat equationFor other models we refer the reader to [5ndash7]
For most classical partial differential equations thereconstruction of source functions from the final data or apartial boundary data is an inverse problemwithmany appli-cations in several branches of sciences and engineering suchas geophysical prospecting and pollutant detection [8ndash12]
The main difficulty of inverse source identification prob-lems is that they are ill-posed that is even if a solution existsit does not depend continuously on the data in other wordssmall error in the data measurement can induce enormouserror to the solution Thus special regularization methodsthat restore the stability with respect to measurements errorsare needed In the present work we focus on an iterativemethod proposed by Kozlov and Mazrsquoya [13 14] for solvingthe problem it is based on solving a sequence of well-posedboundary value problems such that the sequence of solutionsconverges to the solution for the original problem It hasbeen successfully used for solving various classes of ill-posedelliptic parabolic and hyperbolic problems [5 15ndash21]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 713403 9 pageshttpdxdoiorg1011552015713403
2 Mathematical Problems in Engineering
We note that although the interest in inverse problem hasrapidly increased during this decade the literature devoted tothe class of problems (1) is quite scarce
The paper is organized as follows Section 2 gives sometoolswhich are useful for this study in Section 3we introducesome basic results and we show the ill-posedness of theinverse problem Section 4 gives a regularization solutionand error estimation between the approximate solution andthe exact one the numerical implementation is describedin Section 5 to illustrate the accuracy and efficiency of thismethod
2 Preliminaries
Let (120593119899)119899ge1
sub 119867 be an orthonormal eigenbasis correspondingto the eigenvalues (120582
119899)119899ge1
such that
119860120593119899= 120582
119899120593
119899 119899 isin N
lowast
0 lt 1205821le 120582
2sdot sdot sdot le sdot sdot sdot lim
119899 rarr infin
120582119899= +infin
120585 =
infin
sum
119899=1
119864119899120585
119864119899120585 = (120585 120593
119899) 120593
119899 forall120585 isin 119867
(4)
We denote by 119879(119905) = 119890minus119905119860
119905ge0
the analytic semigroupgenerated by minus119860 on119867
We achieve this section by a result concerning nonexpansiveoperators
Definition 1 A linear bounded operator 119871 119867 rarr 119867 is callednonexpansive if 119871 le 1
Let 119871 be an nonexpansive operator to solve the equation
(119868 minus 119871) 120593 = 120595 (8)
we state a convergence theorem for a successive approxima-tion method
Theorem 2 (see [22] p 66) Let 119871 be a nonexpansive self-adjoint positive operator on 119867 Let 120595 isin 119867 be such that (8)has a solution If 1 is not eigenvalue of 119871 then the successiveapproximations
120593119899+1
= 119871120593119899+ 120595 119899 = 0 1 2 (9)
converge to a solution to (8) for any initial data 1205930isin 119867
Moreover 119871119899120593 rarr 0 for every 120593 isin 119867 as 119899 rarr infin
3 Basic Results
31 The Direct Problem Let 119885 = 119863(119860) times 119867 with the norm119880
2
119885= 119860120585
1
2+ 120585
2
2 119880 = (1205851
1205852
) isin 119885For a given 119891 isin 119867 consider the direct problem
32 Ill-Posedness of the Inverse Problem Now we wish tosolve the inverse problem that is find the source term 119891 inthe system (1) Making use of the supplementary condition(2) and defining the operator 119870(119879) 119891 rarr 119892 we have
It is easy to see that 119870(119879) is a self-adjoint compact linearoperator On the other hand
119892 =
infin
sum
119899=1
119864119899119892 =
infin
sum
119899=1
120590119899119864
119899119891 (19)
so
120590119899119864
119899119891 = 119864
119899119892 (20)
which implies
119864119899119891 =
1
120590119899
119864119899119892 (21)
and therefore
119891 = 119870 (119879)minus1119892 =
infin
sum
119899=1
1
120590119899
119864119899119892 (22)
Note that 1120590119899rarr infin as 119899 rarr infin so the inverse problem is
ill-posed that is the solution does not depend continuouslyon the given data Hence this problem cannot be solved byusing classical numerical methods
Remark 4 As many boundary inverse value problems forpartial differential equations which are ill-posed the studyof the problem (1) is reduced to the study of the equation119870(119879)119891 = 119892 where 119870(119879) is a compact self-adjoint operatorin the Hilbert space119867 This equation can be rewritten in thefollowing way
where 120574 is a positive number satisfying 120574 lt 1119870(119879)In the next section we will show that the operator 119871 is
nonexpansive and 1 is not eigenvalue of 119871 so it follows fromTheorem 2 that (119891
119899)119899isinNlowast converges and (119868 minus 120574119870(119879))
