Research ArticleStability Analysis of Additive Runge-Kutta Methods forDelay-Integro-Differential Equations
Hongyu Qin 1 ZhiyongWang2 Fumin Zhu 3 and JinmingWen4
1Wenhua College Wuhan 430074 China2School of Mathematical Sciences University of Electronic Science and Technology of China Sichuan 611731 China3College of Economics Shenzhen University Shenzhen 518060 China4Department of Electrical and Computer Engineering University of Toronto Toronto Canada M5S3G4
Correspondence should be addressed to Fumin Zhu zhufuminszueducn
Received 3 March 2018 Accepted 6 May 2018 Published 11 June 2018
Academic Editor Gaston Mandata Nrsquoguerekata
Copyright copy 2018 Hongyu Qin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper is concerned with stability analysis of additive Runge-Kutta methods for delay-integro-differential equations We showthat if the additive Runge-Kutta methods are algebraically stable the perturbations of the numerical solutions are controlled by theinitial perturbations from the system and the methods
1 Introduction
Spatial discretization of many nonlinear parabolic problemsusually gives a class of ordinary differential equations whichhave the stiff part and the nonstiff part see eg [1ndash5] Insuch cases the most widely used time-discretizations are thespecial organized numerical methods such as the implicit-explicit numerical methods [6 7] the additive Runge-Kuttamethods [8ndash12] and the linearized methods [13 14] Whenapplying the split numerical methods to numerically solvethe equations it is important to investigate the stability of thenumerical methods
In this paper it is assumed that the spatial discretizationof time-dependent partial differential equations yields thefollowing nonlinear delay-integro-differential equations
119910 (119905) = 120595 (119905) minus 120591 le 119905 le 0
(1)
Here 120591 is a positive delay term 120595(119905) is continuous 119891[1][1199050 +infin]times119883 rarr 119883 and119891[2] [1199050 +infin]times119883times119883times119883 rarr 119883 suchthat problem (1) owns a unique solution where119883 is a real orcomplex Hilbert space Particularly when 119892 equiv 0 problem (1)is reduced to the nonlinear delay differential equationsWhenthe delay term 120591 = 0 problem (1) is reduced to the ordinarydifferential equations
The investigation on stability analysis of different numer-ical methods for problem (1) has fascinated generations ofresearchers For example Torelli [15] considered stabilityof Euler methods for the nonautonomous nonlinear delaydifferential equations Hout [16] studied the stability ofRunge-Kutta methods for systems of delay differential equa-tions Baker and Ford [17] discussed stability of continuousRunge-Kutta methods for integrodifferential systems withunbounded delays Zhang and Vandewalle [18] discussed thestability of the general linear methods for integrodifferentialequations with memory Li and Zhang obtained the stabilityand convergence of the discontinuous Galerkin methodsfor nonlinear delay differential equations [19 20] Morereferences for this topic can be found in [21ndash30]However fewworks have been found on the stability of splitting methodsfor the proposed methods
In the present work we present the additive Runge-Kutta methods with some appropriate quadrature rules
HindawiInternational Journal of Differential EquationsVolume 2018 Article ID 8241784 5 pageshttpsdoiorg10115520188241784
2 International Journal of Differential Equations
to numerically solve the nonlinear delay-integrodifferentialequations (1) It is shown that if the additive Runge-Kuttamethods are algebraically stable the obtained numericalsolutions are globally and asymptotically stable under thegiven assumptions respectively The rest of the paper isorganized as follows In Section 2 we present the numericalmethods for problems (1) In Section 3 we consider stabilityanalysis of the numerical schemes Finally we present someextensions in Section 4
2 The Numerical Methods
In this section we present the additive Runge-Kutta methodswith the appropriate quadrature rules to numerically solveproblem (1)
The coefficients of the additive Runge-Kutta methods canbe organized in Buther tableau as follows (cf [31])
where 119888 = [119888119897 sdot sdot sdot 119888119904]119879 119887[119896] = [119887[119896]1 sdot sdot sdot 119887[119896]119904 ]119879 and 119860[119896] =(119886[119896]119894119895 )119904119894119895=1 for 119896 = 1 2Then the presented ARKMs for problem (1) can be
where 119910119899 and 119910(119899)119894 are approximations to the analytic solution119910(119905119899) and 119910(119905119899 + 119888119894ℎ) respectively 119910119899 = 120595(119905119899) for 119899 le 0 119910(119899)119894 =120595(119905119899+119888119894ℎ) for 119905119899+119888119894ℎ le 0 and 119910(119899)119894 denotes the approximationto int119905119899+119888119894ℎ119905119899+119888119894ℎminus120591
119892(119905119899 + 119888119894ℎ 120585 119910(120585))119889120585 which can be computed bysome appropriate quadrature rules
For example we usually adopt the repeated Simpsonrsquos rule orNewton-Cotes rule etc according to the requirement of theconvergence of the method (cf [18])
3 Stability Analysis
In this section we consider the numerical stability of theproposed methods First we introduce a perturbed problemwhose solution satisfies
119910 (119905) = 120601 (119905) minus 120591 le 119905 le 0
(5)
It is assumed that there exist some inner product lt sdot sdot gt andthe induced norm sdot such that
Re ⟨119910 minus 119911 119891[1] (119905 119910) minus 119891[1] (119905 119911)⟩ le 120572 1003817100381710038171003817119910 minus 11991110038171003817100381710038172 Re ⟨119910 minus 119911 119891[2] (119905 119910 1199061 V1) minus 119891[2] (119905 119911 1199062 V2)⟩
