Research ArticleDouble Laplace Transform Method for Solving Space andTime Fractional Telegraph Equations
Ranjit R Dhunde1 and G L Waghmare2
1Department of Mathematics Datta Meghe Institute of Engineering Technology and Research Wardha India2Department of Mathematics Government Science College Gadchiroli India
Correspondence should be addressed to Ranjit R Dhunde ranjitdhunderediffmailcom
Received 17 June 2016 Revised 23 September 2016 Accepted 5 October 2016
Academic Editor Irena Lasiecka
Copyright copy 2016 R R Dhunde and G L Waghmare This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Double Laplace transform method is applied to find exact solutions of linearnonlinear space-time fractional telegraph equationsin terms of Mittag-Leffler functions subject to initial and boundary conditions Furthermore we give illustrative examples todemonstrate the efficiency of the method
1 Introduction
The telegraph equation developed byOliverHeaviside in 1880is widely used in Science and Engineering Its applicationsarise in signal analysis for transmission and propagation ofelectrical signals and also modelling reaction diffusion
In recent years great interest has been developed in frac-tional differential equation because of its frequent appearancein fluid mechanics mathematical biology electrochemistryand physics A space-time fractional telegraph equation isobtained from the classical telegraph equation by replacingthe time and space derivative terms by fractional derivatives
There are various methods developed to solve fractionaltelegraph equations Orsingher and Xuelei [1] and Orsingherand Beghin [2] considered the space and time fractional tele-graph equations and obtained the Fourier transform of theirfundamental solution Momani [3] and Garg and Sharma [4]used Adomian decomposition method developed by Ado-mian in [5] for solving homogeneous and nonhomogeneousspace-time fractional telegraph equation Chen et al [6]implemented separation of variables method for deriving theanalytical solutions of the nonhomogeneous time fractionaltelegraph equation under Dirichlet Neumann and Robinboundary conditions Huang [7] considered the combineFourier-Laplace transform to solve time fractional telegraphequation
Variational iteration method is proposed by He [8] andused by Sevimlican [9] for solving space and time fractionaltelegraph equations Yildirim in [10] used a homotopy per-turbation method and Das et al in [11] used a homotopyanalysis method to obtain approximate analytical solutionof fractional telegraph equation Garg et al [12] and Galue[13] used generalised differential transformmethod to derivethe solution of space-time fractional telegraph equations interms ofMittag-Leffler functions Srivastava et al [14] appliedreduced differential transform method to solve Caputo timefractional order hyperbolic telegraph equation
In recent years significant attention has been givenby many authors towards the study of fractional telegraphequations by using single Laplace transform combined withvariational iterationmethod homotopy analysismethod andhomotopy perturbation method Khan et al [15] Kumar etal [16] and Prakash [17] applied a combination of singleLaplace transform and homotopy perturbation method toobtain analytic and approximate solutions of the space-time fractional telegraph equations Alawad et al [18] useda combination of single Laplace transform and variationaliteration method for finding exact solutions of space-timefractional telegraph equations in terms of Mittag-Lefflerfunctions Kumar in [19] coupled single Laplace transformand homotopy analysis method for the solution of space-fractional telegraph equation
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2016 Article ID 1414595 7 pageshttpdxdoiorg10115520161414595
2 International Journal of Mathematics and Mathematical Sciences
To our knowledge solving fractional partial differentialequations using the double Laplace transform is still seenin very little proportionate or no work is available in theliterature So the main objective of this paper is to findthe exact solutions of homogeneous and nonhomogeneousspace-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditionsby means of double Laplace transform