119899119891 rarr 0
for every 119891 isin 119867 as 119899 rarr infin
4 Iterative Procedure andConvergence Results
The alternating iterative method is based on reducing theill-posed problem (1) to a sequence of well-posed boundaryvalue problems and consists of the following steps
First we start by letting 1198910isin 119867 be arbitrary the initial
approximation 1199060is the solution to the direct problem
11990610158401015840
0+ 2119860119906
1015840
0+ 119860
2119906
0= 119891
0 0 lt 119905 lt 119879
1199060(0) = 0
1199061015840
0(0) = 0
(24)
Then if the pair (119891119896 119906
119896) has been constructed let
119891119896+1
= 119891119896minus 120574 (119906
119896(119879) minus 119892) (25)
where 120574 is such that
0 lt 120574 lt1
119870 (119879) (26)
and 119870(119879) = sup119899isinNlowast(1 minus (1 + 119879120582119899
)119890minus120582119899119879)120582
2
119899
Finally we get 119906119896+1
by solving the problem
11990610158401015840
119896+1+ 2119860119906
1015840
119896+1+ 119860
2119906
119896+1= 119891
119896+1 0 lt 119905 lt 119879
119906119896+1
(0) = 0
1199061015840
119896+1(0) = 0
(27)
Let us iterate backwards in (25) to obtain
119891119896+1
= 119891119896minus 120574119870 (119879) 119891
119896+ 120574119892 = (119868 minus 120574119870 (119879)) 119891
119896+ 120574119892
= (119868 minus 120574119870 (119879))119896+1
1198910+ 120574
119896
sum
119895=0
(119868 minus 120574119870 (119879))119895
119892
(28)
Now we introduce some properties and tools which areuseful for our main theorems
Lemma 5 The norm of the operator 119870(119905) is given by
119870 (119905) = sup119899isinNlowast
(1 minus (1 + 119905120582119899) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1199051205821) 119890
minus1205821119905)
1205822
1
(29)
Proof We aim to find the supremum of the function(1 minus (1 + 119905120582
119899)119890
minus120582119899119905)120582
2
119899 119899 isin Nlowast and for this purpose fix 119905
let 120583 = 120582119905 and define the function
1198661(120583) =
(1 minus (1 + 120583) 119890minus120583)
1205832 for 120583 ge 120583
1= 120582
1119905 (30)
We compute
1198661015840
1(120583) =
(1205832+ 2120583 + 2) 119890
minus120583minus 2
1205833 (31)
4 Mathematical Problems in Engineering
Put
ℎ (120583) = (1205832+ 2120583 + 2) 119890
minus120583minus 2 (32)
Hence
1198661015840
1(120583) =
ℎ (120583)
1205833 (33)
To study the monotony of1198661 it suffices to determine the sign
of ℎ We have
ℎ1015840(120583) = minus120583
2119890
minus120583lt 0 forall120583 gt 0 (34)
and then ℎ is decreasing moreover ℎ(120583) sub ] minus 2 0[forall120583 gt 0 Hence 1198661015840
1(120583) lt 0 forall120583 ge 120583
1 which implies that 119866
1
is decreasing and
sup120583ge1205831
1198661(120583) = 119866
1(120583
1) (35)
Therefore
sup119899ge1
(1 minus (1 + 120582119899119905) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1205821119905) 119890
minus1205821119905)
1205822
1
(36)
Proposition 6 For the linear operator 119871 = 119868minus120574119870(119879) one hasthe following properties
(1) 119871 is positive and self-adjoint(2) 119871 is nonexpansive(3) 1 is not an eigenvalue of 119871
Proof Form properties of operator 119860 and the definition of119871 it follows that 119871 is self-adjoint and nonexpansive positiveoperator and from the inequality
0 lt 1 minus 120574
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822lt 1 for 120582 isin 120590 (119860) (37)
it follows that the point spectrum of 119871 120590119901(119871) sub ]0 1[ Then 1
is not eigenvalue of the operator 119871
Lemma 7 If 120582 gt 0 one has the estimates
1
1 + 1205822le max( 3
1198792 1)
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822 (38)
0 lt
(1 minus (1 + 119905120582) 119890minus120582119905)
1205822lt 119879
2 forall119905 isin [0 119879] (39)
Proof To establish (38) let us first prove that
1
3 + 1205832le(1 minus (1 + 120583) 119890
minus120583)
1205832 forall120583 gt 0 (40)
which is equivalent to prove that
1198662(120583) = 3 minus (3 + 120583
2) (1 + 120583) 119890
minus120583ge 0 forall120583 gt 0 (41)
We have
1198661015840
2(120583) = 120583 (120583 minus 1)
2
119890minus120583
ge 0 forall120583 gt 0 (42)
Then 1198662is nondecreasing and it follows that 119866
2(120583) sub ]0 3[
So 1198662(120583) ge 0 forall120583 gt 0
Choosing 120583 = 119879120582 in (40) we obtain
1
3 + (119879120582)2le
(1 minus (119879120582 + 1) 119890minus119879120582
)
(119879120582)2
(43)
So
1198792
max (3 1198792) (1 + 1205822)le
(1 minus (1 + 119879120582) 119890minus119879120582
)
1205822 (44)
From (44) we deduce (38)Now we prove the estimate (39) It is easy to verify that
1198663(120583) = (1 minus (1 + 120583) 119890
minus120583) minus 120583
2lt 0 forall120583 gt 0 (45)
Then if we choose 120583 = 119905120582 we get
(1 minus (1 + 119905120582) 119890minus119905120582) lt 119905
Setting 1206011015840(120573) = 0 it follows that 120573lowast