le 1205731 1003817100381710038171003817119910 minus 11991110038171003817100381710038172 + 1205732 10038171003817100381710038171199061 minus 119906210038171003817100381710038172 + 120574 1003817100381710038171003817V1 minus V210038171003817100381710038172 1003817100381710038171003817119892 (119905 V 1199041) minus 119892 (119905 V 1199042)1003817100381710038171003817 le 120579 10038171003817100381710038171199041 minus 11990421003817100381710038171003817
(6)
where 120572 lt 0 1205731 lt 0 1205732 gt 0 120574 gt 0 and 120579 gt 0 are constantsIt is remarkable that the conditions can be equivalent to theassumptions in [32 33] (see [34] 119877119890119898119886119903119896 21)Definition 1 (cf [9]) An additive Runge-Kutta method iscalled algebraically stable if the matrices
119861] fl diag (119887[]]1 sdot sdot sdot 119887[]]119904 ) V = 1 2119872]120583 fl 119861]119860[120583] + 119860[]]119879119861120583 minus 119887[]]119887[120583]119879 (7)
are nonnegative
Theorem 2 Assume an additive Runge-Kutta method isalgebraically stable and 1205731 + 1205732 + 4120574120591212057821205792 lt 0 where 120578 =max1199011 1199012 sdot sdot sdot 119901119896 Then it holds that
1003817100381710038171003817119910119899 minus 1199111198991003817100381710038171003817 le radic(1 + 2 119904sum119894=1
1003817100381710038171003817120595 (119905) minus 120601 (119905)1003817100381710038171003817 (8)
where 119910119899 and 119911119899 are numerical approximations to problems (1)and (5) respectively
International Journal of Differential Equations 3
Proof Let 119910119899 119910(119899)119894 119910(119899)119894 ) and 119911119899 119911(119899)119894 (119899)119894 ) be twosequences of approximations to problems (1) and (5)respectively by ARKMs with the same stepsize ℎ and write
119880(119899)119894 = 119910(119899)119894 minus 119911(119899)119894 (119899)119894 = 119910(119899)119894 minus (119899)119894 119880(119899)0 = 119910119899 minus 119911119899119882[1]119894 = ℎ [119891[1] (119905119899 + 119888[1]119894 ℎ 119910(119899)119894 )
Re ⟨119880(119899)119894 119882[1]119894 ⟩ le 120572ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 (13)
and
Re ⟨119880(119899)119894 119882[2]119894 ⟩ le 1205731ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 + 1205732ℎ 10038171003817100381710038171003817119880(119899minus119898)119894 100381710038171003817100381710038172+ 120574ℎ 10038171003817100381710038171003817(119899)119894 100381710038171003817100381710038172
(14)
Together with (11) (12) (13) and (14) we get
10038171003817100381710038171003817119880(119899+1)0 100381710038171003817100381710038172 le 10038171003817100381710038171003817119880(119899)0 100381710038171003817100381710038172 + 2 119904sum119894=1
Remark 4 In [35] Yuan et al also discussed nonlinearstability of additive Runge-Kutta methods for multidelay-integro-differential equations However the main results aredifferent The main reason is that the results in [35] implythat the perturbations of the numerical solutions tend toinfinity when the time increase while the stability resultsin present paper indicate that the perturbations of thenumerical solutions are independent of the time Besides theasymptotical stability of the methods is also discussed in thepresent paper
4 Conclusion
The additive Runge-Kutta methods with some appropriatequadrature rules are applied to solve the delay-integro-differential equations It is shown that if the additive Runge-Kutta methods are algebraically stable the obtained numer-ical solutions can be globally and asymptotically stablerespectively In the future works we will apply the methodsto solve more real-world problems
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported in part by the National NaturalScience Foundation of China (71601125)
References
[1] V Thomee Galerkin Finite Element Methods for ParabolicProblems Springer Berlin Germany 1997
[2] J WuTheory and Applications of Partial Functional-DifferentialEquations Springer New York NY USA 1996
International Journal of Differential Equations 5
[3] J R Cannon and Y Lin ldquoNon-classical H1 projection andGalerkin methods for non-linear parabolic integro-differentialequationsrdquo Calcolo vol 25 pp 187ndash201 1988
[4] D Li and J Wang ldquoUnconditionally optimal error analysis ofcrank-nicolson galerkin fems for a strongly nonlinear parabolicsystemrdquo Journal of Scientific Computing vol 72 no 2 pp 892ndash915 2017
[5] B Li and W Sun ldquoError analysis of linearized semi-implicitgalerkin finite element methods for nonlinear parabolic equa-tionsrdquo International Journal of Numerical Analysis amp Modelingvol 10 no 3 pp 622ndash633 2013
[6] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995
[7] G Akrivis and B Li ldquoMaximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolicequationsrdquo SIAM Journal on Numerical Analysis 2017
[8] I Higueras ldquoStrong stability for additive Runge-Kutta meth-odsrdquo SIAM Journal on Numerical Analysis vol 44 no 4 pp1735ndash1758 2006
[9] A Araujo ldquoA note on B-stability of splitting methodsrdquo Comput-ing and Visualization in Science vol 26 no 2-3 pp 53ndash57 2004
[10] C A Kennedy and M H Carpenter ldquoAdditive Runge-Kuttaschemes for convection-diffusion-reaction equationsrdquo AppliedNumerical Mathematics vol 44 no 1-2 pp 139ndash181 2003
[11] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 pp 201ndash212 2008
[12] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006
[13] D Li C Zhang and M Ran ldquoA linear finite differencescheme for generalized time fractional Burgers equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp6069ndash6081 2016
[14] D Li J Wang and J Zhang ldquoUnconditionally convergentL1-Galerkin FEMs for nonlinear time-fractional Schrodingerequationsrdquo SIAM Journal on Scientific Computing vol 39 no6 pp A3067ndashA3088 2017
[15] L Torelli ldquoStability of numerical methods for delay differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 25 no 1 pp 15ndash26 1989
[16] K J inrsquot Hout ldquoStability analysis of Runge-Kutta methodsfor systems of delay differential equationsrdquo IMA Journal ofNumerical Analysis vol 17 no 1 pp 17ndash27 1997