2 A Brief Introduction ofDouble Laplace Transform andCaputo Fractional Derivative
Let 119891(119909 119905) be a function of two variables 119909 and 119905 definedin the positive quadrant of the 119909119905-plane The double Laplacetransform of the function 119891(119909 119905) as given by Sneddon [20] isdefined by
The inverse double Laplace transform 119871119909minus1119871 119905minus1119891(119901 119904) =119891(119909 119905) is defined as in [21 22] by the complex double integralformula
where 119891(119901 119904) must be an analytic function for all 119901 and 119904 inthe region defined by the inequalities Re119901 ge 119888 and Re 119904 ge 119889where 119888 and 119889 are real constants to be chosen suitably
The double Laplace transform formulas for the partialderivatives of an arbitrary integer order as in [23] are
Taking double Laplace transform of ℎ(119909 119905) we haveℎ (119901 119904) = minus21199013119904 minus 11199011199042 + 1119901119904 (27)
Substituting above in (15) we get solution of (23)
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [119901120572minus1 11199042minus 119904 2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 1119901 minus
2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 21199013119904 minus 11199011199042
+ 1119901119904]]
(28)
Rearranging we have
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [[ 119901120572
1199011199042 minus 1119901minus 11199011199042 minus 1119901119904] + [
minus2119904 (1199012 minus 1)1199013 (119901120572 minus 1) minus
2 (1199012 minus 1)1199013 (119901120572 minus 1)
minus 21199013119904 + 2119901119904]]]
(29)
Simplifying we obtain
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 11199011199042 +2 (1199012 minus 1)1199013119904 (119901120572 minus 1)]
119906 (119909 119905) = 119905 + 1199092 minus 2 minus 211990921198641205723 (119909120572) + 21198641205721 (119909120572) (30)
which agrees with the solution already obtained in [18] for120574 = 1For 120572 = 2 then 119906(119909 119905) = 119905 + 1199092
Example 3 By substituting 119886 = 1 119887 = 1 119888 = 0 120572 = 2 120574 = 1and ℎ(119909 119905) = minus2119905(1199092 minus 119909)(1199051minus120573Γ(3 minus 120573) + 1) + 21199052 in (9)
120597120573119906 (119909 119905)120597119905120573 + 120597119906 (119909 119905)120597119905= 1205972119906 (119909 119905)1205971199092 + 2119905 (1199092 minus 119909)( 1199051minus120573Γ (3 minus 120573) + 1) minus 21199052
1 lt 120573 le 2 119909 119905 ge 0(31)
subject to the initial and boundary conditions119906 (119909 0) = 1198911 (119909) = 0119906119905 (119909 0) = 1198912 (119909) = 0 (32)
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
2 International Journal of Mathematics and Mathematical Sciences
To our knowledge solving fractional partial differentialequations using the double Laplace transform is still seenin very little proportionate or no work is available in theliterature So the main objective of this paper is to findthe exact solutions of homogeneous and nonhomogeneousspace-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditionsby means of double Laplace transform
2 A Brief Introduction ofDouble Laplace Transform andCaputo Fractional Derivative
Let 119891(119909 119905) be a function of two variables 119909 and 119905 definedin the positive quadrant of the 119909119905-plane The double Laplacetransform of the function 119891(119909 119905) as given by Sneddon [20] isdefined by
The inverse double Laplace transform 119871119909minus1119871 119905minus1119891(119901 119904) =119891(119909 119905) is defined as in [21 22] by the complex double integralformula
where 119891(119901 119904) must be an analytic function for all 119901 and 119904 inthe region defined by the inequalities Re119901 ge 119888 and Re 119904 ge 119889where 119888 and 119889 are real constants to be chosen suitably
The double Laplace transform formulas for the partialderivatives of an arbitrary integer order as in [23] are
