On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method

Haci Mehmet Baskonus ; Hasan Bulut

Open Mathematics, Tome 13 (2015), / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper, we apply the Fractional Adams-Bashforth-Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Then, we construct a table including numerical results for both fractional differential equations. Then, we draw two dimensional surfaces of numerical solutions and analytical solutions by considering the suitable values of parameters. Finally, we use the L2 nodal norm and L∞ maximum nodal norm to evaluate the accuracy of method used in this paper.

Détail

Publié le : 2015-01-01

Zbl 1338.35089

EUDML-ID : urn:eudml:doc:275845

Bibliographie

[1] J. Cao and C. Xu, A high order schema for the numerical solution of the fractional ordinary differential equations, Journal of Computational Physics, 238(2013), 154-168, 2013. | Zbl 1286.65092

[2] G.C. Wu, D. Baleanu and Z.G. Deng, Variational iteration method as a kernel constructive technique, Applied Mathematical Modelling, 39(15), 4378-4384, 2015.[Crossref][WoS]

[3] Z. F. Kocak, H. Bulut, and G. Yel, The solution of fractional wave equation by using modified trial equation method and homotopy analysis method, AIP Conference Proceedings, 1637, 504-512, 2014.

[4] A. Esen, Y. Ucar, N. Yagmurlu and O. Tasbozan, A galerkin finite element method to solve fractional diffusion and fractional Diffusion-Wave equations, Mathematical Modelling and Analysis, 18(2), 260-273, 2013. | Zbl 1266.65026

[5] D. Baleanu, B. Guvenc and J.A. Tenreiro-Machado, New Trends in Nanotechnology and Fractional Calculus Applications; Springer: New York, NY, USA, 2010. | Zbl 1196.65021

[6] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Mathematics of Computation, 45, 463-469, 1985. | Zbl 0584.65090

[7] P. Goswami and F.B.M. Belgacem, Solving Special fractional Differential equations by Sumudu transform, AIP Conference. Proceedings 1493, 111-115, 2012.

[8] A. Atangana, Convergence and Stability Analysis of A Novel Iteration Method for Fractional Biological Population Equation, Neural Computing and Applications, 25(5), 1021-1030, 2014.

[9] R.S. Dubey, B. Saad, T. Alkahtani and A. Atangana, Analytical Solution of Space-Time Fractional Fokker-Planck Equation by Homotopy Perturbation Sumudu Transform Method, Mathematical Problems in Engineering, 2014, Article ID 780929, 7 pages, 2014.

[10] S. Abbasbandy and A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems, Numerical Algorithms, 54(4), 521-532, 2010.[Crossref][WoS] | Zbl 1197.65096

[11] L. Song and H. Zhang, Solving the fractional BBM-Burgers equation using the homotopy analysis method, Chaos Solitons Fractals, 40, 1616-1622, 2009.[Crossref][WoS] | Zbl 1198.65205

[12] A. Atangana, Numerical solution of space-time fractional derivative of groundwater flow equation, Proceedings of the International Conference of Algebra and Applied Analysis, 6(2), 20 pages, 2012.

[13] H. Jafari and S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics Letters A, 370(5-6), 388-396, 2007.[WoS] | Zbl 1209.65111

[14] Q.K. Katatbeh and F.B.M. Belgacem, Applications of the Sumudu Transform to Fractional Diffirential Equations, Nonlinear Studies, 18(1), 99-112, 2011. | Zbl 1223.44001

[15] K.B. Oldham and J. Spanier, The Fractional Calculus. Academic, New York, 1974. | Zbl 0292.26011

[16] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.

[17] V. E. Tarasov, Fractional Dynamics; Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, USA, 2010. | Zbl 1214.81004

[18] C. Li and G. Peng, Chaos in Chen’s system with a fractional order, Chaos, Solitons & Fractals, 22, 443-450, 2004.[Crossref] | Zbl 1060.37026

[19] C. Li and W. Deng, Chaos synchronization of fractional-order differential systems, International Journal of Modern Physics B, 20, 791-803, 2006.[Crossref] | Zbl 1101.37025

[20] K. Diethelm, N.J. Ford, A.D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Computer Methods in Applied Mechanics and Engineering, 194, 743-773, 2005. | Zbl 1119.65352

[21] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, International Journal of Computer Mathematics, 87(10), 2281-2290, 2010.[WoS] | Zbl 1206.65197

[22] K. Diethelm, N.J. Ford and A. D. Freed, A predictor corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3-22, 2002.[Crossref] | Zbl 1009.65049

[23] K. Diethelm, N.J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36, 31-52, 2004. | Zbl 1055.65098

[24] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265, 229-248, 2002. | Zbl 1014.34003

[25] I. Petras, Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab in Engineering Education and Research using Matlab, In Tech, Rijeka, Croatia, 239-264, 2011.

[26] A. Atangana, Exact solution of the time-fractional underground water flowing within a leaky aquifer equation Vibration and Control, 1-8, 2014.

[27] K.A. Gepreel, The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations, Applied Mathematics Letters, 24(8), 1428-1434, 2011.[WoS][Crossref] | Zbl 1219.35347

[28] A. Atangana and D. Baleanu, Nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudu transform, Abstract and Applied Analysis, 9 pages, 2013. | Zbl 1275.65066

[29] A. Atangana and N. Bildik, The Use of Fractional Order Derivative to Predict the Groundwater Flow, Mathematical Problems in Engineering, 2013, Article ID 543026, 9 pages, 2013. | Zbl 1296.76144

[30] Z. Hammouch and T. Mekkaoui, Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1, 206-212, 2012.

[31] K. Diethelm and A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: Forschung und wissenschaftliches Rechnen: Beiträge zum Heinz- Billing-Preis 1998, eds. S. Heinzel and T. Plesser (Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen, 1999) pp. 57-71.

[32] H. M. Baskonus, T. Mekkaoui, Z. Hammouch and H. Bulut, Active Control of a Chaotic Fractional Order Economic System, Entropy, 17, 5771-5783, 2015.[Crossref][WoS]

[33] Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Studies, 2015, To appear. | Zbl 1335.34097

[34] R.S. Dubey, P. Goswami and F.B.M. Belgacem, Generalized Time-Fractional Telegraph Equation Analytical Solution by Sumudu and Fourier, Journal of Fractional Calculus and Applications, 5(2), 52-58, 2014.