[17] C T Baker and A Tang ldquoStability analysis of continuousimplicit Runge-Kutta methods for Volterra integro-differentialsystemswith unbounded delaysrdquoAppliedNumericalMathemat-ics vol 24 no 2-3 pp 153ndash173 1997
[18] C Zhang and S Vandewalle ldquoGeneral linear methods forVolterra integro-differential equations with memoryrdquo SIAMJournal on Scientific Computing vol 27 no 6 pp 2010ndash20312006
[19] D Li and C Zhang ldquoNonlinear stability of discontinuousGalerkin methods for delay differential equationsrdquo AppliedMathematics Letters vol 23 no 4 pp 457ndash461 2010
[20] D Li and C Zhang ldquoLinfin error estimates of discontinuousGalerkin methods for delay differential equationsrdquo AppliedNumerical Mathematics vol 82 pp 1ndash10 2014
[21] V K Barwell ldquoSpecial stability problems for functional differ-ential equationsrdquo BIT vol 15 pp 130ndash135 1975
[22] A Bellen and M Zennaro ldquoStrong contractivity properties ofnumerical methods for ordinary and delay differential equa-tionsrdquo Applied Numerical Mathematics vol 9 no 3-5 pp 321ndash346 1992
[23] K Burrage ldquoHigh order algebraically stable Runge-Kutta meth-odsrdquo BIT vol 18 no 4 pp 373ndash383 1978
[24] K Burrage and J C Butcher ldquoNonlinear stability of a generalclass of differential equation methodsrdquo BIT vol 20 no 2 pp185ndash203 1980
[25] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983
[26] K Dekker and J G Verwer Stability of Runge-Kutta Methodsfor Stiff Nonlinear Differential Equations North-Holland Pub-lishing Amsterdam The Netherlands 1984
[27] L Ferracina and M N Spijker ldquoStrong stability of singly-diagonally-implicit Runge-Kutta methodsrdquo Applied NumericalMathematics vol 58 no 11 pp 1675ndash1686 2008
[28] K J inrsquot Hout and M N Spijker ldquoThe 120579-methods in thenumerical solution of delay differential equationsrdquo in TheNumerical Treatment of Differential Equations K Strehmel Edvol 121 pp 61ndash67 1991
[29] M Zennaro ldquoAsymptotic stability analysis of Runge-Kuttamethods for nonlinear systems of delay differential equationsrdquoNumerische Mathematik vol 77 no 4 pp 549ndash563 1997
[30] D Li C Zhang and W Wang ldquoLong time behavior of non-Fickian delay reaction-diffusion equationsrdquoNonlinear AnalysisReal World Applications vol 13 no 3 pp 1401ndash1415 2012
[31] B Garcia-Celayeta I Higueras and T Roldan ldquoContrac-tivitymonotonicity for additive Range-kutta methods Innerproduct normsrdquo Applied Numerical Mathematics vol 56 no 6pp 862ndash878 2006
[32] C Huang ldquoDissipativity of one-leg methods for dynamicalsystems with delaysrdquo Applied Numerical Mathematics vol 35no 1 pp 11ndash22 2000
[33] C Zhang and S Zhou ldquoNonlinear stability and D-convergenceof Runge-Kutta methods for delay differential equationsrdquo Jour-nal of Computational and Applied Mathematics vol 85 no 2pp 225ndash237 1997
[34] C Huang S Li H Fu and G Chen ldquoNonlinear stability ofgeneral linear methods for delay differential equationsrdquo BITNumerical Mathematics vol 42 no 2 pp 380ndash392 2002
[35] H Yuan J Zhao and Y Xu ldquoNonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 854517 22 pages 2012
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Submit your manuscripts atwwwhindawicom
2 International Journal of Differential Equations
to numerically solve the nonlinear delay-integrodifferentialequations (1) It is shown that if the additive Runge-Kuttamethods are algebraically stable the obtained numericalsolutions are globally and asymptotically stable under thegiven assumptions respectively The rest of the paper isorganized as follows In Section 2 we present the numericalmethods for problems (1) In Section 3 we consider stabilityanalysis of the numerical schemes Finally we present someextensions in Section 4
2 The Numerical Methods
In this section we present the additive Runge-Kutta methodswith the appropriate quadrature rules to numerically solveproblem (1)
The coefficients of the additive Runge-Kutta methods canbe organized in Buther tableau as follows (cf [31])
where 119888 = [119888119897 sdot sdot sdot 119888119904]119879 119887[119896] = [119887[119896]1 sdot sdot sdot 119887[119896]119904 ]119879 and 119860[119896] =(119886[119896]119894119895 )119904119894119895=1 for 119896 = 1 2Then the presented ARKMs for problem (1) can be
where 119910119899 and 119910(119899)119894 are approximations to the analytic solution119910(119905119899) and 119910(119905119899 + 119888119894ℎ) respectively 119910119899 = 120595(119905119899) for 119899 le 0 119910(119899)119894 =120595(119905119899+119888119894ℎ) for 119905119899+119888119894ℎ le 0 and 119910(119899)119894 denotes the approximationto int119905119899+119888119894ℎ119905119899+119888119894ℎminus120591
119892(119905119899 + 119888119894ℎ 120585 119910(120585))119889120585 which can be computed bysome appropriate quadrature rules
For example we usually adopt the repeated Simpsonrsquos rule orNewton-Cotes rule etc according to the requirement of theconvergence of the method (cf [18])
3 Stability Analysis
In this section we consider the numerical stability of theproposed methods First we introduce a perturbed problemwhose solution satisfies
119910 (119905) = 120601 (119905) minus 120591 le 119905 le 0
(5)
It is assumed that there exist some inner product lt sdot sdot gt andthe induced norm sdot such that
Re ⟨119910 minus 119911 119891[1] (119905 119910) minus 119891[1] (119905 119911)⟩ le 120572 1003817100381710038171003817119910 minus 11991110038171003817100381710038172 Re ⟨119910 minus 119911 119891[2] (119905 119910 1199061 V1) minus 119891[2] (119905 119911 1199062 V2)⟩
le 1205731 1003817100381710038171003817119910 minus 11991110038171003817100381710038172 + 1205732 10038171003817100381710038171199061 minus 119906210038171003817100381710038172 + 120574 1003817100381710038171003817V1 minus V210038171003817100381710038172 1003817100381710038171003817119892 (119905 V 1199041) minus 119892 (119905 V 1199042)1003817100381710038171003817 le 120579 10038171003817100381710038171199041 minus 11990421003817100381710038171003817