= 120572(2119896 + 120572)120574 is thecritical point of 120601 It is easy to see that the maximum of 120601 isattained at 120573lowast So
sup0le120573le119879
2
120601 (120573) le 120601 (120573lowast) = (1 minus 120574120573
lowast)2119896
(120573lowast)120572
le (120573lowast)120572
= (120572
(2119896 + 120572) 120574)
120572
(61)
and hence
sup0le120573le119879
2
120601 (120573) le (120572
2120574)
120572
119896minus120572 (62)
Combining (58) and (62) we obtain
sup119905isin[0119879]
1003817100381710038171003817119906119896(119905) minus 119906 (119905)
1003817100381710038171003817
2
le 1198794(120572
2120574max ( 3
1198792 1))
120572
(1
119896)
120572
1198642
(63)
Since in practice the measured data 119892 is never knownexactly but only up to an error of say 120575 gt 0 it is our aimto solve the equation 119870(119879)119891 = 119892 from the knowledge of aperturbed right-hand side 119892120575 satisfying
10038171003817100381710038171003817119892 minus 119892
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
We note that although the interest in inverse problem hasrapidly increased during this decade the literature devoted tothe class of problems (1) is quite scarce
The paper is organized as follows Section 2 gives sometoolswhich are useful for this study in Section 3we introducesome basic results and we show the ill-posedness of theinverse problem Section 4 gives a regularization solutionand error estimation between the approximate solution andthe exact one the numerical implementation is describedin Section 5 to illustrate the accuracy and efficiency of thismethod
2 Preliminaries
Let (120593119899)119899ge1
sub 119867 be an orthonormal eigenbasis correspondingto the eigenvalues (120582
119899)119899ge1
such that
119860120593119899= 120582
119899120593
119899 119899 isin N
lowast
0 lt 1205821le 120582
2sdot sdot sdot le sdot sdot sdot lim
119899 rarr infin
120582119899= +infin
120585 =
infin
sum
119899=1
119864119899120585
119864119899120585 = (120585 120593
119899) 120593
119899 forall120585 isin 119867
(4)
We denote by 119879(119905) = 119890minus119905119860
119905ge0
the analytic semigroupgenerated by minus119860 on119867
We achieve this section by a result concerning nonexpansiveoperators
Definition 1 A linear bounded operator 119871 119867 rarr 119867 is callednonexpansive if 119871 le 1
Let 119871 be an nonexpansive operator to solve the equation
(119868 minus 119871) 120593 = 120595 (8)
we state a convergence theorem for a successive approxima-tion method
Theorem 2 (see [22] p 66) Let 119871 be a nonexpansive self-adjoint positive operator on 119867 Let 120595 isin 119867 be such that (8)has a solution If 1 is not eigenvalue of 119871 then the successiveapproximations
120593119899+1
= 119871120593119899+ 120595 119899 = 0 1 2 (9)
converge to a solution to (8) for any initial data 1205930isin 119867
Moreover 119871119899120593 rarr 0 for every 120593 isin 119867 as 119899 rarr infin
3 Basic Results
31 The Direct Problem Let 119885 = 119863(119860) times 119867 with the norm119880
2
119885= 119860120585
1
2+ 120585
2
2 119880 = (1205851
1205852
) isin 119885For a given 119891 isin 119867 consider the direct problem
32 Ill-Posedness of the Inverse Problem Now we wish tosolve the inverse problem that is find the source term 119891 inthe system (1) Making use of the supplementary condition(2) and defining the operator 119870(119879) 119891 rarr 119892 we have
It is easy to see that 119870(119879) is a self-adjoint compact linearoperator On the other hand
119892 =
infin
sum
119899=1
119864119899119892 =
infin
sum
119899=1
120590119899119864
119899119891 (19)
so
120590119899119864
119899119891 = 119864
119899119892 (20)
which implies
119864119899119891 =
1
120590119899
119864119899119892 (21)
and therefore
119891 = 119870 (119879)minus1119892 =
infin
sum
119899=1
1
120590119899
119864119899119892 (22)
Note that 1120590119899rarr infin as 119899 rarr infin so the inverse problem is
ill-posed that is the solution does not depend continuouslyon the given data Hence this problem cannot be solved byusing classical numerical methods
Remark 4 As many boundary inverse value problems forpartial differential equations which are ill-posed the studyof the problem (1) is reduced to the study of the equation119870(119879)119891 = 119892 where 119870(119879) is a compact self-adjoint operatorin the Hilbert space119867 This equation can be rewritten in thefollowing way
where 120574 is a positive number satisfying 120574 lt 1119870(119879)In the next section we will show that the operator 119871 is
nonexpansive and 1 is not eigenvalue of 119871 so it follows fromTheorem 2 that (119891
119899)119899isinNlowast converges and (119868 minus 120574119870(119879))
119899119891 rarr 0
for every 119891 isin 119867 as 119899 rarr infin
4 Iterative Procedure andConvergence Results
The alternating iterative method is based on reducing theill-posed problem (1) to a sequence of well-posed boundaryvalue problems and consists of the following steps