Taking double Laplace transform of ℎ(119909 119905) we haveℎ (119901 119904) = minus21199013119904 minus 11199011199042 + 1119901119904 (27)
Substituting above in (15) we get solution of (23)
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [119901120572minus1 11199042minus 119904 2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 1119901 minus
2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 21199013119904 minus 11199011199042
+ 1119901119904]]
(28)
Rearranging we have
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [[ 119901120572
1199011199042 minus 1119901minus 11199011199042 minus 1119901119904] + [
minus2119904 (1199012 minus 1)1199013 (119901120572 minus 1) minus
2 (1199012 minus 1)1199013 (119901120572 minus 1)
minus 21199013119904 + 2119901119904]]]
(29)
Simplifying we obtain
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 11199011199042 +2 (1199012 minus 1)1199013119904 (119901120572 minus 1)]
119906 (119909 119905) = 119905 + 1199092 minus 2 minus 211990921198641205723 (119909120572) + 21198641205721 (119909120572) (30)
which agrees with the solution already obtained in [18] for120574 = 1For 120572 = 2 then 119906(119909 119905) = 119905 + 1199092
Example 3 By substituting 119886 = 1 119887 = 1 119888 = 0 120572 = 2 120574 = 1and ℎ(119909 119905) = minus2119905(1199092 minus 119909)(1199051minus120573Γ(3 minus 120573) + 1) + 21199052 in (9)
120597120573119906 (119909 119905)120597119905120573 + 120597119906 (119909 119905)120597119905= 1205972119906 (119909 119905)1205971199092 + 2119905 (1199092 minus 119909)( 1199051minus120573Γ (3 minus 120573) + 1) minus 21199052
1 lt 120573 le 2 119909 119905 ge 0(31)
subject to the initial and boundary conditions119906 (119909 0) = 1198911 (119909) = 0119906119905 (119909 0) = 1198912 (119909) = 0 (32)
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
Taking double Laplace transform of ℎ(119909 119905) we haveℎ (119901 119904) = minus21199013119904 minus 11199011199042 + 1119901119904 (27)
Substituting above in (15) we get solution of (23)
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [119901120572minus1 11199042minus 119904 2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 1119901 minus
2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 21199013119904 minus 11199011199042
+ 1119901119904]]
(28)
Rearranging we have
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [[ 119901120572
1199011199042 minus 1119901minus 11199011199042 minus 1119901119904] + [
minus2119904 (1199012 minus 1)1199013 (119901120572 minus 1) minus
2 (1199012 minus 1)1199013 (119901120572 minus 1)
minus 21199013119904 + 2119901119904]]]
(29)
Simplifying we obtain
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 11199011199042 +2 (1199012 minus 1)1199013119904 (119901120572 minus 1)]
119906 (119909 119905) = 119905 + 1199092 minus 2 minus 211990921198641205723 (119909120572) + 21198641205721 (119909120572) (30)
which agrees with the solution already obtained in [18] for120574 = 1For 120572 = 2 then 119906(119909 119905) = 119905 + 1199092
Example 3 By substituting 119886 = 1 119887 = 1 119888 = 0 120572 = 2 120574 = 1and ℎ(119909 119905) = minus2119905(1199092 minus 119909)(1199051minus120573Γ(3 minus 120573) + 1) + 21199052 in (9)
120597120573119906 (119909 119905)120597119905120573 + 120597119906 (119909 119905)120597119905= 1205972119906 (119909 119905)1205971199092 + 2119905 (1199092 minus 119909)( 1199051minus120573Γ (3 minus 120573) + 1) minus 21199052
1 lt 120573 le 2 119909 119905 ge 0(31)
subject to the initial and boundary conditions119906 (119909 0) = 1198911 (119909) = 0119906119905 (119909 0) = 1198912 (119909) = 0 (32)
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
Taking double Laplace transform of ℎ(119909 119905) we haveℎ (119901 119904) = minus21199013119904 minus 11199011199042 + 1119901119904 (27)
Substituting above in (15) we get solution of (23)
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [119901120572minus1 11199042minus 119904 2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 1119901 minus