(6)
where 120572 lt 0 1205731 lt 0 1205732 gt 0 120574 gt 0 and 120579 gt 0 are constantsIt is remarkable that the conditions can be equivalent to theassumptions in [32 33] (see [34] 119877119890119898119886119903119896 21)Definition 1 (cf [9]) An additive Runge-Kutta method iscalled algebraically stable if the matrices
119861] fl diag (119887[]]1 sdot sdot sdot 119887[]]119904 ) V = 1 2119872]120583 fl 119861]119860[120583] + 119860[]]119879119861120583 minus 119887[]]119887[120583]119879 (7)
are nonnegative
Theorem 2 Assume an additive Runge-Kutta method isalgebraically stable and 1205731 + 1205732 + 4120574120591212057821205792 lt 0 where 120578 =max1199011 1199012 sdot sdot sdot 119901119896 Then it holds that
1003817100381710038171003817119910119899 minus 1199111198991003817100381710038171003817 le radic(1 + 2 119904sum119894=1
1003817100381710038171003817120595 (119905) minus 120601 (119905)1003817100381710038171003817 (8)
where 119910119899 and 119911119899 are numerical approximations to problems (1)and (5) respectively
International Journal of Differential Equations 3
Proof Let 119910119899 119910(119899)119894 119910(119899)119894 ) and 119911119899 119911(119899)119894 (119899)119894 ) be twosequences of approximations to problems (1) and (5)respectively by ARKMs with the same stepsize ℎ and write
119880(119899)119894 = 119910(119899)119894 minus 119911(119899)119894 (119899)119894 = 119910(119899)119894 minus (119899)119894 119880(119899)0 = 119910119899 minus 119911119899119882[1]119894 = ℎ [119891[1] (119905119899 + 119888[1]119894 ℎ 119910(119899)119894 )
Re ⟨119880(119899)119894 119882[1]119894 ⟩ le 120572ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 (13)
and
Re ⟨119880(119899)119894 119882[2]119894 ⟩ le 1205731ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 + 1205732ℎ 10038171003817100381710038171003817119880(119899minus119898)119894 100381710038171003817100381710038172+ 120574ℎ 10038171003817100381710038171003817(119899)119894 100381710038171003817100381710038172
(14)
Together with (11) (12) (13) and (14) we get
10038171003817100381710038171003817119880(119899+1)0 100381710038171003817100381710038172 le 10038171003817100381710038171003817119880(119899)0 100381710038171003817100381710038172 + 2 119904sum119894=1
Remark 4 In [35] Yuan et al also discussed nonlinearstability of additive Runge-Kutta methods for multidelay-integro-differential equations However the main results aredifferent The main reason is that the results in [35] implythat the perturbations of the numerical solutions tend toinfinity when the time increase while the stability resultsin present paper indicate that the perturbations of thenumerical solutions are independent of the time Besides theasymptotical stability of the methods is also discussed in thepresent paper
4 Conclusion
The additive Runge-Kutta methods with some appropriatequadrature rules are applied to solve the delay-integro-differential equations It is shown that if the additive Runge-Kutta methods are algebraically stable the obtained numer-ical solutions can be globally and asymptotically stablerespectively In the future works we will apply the methodsto solve more real-world problems
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported in part by the National NaturalScience Foundation of China (71601125)
References
[1] V Thomee Galerkin Finite Element Methods for ParabolicProblems Springer Berlin Germany 1997
[2] J WuTheory and Applications of Partial Functional-DifferentialEquations Springer New York NY USA 1996
International Journal of Differential Equations 5
[3] J R Cannon and Y Lin ldquoNon-classical H1 projection andGalerkin methods for non-linear parabolic integro-differentialequationsrdquo Calcolo vol 25 pp 187ndash201 1988
[4] D Li and J Wang ldquoUnconditionally optimal error analysis ofcrank-nicolson galerkin fems for a strongly nonlinear parabolicsystemrdquo Journal of Scientific Computing vol 72 no 2 pp 892ndash915 2017
[5] B Li and W Sun ldquoError analysis of linearized semi-implicitgalerkin finite element methods for nonlinear parabolic equa-tionsrdquo International Journal of Numerical Analysis amp Modelingvol 10 no 3 pp 622ndash633 2013
[6] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995
[7] G Akrivis and B Li ldquoMaximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolicequationsrdquo SIAM Journal on Numerical Analysis 2017
[8] I Higueras ldquoStrong stability for additive Runge-Kutta meth-odsrdquo SIAM Journal on Numerical Analysis vol 44 no 4 pp1735ndash1758 2006
[9] A Araujo ldquoA note on B-stability of splitting methodsrdquo Comput-ing and Visualization in Science vol 26 no 2-3 pp 53ndash57 2004
[10] C A Kennedy and M H Carpenter ldquoAdditive Runge-Kuttaschemes for convection-diffusion-reaction equationsrdquo AppliedNumerical Mathematics vol 44 no 1-2 pp 139ndash181 2003
[11] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 pp 201ndash212 2008
[12] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006
[13] D Li C Zhang and M Ran ldquoA linear finite differencescheme for generalized time fractional Burgers equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp6069ndash6081 2016
[14] D Li J Wang and J Zhang ldquoUnconditionally convergentL1-Galerkin FEMs for nonlinear time-fractional Schrodingerequationsrdquo SIAM Journal on Scientific Computing vol 39 no6 pp A3067ndashA3088 2017
[15] L Torelli ldquoStability of numerical methods for delay differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 25 no 1 pp 15ndash26 1989
[16] K J inrsquot Hout ldquoStability analysis of Runge-Kutta methodsfor systems of delay differential equationsrdquo IMA Journal ofNumerical Analysis vol 17 no 1 pp 17ndash27 1997