First we start by letting 1198910isin 119867 be arbitrary the initial
approximation 1199060is the solution to the direct problem
11990610158401015840
0+ 2119860119906
1015840
0+ 119860
2119906
0= 119891
0 0 lt 119905 lt 119879
1199060(0) = 0
1199061015840
0(0) = 0
(24)
Then if the pair (119891119896 119906
119896) has been constructed let
119891119896+1
= 119891119896minus 120574 (119906
119896(119879) minus 119892) (25)
where 120574 is such that
0 lt 120574 lt1
119870 (119879) (26)
and 119870(119879) = sup119899isinNlowast(1 minus (1 + 119879120582119899
)119890minus120582119899119879)120582
2
119899
Finally we get 119906119896+1
by solving the problem
11990610158401015840
119896+1+ 2119860119906
1015840
119896+1+ 119860
2119906
119896+1= 119891
119896+1 0 lt 119905 lt 119879
119906119896+1
(0) = 0
1199061015840
119896+1(0) = 0
(27)
Let us iterate backwards in (25) to obtain
119891119896+1
= 119891119896minus 120574119870 (119879) 119891
119896+ 120574119892 = (119868 minus 120574119870 (119879)) 119891
119896+ 120574119892
= (119868 minus 120574119870 (119879))119896+1
1198910+ 120574
119896
sum
119895=0
(119868 minus 120574119870 (119879))119895
119892
(28)
Now we introduce some properties and tools which areuseful for our main theorems
Lemma 5 The norm of the operator 119870(119905) is given by
119870 (119905) = sup119899isinNlowast
(1 minus (1 + 119905120582119899) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1199051205821) 119890
minus1205821119905)
1205822
1
(29)
Proof We aim to find the supremum of the function(1 minus (1 + 119905120582
119899)119890
minus120582119899119905)120582
2
119899 119899 isin Nlowast and for this purpose fix 119905
let 120583 = 120582119905 and define the function
1198661(120583) =
(1 minus (1 + 120583) 119890minus120583)
1205832 for 120583 ge 120583
1= 120582
1119905 (30)
We compute
1198661015840
1(120583) =
(1205832+ 2120583 + 2) 119890
minus120583minus 2
1205833 (31)
4 Mathematical Problems in Engineering
Put
ℎ (120583) = (1205832+ 2120583 + 2) 119890
minus120583minus 2 (32)
Hence
1198661015840
1(120583) =
ℎ (120583)
1205833 (33)
To study the monotony of1198661 it suffices to determine the sign
of ℎ We have
ℎ1015840(120583) = minus120583
2119890
minus120583lt 0 forall120583 gt 0 (34)
and then ℎ is decreasing moreover ℎ(120583) sub ] minus 2 0[forall120583 gt 0 Hence 1198661015840
1(120583) lt 0 forall120583 ge 120583
1 which implies that 119866
1
is decreasing and
sup120583ge1205831
1198661(120583) = 119866
1(120583
1) (35)
Therefore
sup119899ge1
(1 minus (1 + 120582119899119905) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1205821119905) 119890
minus1205821119905)
1205822
1
(36)
Proposition 6 For the linear operator 119871 = 119868minus120574119870(119879) one hasthe following properties
(1) 119871 is positive and self-adjoint(2) 119871 is nonexpansive(3) 1 is not an eigenvalue of 119871
Proof Form properties of operator 119860 and the definition of119871 it follows that 119871 is self-adjoint and nonexpansive positiveoperator and from the inequality
0 lt 1 minus 120574
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822lt 1 for 120582 isin 120590 (119860) (37)
it follows that the point spectrum of 119871 120590119901(119871) sub ]0 1[ Then 1
is not eigenvalue of the operator 119871
Lemma 7 If 120582 gt 0 one has the estimates
1
1 + 1205822le max( 3
1198792 1)
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822 (38)
0 lt
(1 minus (1 + 119905120582) 119890minus120582119905)
1205822lt 119879
2 forall119905 isin [0 119879] (39)
Proof To establish (38) let us first prove that
1
3 + 1205832le(1 minus (1 + 120583) 119890
minus120583)
1205832 forall120583 gt 0 (40)
which is equivalent to prove that
1198662(120583) = 3 minus (3 + 120583
2) (1 + 120583) 119890
minus120583ge 0 forall120583 gt 0 (41)
We have
1198661015840
2(120583) = 120583 (120583 minus 1)
2
119890minus120583
ge 0 forall120583 gt 0 (42)
Then 1198662is nondecreasing and it follows that 119866
2(120583) sub ]0 3[
So 1198662(120583) ge 0 forall120583 gt 0
Choosing 120583 = 119879120582 in (40) we obtain
1
3 + (119879120582)2le
(1 minus (119879120582 + 1) 119890minus119879120582
)
(119879120582)2
(43)
So
1198792
max (3 1198792) (1 + 1205822)le
(1 minus (1 + 119879120582) 119890minus119879120582
)
1205822 (44)
From (44) we deduce (38)Now we prove the estimate (39) It is easy to verify that
1198663(120583) = (1 minus (1 + 120583) 119890
minus120583) minus 120583
2lt 0 forall120583 gt 0 (45)
Then if we choose 120583 = 119905120582 we get
(1 minus (1 + 119905120582) 119890minus119905120582) lt 119905
Setting 1206011015840(120573) = 0 it follows that 120573lowast
= 120572(2119896 + 120572)120574 is thecritical point of 120601 It is easy to see that the maximum of 120601 isattained at 120573lowast So
sup0le120573le119879
2
120601 (120573) le 120601 (120573lowast) = (1 minus 120574120573
