Convergence analysis and approximate solution of fractional differential equations

Bhausaheb R. Sontakke; Rajashri Pandit

doi:10.26637/MJM0702/0029

Abstract

This paper investigates the iterative solution of linear and nonlinear fractional partial differential equations using fractional Adomian decomposition method (ADM). We also establish uniqueness and convergence criteria for obtaining approximate solution. To illustrate applicability of present technique, solutions of some test problems and their graphical representation are done by Mathematica software.

Keywords:

Fractional differential equation, Adomian Decomposition Method, Uniqueness, Convergence

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Bhausaheb R. Sontakke Department of Mathematics, Pratishthan Mahavidalaya , Paithan, Aurangabad (M.S.), India.
Rajashri Pandit Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, India (M.S.)

Pages: 338-344
Date Published: 01-04-2019
Vol. 7 No. 02 (2019): Malaya Journal of Matematik (MJM)

K. A. Abro and I. Khan, Analysis of the heat and mass transfer in the MHD flow of a generalized Casson fluid in a porous space via non integer order derivatives without a singular kernel, Chinese Journal of Physics, 55 (2017) 1583-1595

G. Adomian, Solving frontier problems of physics, the Decomposition Method, Kluwer Academic Publishers, Boston, (1994).

G. Adomian, A review of the decomposition method in applied mathematics, J.Math., 135(1988), 501-544.

G. Adomian and R. Rach, Modified Adomian polynomials. Mathematical and computer modelling, 24(11)(1996), 39-46.

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, 2016, https://arxiv.org/abs/1602.03408.

J. Biazar and M. Pourabd, A Maple program for computing Adomian polynomials, International Mathematical Forum, 1(39) (2006)1919-1924.

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism , Pure and Applied Geophysics, 91(8)(1971), 134-147.

P. P. Cárdenas Alzate, J. R. González Granada and J. F. Trujillo Valencia, An Iterative method for solving delay differential equations applied to biological models, Applied Mathematical Sciences, 11(26)(2017), 1287 - 1295.

D. B. Dhaigude and G. A. Birajdar, Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method, Advances in Applied Mathematics and Mechanics, 6(1)(2014), 107-119.

J.S. Duan, R. Rach and A. M. Wazwaz, A new modified Adomian decomposition method for higher-order nonlinear dynamical systems. Comput. Model. Eng. Sci.,

J.S. Duan, R. Rach and A.M. Wazwaz, A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, 5 (2015), 59-74.

D.J.Evans and K.R.Raslan, The Adomian decomposition method for solving Delay differential equations, Inter national Journal of Computer Mathematics, (2004), 1-6

J.F. Gómez-Aguilar, H. Yépez-Martínez, J. TorresJiménez, T. Córdova-Fraga, R.F. Escobar-Jiménez and V.H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. $A d$ vances in Difference Equations, 68(2017), 1-18.

J.Hristov, Subdiffusion model with time-dependent diffusion coefficient: integral-balance solution and analysis, Thermal Science, 21(2017) 69-80.

J. H. He, Variational iteration method a kind of nonlinear analytical technique some examples. International Journal of Nonlinear Mechanics, 34(4)(1999), 699-708.

J. H. He, Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, $mathbf{1 7 8}(1999), 257-262$.

G.W. Leibniz, Mathematische Schiften, George Olms Verlagsbuchhandlung,Hildesheim, (1962).

S. J. Liao, On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147(2004), 499-513.

Q. Mdallal K. A. Abro and Ilyas Khan, Analytical Solutions of Fractional Walter's B Fluid with Applications, Complexity, (2018) 2018, Article ID 8131329, 10 pages.

${ }^{text {[20] K. K. Siller and B. Ross, An introduction to the Frac- }}$ tional calculus and Fractional differential equation, John Wiley and sons Inc. Newyork, (1993).

V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. YépezMartínez, D. Baleanu, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Advances in Difference Equations, 164(2016), 1-17.

K. Nouri, Study on efficiency of the Adomian decomposition method for stochastic differential equations, Int. J. Nonlinear Anal. Appl., 8(1)(2017), 61-68.

K. B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, (1974).

F.O. Ogunfiditimi, Numerical solution of delay differential equations using the Adomian decomposition method, The Int. Journal of Engineering and Science, 4(5)(2015), 18-23.

B. Ross, A brief history and exposition of the fundamental theory of the fractional calculus, Lecture Notes in Mathematics, 457(1975), 1-36.

S. G. Samko, A. A. Kilbas and O. I. Martichev,Integrals and Derivatives of the Fractional order and Some of their Applications, Tekhnika, Minsk, (1987).

L. N. Song and H. Q. Zhang, Solving the fractional BBMBurgers equation using the homotopy analysis method. Chaos Solitons Fract., 40(2009), 1616-1622.

B. R. Sontakke and A. S. Shaikh, Properties of Caputo operator and its applications to linear fractional differential equations. International Journal of Engineering Research and Applications, 5(5)(2015), 22-27.

B. R. Sontakke and A. Shaikh, Numerical Solutions of time fractional Fornberg-Whitam and modified Fornberg-Whitam equations using new iterative method. Asian Journal of Mathematics and Computer Research, $13(2)(2016), 66-76$.

B. R. Sontakke and A. Shaikh, Solving time fractional Sharma-Tasso-Olever equation using fractional complex transform with iterative method. British Journal of Mathematics and Computer Science, 19(1)(2016), 1-10.

B. R. Sontakke and A. Shaikh, Approximate solutions of time fractional Kawahara and modified Kawahara equations by fractional complex transform. Communications in numerical analysis, $mathbf{2 ( 2 )}(2016), 218-229$.

K.C. Takale, Fractional order Finite fifference scheme for space fractional Boussinesq'S equation and its application. International Journal of Mathematical Sciences and Applications, 3(1)(2013), 343-353.

H. Yépez-Martíneza, J. F. Gómez-Aguilar, I.O. Sosaa, J.M. Reyesa and J. Torres-Jiménezc, The Feng's first integral method applied to the nonlinear $mathrm{mKdV}$ space-time fractional partial differential equation. Revista Mexicana de Física, 62(2016), 310-316.

B. Zheng, (G0/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoratical Physics, $mathbf{5 8}(2012), 623-630$