[17] C T Baker and A Tang ldquoStability analysis of continuousimplicit Runge-Kutta methods for Volterra integro-differentialsystemswith unbounded delaysrdquoAppliedNumericalMathemat-ics vol 24 no 2-3 pp 153ndash173 1997
[18] C Zhang and S Vandewalle ldquoGeneral linear methods forVolterra integro-differential equations with memoryrdquo SIAMJournal on Scientific Computing vol 27 no 6 pp 2010ndash20312006
[19] D Li and C Zhang ldquoNonlinear stability of discontinuousGalerkin methods for delay differential equationsrdquo AppliedMathematics Letters vol 23 no 4 pp 457ndash461 2010
[20] D Li and C Zhang ldquoLinfin error estimates of discontinuousGalerkin methods for delay differential equationsrdquo AppliedNumerical Mathematics vol 82 pp 1ndash10 2014
[21] V K Barwell ldquoSpecial stability problems for functional differ-ential equationsrdquo BIT vol 15 pp 130ndash135 1975
[22] A Bellen and M Zennaro ldquoStrong contractivity properties ofnumerical methods for ordinary and delay differential equa-tionsrdquo Applied Numerical Mathematics vol 9 no 3-5 pp 321ndash346 1992
[23] K Burrage ldquoHigh order algebraically stable Runge-Kutta meth-odsrdquo BIT vol 18 no 4 pp 373ndash383 1978
[24] K Burrage and J C Butcher ldquoNonlinear stability of a generalclass of differential equation methodsrdquo BIT vol 20 no 2 pp185ndash203 1980
[25] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983
[26] K Dekker and J G Verwer Stability of Runge-Kutta Methodsfor Stiff Nonlinear Differential Equations North-Holland Pub-lishing Amsterdam The Netherlands 1984
[27] L Ferracina and M N Spijker ldquoStrong stability of singly-diagonally-implicit Runge-Kutta methodsrdquo Applied NumericalMathematics vol 58 no 11 pp 1675ndash1686 2008
[28] K J inrsquot Hout and M N Spijker ldquoThe 120579-methods in thenumerical solution of delay differential equationsrdquo in TheNumerical Treatment of Differential Equations K Strehmel Edvol 121 pp 61ndash67 1991
[29] M Zennaro ldquoAsymptotic stability analysis of Runge-Kuttamethods for nonlinear systems of delay differential equationsrdquoNumerische Mathematik vol 77 no 4 pp 549ndash563 1997
[30] D Li C Zhang and W Wang ldquoLong time behavior of non-Fickian delay reaction-diffusion equationsrdquoNonlinear AnalysisReal World Applications vol 13 no 3 pp 1401ndash1415 2012
[31] B Garcia-Celayeta I Higueras and T Roldan ldquoContrac-tivitymonotonicity for additive Range-kutta methods Innerproduct normsrdquo Applied Numerical Mathematics vol 56 no 6pp 862ndash878 2006
[32] C Huang ldquoDissipativity of one-leg methods for dynamicalsystems with delaysrdquo Applied Numerical Mathematics vol 35no 1 pp 11ndash22 2000
[33] C Zhang and S Zhou ldquoNonlinear stability and D-convergenceof Runge-Kutta methods for delay differential equationsrdquo Jour-nal of Computational and Applied Mathematics vol 85 no 2pp 225ndash237 1997
[34] C Huang S Li H Fu and G Chen ldquoNonlinear stability ofgeneral linear methods for delay differential equationsrdquo BITNumerical Mathematics vol 42 no 2 pp 380ndash392 2002
[35] H Yuan J Zhao and Y Xu ldquoNonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 854517 22 pages 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Differential Equations 3
Proof Let 119910119899 119910(119899)119894 119910(119899)119894 ) and 119911119899 119911(119899)119894 (119899)119894 ) be twosequences of approximations to problems (1) and (5)respectively by ARKMs with the same stepsize ℎ and write
119880(119899)119894 = 119910(119899)119894 minus 119911(119899)119894 (119899)119894 = 119910(119899)119894 minus (119899)119894 119880(119899)0 = 119910119899 minus 119911119899119882[1]119894 = ℎ [119891[1] (119905119899 + 119888[1]119894 ℎ 119910(119899)119894 )
Re ⟨119880(119899)119894 119882[1]119894 ⟩ le 120572ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 (13)
and
Re ⟨119880(119899)119894 119882[2]119894 ⟩ le 1205731ℎ 10038171003817100381710038171003817119880(119899)119894 100381710038171003817100381710038172 + 1205732ℎ 10038171003817100381710038171003817119880(119899minus119898)119894 100381710038171003817100381710038172+ 120574ℎ 10038171003817100381710038171003817(119899)119894 100381710038171003817100381710038172
(14)
Together with (11) (12) (13) and (14) we get
10038171003817100381710038171003817119880(119899+1)0 100381710038171003817100381710038172 le 10038171003817100381710038171003817119880(119899)0 100381710038171003817100381710038172 + 2 119904sum119894=1
Remark 4 In [35] Yuan et al also discussed nonlinearstability of additive Runge-Kutta methods for multidelay-integro-differential equations However the main results aredifferent The main reason is that the results in [35] implythat the perturbations of the numerical solutions tend toinfinity when the time increase while the stability resultsin present paper indicate that the perturbations of thenumerical solutions are independent of the time Besides theasymptotical stability of the methods is also discussed in thepresent paper
4 Conclusion
The additive Runge-Kutta methods with some appropriatequadrature rules are applied to solve the delay-integro-differential equations It is shown that if the additive Runge-Kutta methods are algebraically stable the obtained numer-ical solutions can be globally and asymptotically stablerespectively In the future works we will apply the methodsto solve more real-world problems
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported in part by the National NaturalScience Foundation of China (71601125)
References
[1] V Thomee Galerkin Finite Element Methods for ParabolicProblems Springer Berlin Germany 1997
[2] J WuTheory and Applications of Partial Functional-DifferentialEquations Springer New York NY USA 1996
International Journal of Differential Equations 5
[3] J R Cannon and Y Lin ldquoNon-classical H1 projection andGalerkin methods for non-linear parabolic integro-differentialequationsrdquo Calcolo vol 25 pp 187ndash201 1988