2 (1199012 minus 1)1199013 (119901120572 minus 1) minus 21199013119904 minus 11199011199042
+ 1119901119904]]
(28)
Rearranging we have
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904 minus 1) [[ 119901120572
1199011199042 minus 1119901minus 11199011199042 minus 1119901119904] + [
minus2119904 (1199012 minus 1)1199013 (119901120572 minus 1) minus
2 (1199012 minus 1)1199013 (119901120572 minus 1)
minus 21199013119904 + 2119901119904]]]
(29)
Simplifying we obtain
119906 (119909 119905) = 119871minus1119909 119871minus1119905 [ 11199011199042 +2 (1199012 minus 1)1199013119904 (119901120572 minus 1)]
119906 (119909 119905) = 119905 + 1199092 minus 2 minus 211990921198641205723 (119909120572) + 21198641205721 (119909120572) (30)
which agrees with the solution already obtained in [18] for120574 = 1For 120572 = 2 then 119906(119909 119905) = 119905 + 1199092
Example 3 By substituting 119886 = 1 119887 = 1 119888 = 0 120572 = 2 120574 = 1and ℎ(119909 119905) = minus2119905(1199092 minus 119909)(1199051minus120573Γ(3 minus 120573) + 1) + 21199052 in (9)
120597120573119906 (119909 119905)120597119905120573 + 120597119906 (119909 119905)120597119905= 1205972119906 (119909 119905)1205971199092 + 2119905 (1199092 minus 119909)( 1199051minus120573Γ (3 minus 120573) + 1) minus 21199052
1 lt 120573 le 2 119909 119905 ge 0(31)
subject to the initial and boundary conditions119906 (119909 0) = 1198911 (119909) = 0119906119905 (119909 0) = 1198912 (119909) = 0 (32)
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
6 International Journal of Mathematics and Mathematical Sciences
By substituting (55) in (54) and simplifying we obtain
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= 1199011205721199012 (119904 + 1) minus 119904 [ 11199012 minus 21199013 ] + [ 11199012 minus 21199013 ]minus [ 11199012 minus 21199013 ] minus 2119901120572(119904 + 1) 1199013+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
(56)
(119901120572 minus 1199042 minus 119904) 119906 (119901 119904)= [ 1199011205721199012 (119904 + 1) minus 1199041199012 ] minus [ 2119901120572(119904 + 1) 1199013 minus 21199041199013 ]+ 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]
119906 (119901 119904)= [ 11199012 minus 21199013 ] 1(119904 + 1)+ 1(119901120572 minus 1199042 minus 119904)119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)
2]
(57)
Applying inverse double Laplace transform of (57) we get
119906 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [ 1(119901120572 minus 1199042 minus 119904)sdot 119871119909119871 119905 [1199062 (119909 119905) minus 119890minus2119905 (119909 minus 1199092)2]]
(58)
Now we apply the Iterative method as in [26]
119906 (119909 119905) = infinsum119894=0
119906119894 (119909 119905) (59)
Substituting (59) in (58) we get
infinsum119894=0
119906119894 (119909 119905) = (119909 minus 1199092) 119890minus119905 + 119871minus1119909 119871minus1119905 [[1(119901120572 minus 1199042 minus 119904)
We have applied double Laplace transform to obtain exactsolutions of linearnonlinear space-time fractional telegraphequations All of the examples considered show that doubleLaplace transform method is capable of reducing the volumeof computational work as compared to other methods Itmay be concluded that DLT technique solves the problemswithout using Adomian polynomials Lagrange multipliervalue Hersquos polynomials and small parameters
Competing Interests
The authors declare no competing interests regarding thepublication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
International Journal of Mathematics and Mathematical Sciences 7
References
[1] E Orsingher and Z Xuelei ldquoThe space-fractional telegraphequation and the related fractional telegraph processrdquo ChineseAnnals Mathematics B vol 24 no 1 pp 45ndash56 2003
[2] E Orsingher and L Beghin ldquoTime-fractional telegraph equa-tions and telegraph processes with Brownian timerdquo ProbabilityTheory and Related Fields vol 128 no 1 pp 141ndash160 2004
[3] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005