lowast)2119896
(120573lowast)120572
le (120573lowast)120572
= (120572
(2119896 + 120572) 120574)
120572
(61)
and hence
sup0le120573le119879
2
120601 (120573) le (120572
2120574)
120572
119896minus120572 (62)
Combining (58) and (62) we obtain
sup119905isin[0119879]
1003817100381710038171003817119906119896(119905) minus 119906 (119905)
1003817100381710038171003817
2
le 1198794(120572
2120574max ( 3
1198792 1))
120572
(1
119896)
120572
1198642
(63)
Since in practice the measured data 119892 is never knownexactly but only up to an error of say 120575 gt 0 it is our aimto solve the equation 119870(119879)119891 = 119892 from the knowledge of aperturbed right-hand side 119892120575 satisfying
10038171003817100381710038171003817119892 minus 119892
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
32 Ill-Posedness of the Inverse Problem Now we wish tosolve the inverse problem that is find the source term 119891 inthe system (1) Making use of the supplementary condition(2) and defining the operator 119870(119879) 119891 rarr 119892 we have
It is easy to see that 119870(119879) is a self-adjoint compact linearoperator On the other hand
119892 =
infin
sum
119899=1
119864119899119892 =
infin
sum
119899=1
120590119899119864
119899119891 (19)
so
120590119899119864
119899119891 = 119864
119899119892 (20)
which implies
119864119899119891 =
1
120590119899
119864119899119892 (21)
and therefore
119891 = 119870 (119879)minus1119892 =
infin
sum
119899=1
1
120590119899
119864119899119892 (22)
Note that 1120590119899rarr infin as 119899 rarr infin so the inverse problem is
ill-posed that is the solution does not depend continuouslyon the given data Hence this problem cannot be solved byusing classical numerical methods
Remark 4 As many boundary inverse value problems forpartial differential equations which are ill-posed the studyof the problem (1) is reduced to the study of the equation119870(119879)119891 = 119892 where 119870(119879) is a compact self-adjoint operatorin the Hilbert space119867 This equation can be rewritten in thefollowing way
where 120574 is a positive number satisfying 120574 lt 1119870(119879)In the next section we will show that the operator 119871 is
nonexpansive and 1 is not eigenvalue of 119871 so it follows fromTheorem 2 that (119891
119899)119899isinNlowast converges and (119868 minus 120574119870(119879))
119899119891 rarr 0
for every 119891 isin 119867 as 119899 rarr infin
4 Iterative Procedure andConvergence Results
The alternating iterative method is based on reducing theill-posed problem (1) to a sequence of well-posed boundaryvalue problems and consists of the following steps
First we start by letting 1198910isin 119867 be arbitrary the initial
approximation 1199060is the solution to the direct problem
11990610158401015840
0+ 2119860119906
1015840
0+ 119860
2119906
0= 119891
0 0 lt 119905 lt 119879
1199060(0) = 0
1199061015840
0(0) = 0
(24)
Then if the pair (119891119896 119906
119896) has been constructed let
119891119896+1
= 119891119896minus 120574 (119906
119896(119879) minus 119892) (25)
where 120574 is such that
0 lt 120574 lt1
119870 (119879) (26)
and 119870(119879) = sup119899isinNlowast(1 minus (1 + 119879120582119899
)119890minus120582119899119879)120582
2
119899
Finally we get 119906119896+1
by solving the problem
11990610158401015840
119896+1+ 2119860119906
1015840
119896+1+ 119860
2119906
119896+1= 119891
119896+1 0 lt 119905 lt 119879
119906119896+1
(0) = 0
1199061015840
119896+1(0) = 0
(27)
Let us iterate backwards in (25) to obtain
119891119896+1
= 119891119896minus 120574119870 (119879) 119891
119896+ 120574119892 = (119868 minus 120574119870 (119879)) 119891
119896+ 120574119892
= (119868 minus 120574119870 (119879))119896+1
1198910+ 120574
119896
sum
119895=0
(119868 minus 120574119870 (119879))119895
119892
(28)
Now we introduce some properties and tools which areuseful for our main theorems
Lemma 5 The norm of the operator 119870(119905) is given by
119870 (119905) = sup119899isinNlowast
(1 minus (1 + 119905120582119899) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1199051205821) 119890
minus1205821119905)
1205822
1
(29)
Proof We aim to find the supremum of the function(1 minus (1 + 119905120582
119899)119890
minus120582119899119905)120582
2
119899 119899 isin Nlowast and for this purpose fix 119905
let 120583 = 120582119905 and define the function
1198661(120583) =
(1 minus (1 + 120583) 119890minus120583)
1205832 for 120583 ge 120583
1= 120582
1119905 (30)
We compute
1198661015840
1(120583) =
(1205832+ 2120583 + 2) 119890
minus120583minus 2
1205833 (31)
4 Mathematical Problems in Engineering
Put
ℎ (120583) = (1205832+ 2120583 + 2) 119890
minus120583minus 2 (32)
Hence
1198661015840
1(120583) =
ℎ (120583)
1205833 (33)
To study the monotony of1198661 it suffices to determine the sign
of ℎ We have
ℎ1015840(120583) = minus120583
2119890
minus120583lt 0 forall120583 gt 0 (34)
