×
Khan, Hasib; Khan, Aziz; Chen, Wen; Shah, Kamal

Stability analysis and a numerical scheme for fractional Klein-Gordon equations. (English) Zbl 1408.35213

Math. Methods Appl. Sci. 42, No. 2, 723-732 (2019).
The authors present a numerical scheme for the fractional order nonlinear Klein-Gordon equations involving the Caputo’s fractional derivative. The scheme is based on the use of Sumudu decomposition method which is a coupling of Sumudu integral transform and Adomian decomposition method. It is shown that such scheme is efficient in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.
Reviewer: Abdallah Bradji (Annaba)

Cited in 25 Documents

MSC:

35R11 Fractional partial differential equations
35B35 Stability in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems

Keywords:

Adomian decomposition; Caputo’s fractional derivative; Klein-Gordon equation; Sumudu transform
Cite Review PDF
Full Text: DOI

References:

[1] PodlubnyI. Fractional Differential Equations. New York: Academic Press; 1999. · Zbl 0924.34008
[2] KilbasA, SrivastavaH, TrujilloJ. Theory and applications of fractional differential equations. North‐Hollan Math Studies. 2006;204:135‐209. · Zbl 1092.45003
[3] McCluskeyCC. A model of HIV/AIDS with staged progression and amelioration. Appl Math Biosci. 2003;181:1‐6. · Zbl 1008.92032
[4] DiazJI, De‐ThelinF. On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J Math Anal. 1994;25:1085‐111. · Zbl 0808.35066
[5] IchiseM, NagayanagiY, KojimaT. An analog simulation of noninteger order transfer functions for analysis of electrode processes. J Electroanal Chem Interfacial Electrochem. 1971;33(2):253‐65.
[6] LinW. Global existence theory and chaos control of fractional differential equations. J Math Anal Appl. 2007;332:709‐726. · Zbl 1113.37016
[7] EvansLC, GangboW. Differential equations methods for the Monge‐Kantorovich mass transfer problem. American Mathematical Soc. 1999;653. · Zbl 0920.49004
[8] LaskinN. Fractional quantum mechanics. Amer Math Soc. 2000;62(3):709‐726.
[9] JafariH, BaleanuD, KhanH, KhanRA, KhanA. Existence criterion for the solutions of fractional order p‐Laplacian boundary value problems. Bound Value Probl. 2015;164:10pp. · Zbl 1381.34016
[10] Ran‐ChaoW, Xin‐DongH, Li‐PingC. Finite‐time stability of fractional‐order neural networks with delay. Commun Theor Phys. 2013;60(2):189‐193. · Zbl 1284.92016
[11] WangY, HouC. Existence of multiple positive solutions for one‐dimensional p‐Laplacian. J Math Anal Appl. 2013;315:144‐153. · Zbl 1098.34017
[12] ShenT, LiuW, ShenX. Existence and uniqueness of solutions for several BVPS of fractional differential equations with p‐Laplacian operator. Mediterr J Math. 2016;13:4623‐4637. · Zbl 1349.34018
[13] KhanA, LiY, ShahK, KhanTS. On coupled p‐Laplacian fractional differential equations with nonlinear boundary conditions. Complexity. 2017;2017:Article ID 8197610, 9 pages. · Zbl 1373.93157
[14] AdomianG. A review of the decomposition method and some recent results for nonlinear equations. Math Comput Model. 1990;13(7):17‐43. · Zbl 0713.65051
[15] DemirayST, PandirY, BulutH. Generalized Kudryashov method for time‐fractional differential equations. Abstr Appl Anal. 2014;2014:13. · Zbl 1474.35674
[16] TandoganYA, PandirY, GurefeY. Solutions of the nonlinear differential equations by use of modified Kudryashov method. Turk J Math Comput Sci. 2013;2013:7.
[17] MerdanM. On the solutions of nonlinear fractional Klein‐Gordon equation with modified Riemann‐Liouville derivative. Appl Math Comput. 2014;242:877‐888. · Zbl 1334.35393
[18] WatugalaGK. Sumudu transform: A new integral transform to solve differential equations and control engineering problems. Integrated Educ. 1993;24(1):35‐43. · Zbl 0768.44003
[19] WatugalaGK. The Sumudu transform for functions of two variables. Math Eng Industry. 1990;8(4):293‐302. · Zbl 1025.44003
[20] Gomez‐AguilarJF. Analytical and numerical solutions of a nonlinear alcoholism model via variable‐order fractional differential equations. Phys A. 2018;294:52‐75. · Zbl 1514.92041
[21] WeerakoonS. Complex inversion formula for Sumudu transform. Int J Math Edu Sci Tech. 1998;29(4):618‐20. · Zbl 1018.44004
[22] AtanganaA, KilicmanA. The use of Sumudu transform for solving certain nonlinear fractional heat‐like equations. Abstr Appl Anal. 2013;2013:Article ID 737481, 12 pages. · Zbl 1275.65067
[23] KhanH, ChenW, SunH. Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space. Math Meth Appl Sci. 2018;41(9):1‐11.
[24] WazwazAM. Compactons, solitons and periodic solutions for some forms of nonlinear Klein‐Gordon equations. Chaos Solitons Fract. 2006;28(4):1005‐1013. · Zbl 1099.35125
[25] VongS, WangZ. A compact difference scheme for a two dimensional fractional Klein‐Gordon equation with Neumann boundary conditions. J Comput Phys. 2014;274:268‐282. · Zbl 1352.65273
[26] HosseiniK, MayeliP, AnsariR. Modified Kudryashov method for solving the conformable time‐fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Optik. 2017;130:737‐742.
[27] OnateCA, OnyeajuMC, IkotAN, OjonubahJO. Analytical solutions of the Klein‐Gordon equation with a combined potential. Chin J Phys. 2016;54:820‐829. · Zbl 1539.81056
[28] AtanganaA. On the new fractional derivative and application to nonlinear Fisher’s reaction diffusion equation. Appl Math Comput. 2016;273:948‐956. · Zbl 1410.35272
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.