[4] M Garg and A Sharma ldquoSolution of space-time fractional tele-graph equation by Adomian decompositionmethodrdquo Journal ofInequalities and Special Functions vol 2 no 1 pp 1ndash7 2011
[5] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sitionMethod vol 60 Kluwer Academic Publishers DordrechtThe Netherlands 1994
[6] J Chen F Liu and V Anh ldquoAnalytical solution for the time-fractional telegraph equation by the method of separatingvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 338 no 2 pp 1364ndash1377 2008
[7] F Huang ldquoAnalytical solution for the time-fractional telegraphequationrdquo Journal of Applied Mathematics vol 2009 Article ID890158 9 pages 2009
[8] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[9] A Sevimlican ldquoAn approximation to solution of space andtime fractional telegraph equations by Hersquos variational iterationmethodrdquo Mathematical Problems in Engineering vol 2010Article ID 290631 10 pages 2010
[10] A Yildirim ldquoHersquos homotopy perturbation method for solvingthe space- and time-fractional telegraph equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 13 pp2998ndash3006 2010
[11] S Das K Vishal P K Gupta and A Yildirim ldquoAn approxi-mate analytical solution of time-fractional telegraph equationrdquoApplied Mathematics and Computation vol 217 no 18 pp7405ndash7411 2011
[12] M Garg P Manohar and S L Kalla ldquoGeneralized differentialtransformmethod to space-time fractional telegraph equationrdquoInternational Journal of Differential Equations vol 2011 ArticleID 548982 9 pages 2011
[13] L Galue ldquoSolution of some fractional order telegraph equa-tionsrdquo Revista Colombiana de Mathematicas vol 48 no 2 pp247ndash267 2014
[14] VK SrivastavaMKAwasthi andMTamsir ldquoRDTMsolutionof Caputo time fractional-order hyperbolic telegraph equationrdquoAIP Advances vol 3 no 3 2013
[15] Y Khan J Diblik N Faraz and Z Smarda ldquoAn efficientnew perturbative Laplace method for space-time fractionaltelegraph equationsrdquo Advances in Difference Equations vol2012 article 204 2012
[16] D Kumar J Singh and S Kumar ldquoAnalytic and approxi-mate solutions of space-time fractional telegraph equations vialaplace transformrdquoWalailak Journal of Science and Technologyvol 11 no 8 pp 711ndash728 2014
[17] A Prakash ldquoAnalytical method for space-fractional telegraphequation by homotopy perturbation transform methodrdquo Non-linear Engineering vol 5 no 2 pp 123ndash128 2016
[18] F A Alawad E A Yousif and A I Arbab ldquoA new techniqueof Laplace variational iteration method for solving space-time fractional telegraph equationsrdquo International Journal ofDifferential Equations vol 2013 Article ID 256593 10 pages2013
[19] S Kumar ldquoA new analytical modelling for fractional telegraphequation via Laplace transformrdquo Applied Mathematical Mod-elling vol 38 no 13 pp 3154ndash3163 2014
[20] I N SneddonTheUse of Integral Transforms TataMcgrawHill1974
[21] L Debnath and D Bhatta Integral Transforms and TheirApplications CRC Press Taylor amp Francis Group Boca RatonFla USA 3rd edition 2015
[22] L Debnath ldquoThe double Laplace transforms and their prop-erties with applications to functional integral and partialdifferential equationsrdquo International Journal of Applied andComputational Mathematics vol 2 no 2 pp 223ndash241 2016
[23] A M Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013
[24] S Sarwar and M M Rashidi ldquoApproximate solution of two-term fractional-order diffusion wave-diffusion and telegraphmodels arising in mathematical physics using optimal homo-topy asymptotic methodrdquo Waves in Random and ComplexMedia vol 26 no 3 pp 365ndash382 2016
[25] R Joice Nirmala and K Balachandran ldquoAnalysis of solutionsof time fractional telegraph equationrdquo Journal of the KoreanSociety for Industrial and AppliedMathematics vol 18 no 3 pp209ndash224 2014
[26] R R Dhunde and G L Waghmare ldquoDouble Laplace trans-form combined with Iterative method for solving non-lineartelegraph equationrdquo Journal of the Indian Mathematical Societyvol 83 no 3-4 pp 221ndash230 2016