and then ℎ is decreasing moreover ℎ(120583) sub ] minus 2 0[forall120583 gt 0 Hence 1198661015840
1(120583) lt 0 forall120583 ge 120583
1 which implies that 119866
1
is decreasing and
sup120583ge1205831
1198661(120583) = 119866
1(120583
1) (35)
Therefore
sup119899ge1
(1 minus (1 + 120582119899119905) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1205821119905) 119890
minus1205821119905)
1205822
1
(36)
Proposition 6 For the linear operator 119871 = 119868minus120574119870(119879) one hasthe following properties
(1) 119871 is positive and self-adjoint(2) 119871 is nonexpansive(3) 1 is not an eigenvalue of 119871
Proof Form properties of operator 119860 and the definition of119871 it follows that 119871 is self-adjoint and nonexpansive positiveoperator and from the inequality
0 lt 1 minus 120574
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822lt 1 for 120582 isin 120590 (119860) (37)
it follows that the point spectrum of 119871 120590119901(119871) sub ]0 1[ Then 1
is not eigenvalue of the operator 119871
Lemma 7 If 120582 gt 0 one has the estimates
1
1 + 1205822le max( 3
1198792 1)
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822 (38)
0 lt
(1 minus (1 + 119905120582) 119890minus120582119905)
1205822lt 119879
2 forall119905 isin [0 119879] (39)
Proof To establish (38) let us first prove that
1
3 + 1205832le(1 minus (1 + 120583) 119890
minus120583)
1205832 forall120583 gt 0 (40)
which is equivalent to prove that
1198662(120583) = 3 minus (3 + 120583
2) (1 + 120583) 119890
minus120583ge 0 forall120583 gt 0 (41)
We have
1198661015840
2(120583) = 120583 (120583 minus 1)
2
119890minus120583
ge 0 forall120583 gt 0 (42)
Then 1198662is nondecreasing and it follows that 119866
2(120583) sub ]0 3[
So 1198662(120583) ge 0 forall120583 gt 0
Choosing 120583 = 119879120582 in (40) we obtain
1
3 + (119879120582)2le
(1 minus (119879120582 + 1) 119890minus119879120582
)
(119879120582)2
(43)
So
1198792
max (3 1198792) (1 + 1205822)le
(1 minus (1 + 119879120582) 119890minus119879120582
)
1205822 (44)
From (44) we deduce (38)Now we prove the estimate (39) It is easy to verify that
1198663(120583) = (1 minus (1 + 120583) 119890
minus120583) minus 120583
2lt 0 forall120583 gt 0 (45)
Then if we choose 120583 = 119905120582 we get
(1 minus (1 + 119905120582) 119890minus119905120582) lt 119905
Setting 1206011015840(120573) = 0 it follows that 120573lowast
= 120572(2119896 + 120572)120574 is thecritical point of 120601 It is easy to see that the maximum of 120601 isattained at 120573lowast So
sup0le120573le119879
2
120601 (120573) le 120601 (120573lowast) = (1 minus 120574120573
lowast)2119896
(120573lowast)120572
le (120573lowast)120572
= (120572
(2119896 + 120572) 120574)
120572
(61)
and hence
sup0le120573le119879
2
120601 (120573) le (120572
2120574)
120572
119896minus120572 (62)
Combining (58) and (62) we obtain
sup119905isin[0119879]
1003817100381710038171003817119906119896(119905) minus 119906 (119905)
1003817100381710038171003817
2
le 1198794(120572
2120574max ( 3
1198792 1))
120572
(1
119896)
120572
1198642
(63)
Since in practice the measured data 119892 is never knownexactly but only up to an error of say 120575 gt 0 it is our aimto solve the equation 119870(119879)119891 = 119892 from the knowledge of aperturbed right-hand side 119892120575 satisfying
10038171003817100381710038171003817119892 minus 119892
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
To study the monotony of1198661 it suffices to determine the sign
of ℎ We have
ℎ1015840(120583) = minus120583
2119890
minus120583lt 0 forall120583 gt 0 (34)
and then ℎ is decreasing moreover ℎ(120583) sub ] minus 2 0[forall120583 gt 0 Hence 1198661015840
1(120583) lt 0 forall120583 ge 120583
1 which implies that 119866
1
is decreasing and
sup120583ge1205831
1198661(120583) = 119866
1(120583
1) (35)
Therefore
sup119899ge1
(1 minus (1 + 120582119899119905) 119890
minus120582119899119905)
1205822
119899
=
(1 minus (1 + 1205821119905) 119890
minus1205821119905)
1205822
1
(36)
Proposition 6 For the linear operator 119871 = 119868minus120574119870(119879) one hasthe following properties
(1) 119871 is positive and self-adjoint(2) 119871 is nonexpansive(3) 1 is not an eigenvalue of 119871
Proof Form properties of operator 119860 and the definition of119871 it follows that 119871 is self-adjoint and nonexpansive positiveoperator and from the inequality
0 lt 1 minus 120574
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822lt 1 for 120582 isin 120590 (119860) (37)
it follows that the point spectrum of 119871 120590119901(119871) sub ]0 1[ Then 1
is not eigenvalue of the operator 119871
Lemma 7 If 120582 gt 0 one has the estimates
1
1 + 1205822le max( 3
1198792 1)
(1 minus (1 + 119879120582) 119890minus120582119879
)
1205822 (38)
0 lt
(1 minus (1 + 119905120582) 119890minus120582119905)
1205822lt 119879
2 forall119905 isin [0 119879] (39)
Proof To establish (38) let us first prove that
1
3 + 1205832le(1 minus (1 + 120583) 119890