[4] D Li and J Wang ldquoUnconditionally optimal error analysis ofcrank-nicolson galerkin fems for a strongly nonlinear parabolicsystemrdquo Journal of Scientific Computing vol 72 no 2 pp 892ndash915 2017
[5] B Li and W Sun ldquoError analysis of linearized semi-implicitgalerkin finite element methods for nonlinear parabolic equa-tionsrdquo International Journal of Numerical Analysis amp Modelingvol 10 no 3 pp 622ndash633 2013
[6] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995
[7] G Akrivis and B Li ldquoMaximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolicequationsrdquo SIAM Journal on Numerical Analysis 2017
[8] I Higueras ldquoStrong stability for additive Runge-Kutta meth-odsrdquo SIAM Journal on Numerical Analysis vol 44 no 4 pp1735ndash1758 2006
[9] A Araujo ldquoA note on B-stability of splitting methodsrdquo Comput-ing and Visualization in Science vol 26 no 2-3 pp 53ndash57 2004
[10] C A Kennedy and M H Carpenter ldquoAdditive Runge-Kuttaschemes for convection-diffusion-reaction equationsrdquo AppliedNumerical Mathematics vol 44 no 1-2 pp 139ndash181 2003
[11] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 pp 201ndash212 2008
[12] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006
[13] D Li C Zhang and M Ran ldquoA linear finite differencescheme for generalized time fractional Burgers equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp6069ndash6081 2016
[14] D Li J Wang and J Zhang ldquoUnconditionally convergentL1-Galerkin FEMs for nonlinear time-fractional Schrodingerequationsrdquo SIAM Journal on Scientific Computing vol 39 no6 pp A3067ndashA3088 2017
[15] L Torelli ldquoStability of numerical methods for delay differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 25 no 1 pp 15ndash26 1989
[16] K J inrsquot Hout ldquoStability analysis of Runge-Kutta methodsfor systems of delay differential equationsrdquo IMA Journal ofNumerical Analysis vol 17 no 1 pp 17ndash27 1997
[17] C T Baker and A Tang ldquoStability analysis of continuousimplicit Runge-Kutta methods for Volterra integro-differentialsystemswith unbounded delaysrdquoAppliedNumericalMathemat-ics vol 24 no 2-3 pp 153ndash173 1997
[18] C Zhang and S Vandewalle ldquoGeneral linear methods forVolterra integro-differential equations with memoryrdquo SIAMJournal on Scientific Computing vol 27 no 6 pp 2010ndash20312006
[19] D Li and C Zhang ldquoNonlinear stability of discontinuousGalerkin methods for delay differential equationsrdquo AppliedMathematics Letters vol 23 no 4 pp 457ndash461 2010
[20] D Li and C Zhang ldquoLinfin error estimates of discontinuousGalerkin methods for delay differential equationsrdquo AppliedNumerical Mathematics vol 82 pp 1ndash10 2014
[21] V K Barwell ldquoSpecial stability problems for functional differ-ential equationsrdquo BIT vol 15 pp 130ndash135 1975
[22] A Bellen and M Zennaro ldquoStrong contractivity properties ofnumerical methods for ordinary and delay differential equa-tionsrdquo Applied Numerical Mathematics vol 9 no 3-5 pp 321ndash346 1992
[23] K Burrage ldquoHigh order algebraically stable Runge-Kutta meth-odsrdquo BIT vol 18 no 4 pp 373ndash383 1978
[24] K Burrage and J C Butcher ldquoNonlinear stability of a generalclass of differential equation methodsrdquo BIT vol 20 no 2 pp185ndash203 1980
[25] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983
[26] K Dekker and J G Verwer Stability of Runge-Kutta Methodsfor Stiff Nonlinear Differential Equations North-Holland Pub-lishing Amsterdam The Netherlands 1984
[27] L Ferracina and M N Spijker ldquoStrong stability of singly-diagonally-implicit Runge-Kutta methodsrdquo Applied NumericalMathematics vol 58 no 11 pp 1675ndash1686 2008
[28] K J inrsquot Hout and M N Spijker ldquoThe 120579-methods in thenumerical solution of delay differential equationsrdquo in TheNumerical Treatment of Differential Equations K Strehmel Edvol 121 pp 61ndash67 1991
[29] M Zennaro ldquoAsymptotic stability analysis of Runge-Kuttamethods for nonlinear systems of delay differential equationsrdquoNumerische Mathematik vol 77 no 4 pp 549ndash563 1997
[30] D Li C Zhang and W Wang ldquoLong time behavior of non-Fickian delay reaction-diffusion equationsrdquoNonlinear AnalysisReal World Applications vol 13 no 3 pp 1401ndash1415 2012
[31] B Garcia-Celayeta I Higueras and T Roldan ldquoContrac-tivitymonotonicity for additive Range-kutta methods Innerproduct normsrdquo Applied Numerical Mathematics vol 56 no 6pp 862ndash878 2006
[32] C Huang ldquoDissipativity of one-leg methods for dynamicalsystems with delaysrdquo Applied Numerical Mathematics vol 35no 1 pp 11ndash22 2000
[33] C Zhang and S Zhou ldquoNonlinear stability and D-convergenceof Runge-Kutta methods for delay differential equationsrdquo Jour-nal of Computational and Applied Mathematics vol 85 no 2pp 225ndash237 1997
[34] C Huang S Li H Fu and G Chen ldquoNonlinear stability ofgeneral linear methods for delay differential equationsrdquo BITNumerical Mathematics vol 42 no 2 pp 380ndash392 2002
[35] H Yuan J Zhao and Y Xu ldquoNonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 854517 22 pages 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Remark 4 In [35] Yuan et al also discussed nonlinearstability of additive Runge-Kutta methods for multidelay-integro-differential equations However the main results aredifferent The main reason is that the results in [35] implythat the perturbations of the numerical solutions tend toinfinity when the time increase while the stability resultsin present paper indicate that the perturbations of thenumerical solutions are independent of the time Besides theasymptotical stability of the methods is also discussed in thepresent paper
4 Conclusion
The additive Runge-Kutta methods with some appropriatequadrature rules are applied to solve the delay-integro-differential equations It is shown that if the additive Runge-Kutta methods are algebraically stable the obtained numer-ical solutions can be globally and asymptotically stablerespectively In the future works we will apply the methodsto solve more real-world problems