minus120583)
1205832 forall120583 gt 0 (40)
which is equivalent to prove that
1198662(120583) = 3 minus (3 + 120583
2) (1 + 120583) 119890
minus120583ge 0 forall120583 gt 0 (41)
We have
1198661015840
2(120583) = 120583 (120583 minus 1)
2
119890minus120583
ge 0 forall120583 gt 0 (42)
Then 1198662is nondecreasing and it follows that 119866
2(120583) sub ]0 3[
So 1198662(120583) ge 0 forall120583 gt 0
Choosing 120583 = 119879120582 in (40) we obtain
1
3 + (119879120582)2le
(1 minus (119879120582 + 1) 119890minus119879120582
)
(119879120582)2
(43)
So
1198792
max (3 1198792) (1 + 1205822)le
(1 minus (1 + 119879120582) 119890minus119879120582
)
1205822 (44)
From (44) we deduce (38)Now we prove the estimate (39) It is easy to verify that
1198663(120583) = (1 minus (1 + 120583) 119890
minus120583) minus 120583
2lt 0 forall120583 gt 0 (45)
Then if we choose 120583 = 119905120582 we get
(1 minus (1 + 119905120582) 119890minus119905120582) lt 119905
Setting 1206011015840(120573) = 0 it follows that 120573lowast
= 120572(2119896 + 120572)120574 is thecritical point of 120601 It is easy to see that the maximum of 120601 isattained at 120573lowast So
sup0le120573le119879
2
120601 (120573) le 120601 (120573lowast) = (1 minus 120574120573
lowast)2119896
(120573lowast)120572
le (120573lowast)120572
= (120572
(2119896 + 120572) 120574)
120572
(61)
and hence
sup0le120573le119879
2
120601 (120573) le (120572
2120574)
120572
119896minus120572 (62)
Combining (58) and (62) we obtain
sup119905isin[0119879]
1003817100381710038171003817119906119896(119905) minus 119906 (119905)
1003817100381710038171003817
2
le 1198794(120572
2120574max ( 3
1198792 1))
120572
(1
119896)
120572
1198642
(63)
Since in practice the measured data 119892 is never knownexactly but only up to an error of say 120575 gt 0 it is our aimto solve the equation 119870(119879)119891 = 119892 from the knowledge of aperturbed right-hand side 119892120575 satisfying
10038171003817100381710038171003817119892 minus 119892
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
Setting 1206011015840(120573) = 0 it follows that 120573lowast
= 120572(2119896 + 120572)120574 is thecritical point of 120601 It is easy to see that the maximum of 120601 isattained at 120573lowast So
sup0le120573le119879
2
120601 (120573) le 120601 (120573lowast) = (1 minus 120574120573
lowast)2119896
(120573lowast)120572
le (120573lowast)120572
= (120572
(2119896 + 120572) 120574)
120572
(61)
and hence
sup0le120573le119879
2
120601 (120573) le (120572
2120574)
120572
119896minus120572 (62)
Combining (58) and (62) we obtain
sup119905isin[0119879]
1003817100381710038171003817119906119896(119905) minus 119906 (119905)
1003817100381710038171003817
2
le 1198794(120572
2120574max ( 3
1198792 1))
120572
(1
119896)
120572
1198642
(63)
Since in practice the measured data 119892 is never knownexactly but only up to an error of say 120575 gt 0 it is our aimto solve the equation 119870(119879)119891 = 119892 from the knowledge of aperturbed right-hand side 119892120575 satisfying
10038171003817100381710038171003817119892 minus 119892
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
Combining (68) and (71) and passing to the supremum withrespect to 119905 isin [0 119879] we obtain the estimate (65)
Remark 10 If we choose the number of the iterations 119896(120575) sothat 119896(120575) rarr 0 as 120575 rarr 0 we obtain
sup119905isin[0119879]
10038171003817100381710038171003817119906
120575
119896(119905) minus 119906 (119905)
10038171003817100381710038171003817997888rarr 0 as 119896 997888rarr +infin (72)
5 Numerical Implementation
In this section an example is devised for verifying theeffectiveness of the proposed method Consider the problemof finding a pair of functions (119906(119909 119905) 119891(119909)) in the system
It is easy to see that if 119891(119909) = sin120587119909 then
119906 (119909 119905) =
(1 minus (1 + 1205872119905) 119890
minus1205872
119905)
1205874sin (120587119909) (78)
is the exact solution of the problem (73) Consequently119892(119909) = ((1 minus (1 + 120587
2)119890
minus1205872
)1205874)sin(120587119909)
Now we propose to approximate the first and secondspace derivatives by using central difference and we consideran equidistant grid points to a spatial step size 119909
0= 0 lt 119909
1lt
sdot sdot sdot lt 119909119873+1
= 1 (ℎ = 1(119873+1)) where119873 is a positive integerWe get the following semidiscrete problem
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
where 120576 indicates the noise level of the measurements dataand the function randn(sdot) generates arrays of random num-bers whose elements are normally distributed with mean 0variance1205902
= 1 and standard deviation120590 = 1 randn(size(119892))returns an array of random entries that is of the same size as119892The noise level 120575 can bemeasured in the sense of rootmeansquare error (RMSE) according to
120575 =10038171003817100381710038171003817119892
120575minus 119892
100381710038171003817100381710038171198972= (
1
119873 + 1
119873
sum
119894=0
(119892 (119909119894) minus 119892
120575(119909
119894))
2
)
12
(84)
01 02 03 04 05 06 07 08 10 09
01 02 080705 10 0903 0604
Iterative regularization method
0
005
01
015
02
minus05
0
05
1
Error = |exact solution minus approximate solution|
f
e
f
a
e
r
Figure 2 The comparison between the exact solution 119891119890and its
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
In this paper we have extended the iterative method to iden-tify the unknown source term in a second order differentialequation convergence results were established and errorestimates have been obtained under an a priori bound of theexact solution Some numerical tests have been given to verifythe validity of the method
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
All authors read and approved the paper
Acknowledgments
The authors would like to thank the anonymous referees fortheir suggestions
References
[1] H Leiva ldquoA lemmaon1198620-semigroups and applicationsrdquoQuaes-
tiones Mathematicae vol 26 no 3 pp 247ndash265 2003[2] H Leiva Linear Reaction-Diffusion Systems Notas de Mathe-
matica No 185 Mereida 1999[3] L F de Oliveira ldquoOn reaction-diffusion systemsrdquo Electronic
Journal of Differential Equations vol 1998 no 24 10 pages 1998[4] V L Fushchich A S Galitsyn and A S Polubinskii ldquoA new
mathematical model of heat conduction processesrdquo UkrainianMathematical Journal vol 42 no 2 pp 210ndash216 1990
[5] G Bastay ldquoIterative methods for Ill-posed boundary valueproblems Linkoping studies in science and technologyrdquo Dis-sertations 392 Linkoping University Linkoping Sweden 1995
[6] A S Carasso ldquoBochner subordination logarithmic diffusionequations and blind deconvolution of hubble space telescopeimagery and other scientific datardquo SIAM Journal on ImagingSciences vol 3 no 4 pp 954ndash980 2010
[7] L Wang X Zhou and X Wei Heat Conduction MathematicalModels and Analytical Solutions Springer 2008
[8] M Andrle and A El Badia ldquoIdentification of multiple movingpollution sources in surface waters or atmospheric media withboundary observationsrdquo Inverse Problems vol 28 no 7 ArticleID 075009 2012
[9] H W Engl and C Groetsch Inverse and Ill-Posed Problemsvol 4 of Notes and Reports in Mathematics in Science andEngineering Academic press New York NY USA 1987
[10] A KirschAn Introduction to theMathematicalTheory of InverseProblems Springer Heidelberg Germany 1996
[11] D Mace and P Lailly ldquoSolution of the VSP one-dimensionalinverse problemrdquo Geophysical Prospecting vol 34 no 7 pp1002ndash1021 1986
[13] V A Kozlov and V G Mazrsquoya ldquoOn iterative proceduresfor solving ill-posed boundary value problems that preservedifferential equationsrdquo Lenningrad Mathematics Journal vol 1pp 1207ndash1228 1990
[14] V A Kozlov V G Mazrsquoya and A V Fomin ldquoAn iterativemethod for solving the Cauchy problem for elliptic equationsrdquoUSSR Computational Mathematics andMathematical Physicsvol 31 no 1 pp 45ndash52 1991
[15] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems vol 577 ofMathemat-ics and Its Applications Springer Berlin Germany 2004
Mathematical Problems in Engineering 9
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
[16] J Baumeister and A Leitao ldquoOn iterative methods for solvingill-posed problems modeled by partial differential equationsrdquoJournal of Inverse and Ill-Posed Problems vol 9 no 1 pp 13ndash292001
[17] F Berntsson V A Kozlov LMpinganzima and B O TuressonldquoAn alternating iterative procedure for the Cauchy problemfor the Helmholtz equationrdquo Inverse Problems in Science andEngineering vol 22 no 1 pp 45ndash62 2014
[18] A Bouzitouna N Boussetila and F Rebbani ldquoTwo regulariza-tion methods for a class of inverse boundary value problemsof elliptic typerdquo Boundary Value Problems vol 2013 article 1782013
[19] A Lakhdari and N Boussetila ldquoAn iterative regularizationmethod for an abstract ill-posed biparabolic problemrdquo Bound-ary Value Problems vol 2015 article 55 2015
[20] J-GWang and TWei ldquoAn iterativemethod for backward time-fractional diffusion problemrdquo Numerical Methods for PartialDifferential Equations vol 30 no 6 pp 2029ndash2041 2014
[21] H W Zhang and T Wei ldquoTwo iterative methods for a Cauchyproblem of the elliptic equation with variable coefficients in astrip regionrdquo Numerical Algorithms vol 65 no 4 pp 875ndash8922014
[22] M A Krasnoselrsquoskii G M Vainikko P P Zabreiko and YU B Rutitskii Approximate Solutions of Operator EquationsWolters-Noordhoff Publishing Groningen The Netherlands1972
[23] A Pazy Semigroups of Linear Operators and Application toPartial Differential Equations Springer New York NY USA1983
[24] L Elden and V Simoncini ldquoA numerical solution of a Cauchyproblem for an elliptic equation by Krylov subspacesrdquo InverseProblems vol 25 no 6 Article ID 065002 2009