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported in part by the National NaturalScience Foundation of China (71601125)
References
[1] V Thomee Galerkin Finite Element Methods for ParabolicProblems Springer Berlin Germany 1997
[2] J WuTheory and Applications of Partial Functional-DifferentialEquations Springer New York NY USA 1996
International Journal of Differential Equations 5
[3] J R Cannon and Y Lin ldquoNon-classical H1 projection andGalerkin methods for non-linear parabolic integro-differentialequationsrdquo Calcolo vol 25 pp 187ndash201 1988
[4] D Li and J Wang ldquoUnconditionally optimal error analysis ofcrank-nicolson galerkin fems for a strongly nonlinear parabolicsystemrdquo Journal of Scientific Computing vol 72 no 2 pp 892ndash915 2017
[5] B Li and W Sun ldquoError analysis of linearized semi-implicitgalerkin finite element methods for nonlinear parabolic equa-tionsrdquo International Journal of Numerical Analysis amp Modelingvol 10 no 3 pp 622ndash633 2013
[6] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995
[7] G Akrivis and B Li ldquoMaximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolicequationsrdquo SIAM Journal on Numerical Analysis 2017
[8] I Higueras ldquoStrong stability for additive Runge-Kutta meth-odsrdquo SIAM Journal on Numerical Analysis vol 44 no 4 pp1735ndash1758 2006
[9] A Araujo ldquoA note on B-stability of splitting methodsrdquo Comput-ing and Visualization in Science vol 26 no 2-3 pp 53ndash57 2004
[10] C A Kennedy and M H Carpenter ldquoAdditive Runge-Kuttaschemes for convection-diffusion-reaction equationsrdquo AppliedNumerical Mathematics vol 44 no 1-2 pp 139ndash181 2003
[11] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 pp 201ndash212 2008
[12] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006
[13] D Li C Zhang and M Ran ldquoA linear finite differencescheme for generalized time fractional Burgers equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp6069ndash6081 2016
[14] D Li J Wang and J Zhang ldquoUnconditionally convergentL1-Galerkin FEMs for nonlinear time-fractional Schrodingerequationsrdquo SIAM Journal on Scientific Computing vol 39 no6 pp A3067ndashA3088 2017
[15] L Torelli ldquoStability of numerical methods for delay differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 25 no 1 pp 15ndash26 1989
[16] K J inrsquot Hout ldquoStability analysis of Runge-Kutta methodsfor systems of delay differential equationsrdquo IMA Journal ofNumerical Analysis vol 17 no 1 pp 17ndash27 1997
[17] C T Baker and A Tang ldquoStability analysis of continuousimplicit Runge-Kutta methods for Volterra integro-differentialsystemswith unbounded delaysrdquoAppliedNumericalMathemat-ics vol 24 no 2-3 pp 153ndash173 1997
[18] C Zhang and S Vandewalle ldquoGeneral linear methods forVolterra integro-differential equations with memoryrdquo SIAMJournal on Scientific Computing vol 27 no 6 pp 2010ndash20312006
[19] D Li and C Zhang ldquoNonlinear stability of discontinuousGalerkin methods for delay differential equationsrdquo AppliedMathematics Letters vol 23 no 4 pp 457ndash461 2010
[20] D Li and C Zhang ldquoLinfin error estimates of discontinuousGalerkin methods for delay differential equationsrdquo AppliedNumerical Mathematics vol 82 pp 1ndash10 2014
[21] V K Barwell ldquoSpecial stability problems for functional differ-ential equationsrdquo BIT vol 15 pp 130ndash135 1975
[22] A Bellen and M Zennaro ldquoStrong contractivity properties ofnumerical methods for ordinary and delay differential equa-tionsrdquo Applied Numerical Mathematics vol 9 no 3-5 pp 321ndash346 1992
[23] K Burrage ldquoHigh order algebraically stable Runge-Kutta meth-odsrdquo BIT vol 18 no 4 pp 373ndash383 1978
[24] K Burrage and J C Butcher ldquoNonlinear stability of a generalclass of differential equation methodsrdquo BIT vol 20 no 2 pp185ndash203 1980
[25] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983
[26] K Dekker and J G Verwer Stability of Runge-Kutta Methodsfor Stiff Nonlinear Differential Equations North-Holland Pub-lishing Amsterdam The Netherlands 1984
[27] L Ferracina and M N Spijker ldquoStrong stability of singly-diagonally-implicit Runge-Kutta methodsrdquo Applied NumericalMathematics vol 58 no 11 pp 1675ndash1686 2008
[28] K J inrsquot Hout and M N Spijker ldquoThe 120579-methods in thenumerical solution of delay differential equationsrdquo in TheNumerical Treatment of Differential Equations K Strehmel Edvol 121 pp 61ndash67 1991
[29] M Zennaro ldquoAsymptotic stability analysis of Runge-Kuttamethods for nonlinear systems of delay differential equationsrdquoNumerische Mathematik vol 77 no 4 pp 549ndash563 1997
[30] D Li C Zhang and W Wang ldquoLong time behavior of non-Fickian delay reaction-diffusion equationsrdquoNonlinear AnalysisReal World Applications vol 13 no 3 pp 1401ndash1415 2012
[31] B Garcia-Celayeta I Higueras and T Roldan ldquoContrac-tivitymonotonicity for additive Range-kutta methods Innerproduct normsrdquo Applied Numerical Mathematics vol 56 no 6pp 862ndash878 2006
[32] C Huang ldquoDissipativity of one-leg methods for dynamicalsystems with delaysrdquo Applied Numerical Mathematics vol 35no 1 pp 11ndash22 2000
[33] C Zhang and S Zhou ldquoNonlinear stability and D-convergenceof Runge-Kutta methods for delay differential equationsrdquo Jour-nal of Computational and Applied Mathematics vol 85 no 2pp 225ndash237 1997
[34] C Huang S Li H Fu and G Chen ldquoNonlinear stability ofgeneral linear methods for delay differential equationsrdquo BITNumerical Mathematics vol 42 no 2 pp 380ndash392 2002
[35] H Yuan J Zhao and Y Xu ldquoNonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 854517 22 pages 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Differential Equations 5
[3] J R Cannon and Y Lin ldquoNon-classical H1 projection andGalerkin methods for non-linear parabolic integro-differentialequationsrdquo Calcolo vol 25 pp 187ndash201 1988
[4] D Li and J Wang ldquoUnconditionally optimal error analysis ofcrank-nicolson galerkin fems for a strongly nonlinear parabolicsystemrdquo Journal of Scientific Computing vol 72 no 2 pp 892ndash915 2017
[5] B Li and W Sun ldquoError analysis of linearized semi-implicitgalerkin finite element methods for nonlinear parabolic equa-tionsrdquo International Journal of Numerical Analysis amp Modelingvol 10 no 3 pp 622ndash633 2013
[6] U M Ascher S J Ruuth and B T Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995
[7] G Akrivis and B Li ldquoMaximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolicequationsrdquo SIAM Journal on Numerical Analysis 2017
[8] I Higueras ldquoStrong stability for additive Runge-Kutta meth-odsrdquo SIAM Journal on Numerical Analysis vol 44 no 4 pp1735ndash1758 2006
[9] A Araujo ldquoA note on B-stability of splitting methodsrdquo Comput-ing and Visualization in Science vol 26 no 2-3 pp 53ndash57 2004
[10] C A Kennedy and M H Carpenter ldquoAdditive Runge-Kuttaschemes for convection-diffusion-reaction equationsrdquo AppliedNumerical Mathematics vol 44 no 1-2 pp 139ndash181 2003
[11] T Koto ldquoStability of IMEX Runge-Kutta methods for delaydifferential equationsrdquo Journal of Computational and AppliedMathematics vol 211 pp 201ndash212 2008
[12] H Liu and J Zou ldquoSome new additive Runge-Kutta methodsand their applicationsrdquo Journal of Computational and AppliedMathematics vol 190 no 1-2 pp 74ndash98 2006
[13] D Li C Zhang and M Ran ldquoA linear finite differencescheme for generalized time fractional Burgers equationrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp6069ndash6081 2016
[14] D Li J Wang and J Zhang ldquoUnconditionally convergentL1-Galerkin FEMs for nonlinear time-fractional Schrodingerequationsrdquo SIAM Journal on Scientific Computing vol 39 no6 pp A3067ndashA3088 2017
[15] L Torelli ldquoStability of numerical methods for delay differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 25 no 1 pp 15ndash26 1989
[16] K J inrsquot Hout ldquoStability analysis of Runge-Kutta methodsfor systems of delay differential equationsrdquo IMA Journal ofNumerical Analysis vol 17 no 1 pp 17ndash27 1997
[17] C T Baker and A Tang ldquoStability analysis of continuousimplicit Runge-Kutta methods for Volterra integro-differentialsystemswith unbounded delaysrdquoAppliedNumericalMathemat-ics vol 24 no 2-3 pp 153ndash173 1997
[18] C Zhang and S Vandewalle ldquoGeneral linear methods forVolterra integro-differential equations with memoryrdquo SIAMJournal on Scientific Computing vol 27 no 6 pp 2010ndash20312006
[19] D Li and C Zhang ldquoNonlinear stability of discontinuousGalerkin methods for delay differential equationsrdquo AppliedMathematics Letters vol 23 no 4 pp 457ndash461 2010
[20] D Li and C Zhang ldquoLinfin error estimates of discontinuousGalerkin methods for delay differential equationsrdquo AppliedNumerical Mathematics vol 82 pp 1ndash10 2014
[21] V K Barwell ldquoSpecial stability problems for functional differ-ential equationsrdquo BIT vol 15 pp 130ndash135 1975
[22] A Bellen and M Zennaro ldquoStrong contractivity properties ofnumerical methods for ordinary and delay differential equa-tionsrdquo Applied Numerical Mathematics vol 9 no 3-5 pp 321ndash346 1992
[23] K Burrage ldquoHigh order algebraically stable Runge-Kutta meth-odsrdquo BIT vol 18 no 4 pp 373ndash383 1978
[24] K Burrage and J C Butcher ldquoNonlinear stability of a generalclass of differential equation methodsrdquo BIT vol 20 no 2 pp185ndash203 1980
[25] G J Cooper and A Sayfy ldquoAdditive Runge-Kutta methods forstiff ordinary differential equationsrdquo Mathematics of Computa-tion vol 40 no 161 pp 207ndash218 1983
[26] K Dekker and J G Verwer Stability of Runge-Kutta Methodsfor Stiff Nonlinear Differential Equations North-Holland Pub-lishing Amsterdam The Netherlands 1984
[27] L Ferracina and M N Spijker ldquoStrong stability of singly-diagonally-implicit Runge-Kutta methodsrdquo Applied NumericalMathematics vol 58 no 11 pp 1675ndash1686 2008
[28] K J inrsquot Hout and M N Spijker ldquoThe 120579-methods in thenumerical solution of delay differential equationsrdquo in TheNumerical Treatment of Differential Equations K Strehmel Edvol 121 pp 61ndash67 1991
[29] M Zennaro ldquoAsymptotic stability analysis of Runge-Kuttamethods for nonlinear systems of delay differential equationsrdquoNumerische Mathematik vol 77 no 4 pp 549ndash563 1997
[30] D Li C Zhang and W Wang ldquoLong time behavior of non-Fickian delay reaction-diffusion equationsrdquoNonlinear AnalysisReal World Applications vol 13 no 3 pp 1401ndash1415 2012
[31] B Garcia-Celayeta I Higueras and T Roldan ldquoContrac-tivitymonotonicity for additive Range-kutta methods Innerproduct normsrdquo Applied Numerical Mathematics vol 56 no 6pp 862ndash878 2006
[32] C Huang ldquoDissipativity of one-leg methods for dynamicalsystems with delaysrdquo Applied Numerical Mathematics vol 35no 1 pp 11ndash22 2000
[33] C Zhang and S Zhou ldquoNonlinear stability and D-convergenceof Runge-Kutta methods for delay differential equationsrdquo Jour-nal of Computational and Applied Mathematics vol 85 no 2pp 225ndash237 1997
[34] C Huang S Li H Fu and G Chen ldquoNonlinear stability ofgeneral linear methods for delay differential equationsrdquo BITNumerical Mathematics vol 42 no 2 pp 380ndash392 2002
[35] H Yuan J Zhao and Y Xu ldquoNonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 854517 22 pages 2012
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![Third-order Composite Runge Kutta Method for Solving Fuzzy … · Adam Bashford [14], Runge Kutta of order five [15], block methods [16], and Runge-Kutta Method with Harmonic Mean](https://static.documents.pub/doc/80x56/5e2750b6a2f1ce49c1270795/third-order-composite-runge-kutta-method-for-solving-fuzzy-adam-bashford-14-runge.jpg)
