Approximate Methods For Solving Fractional Differential Equations


  • Educational Directorate of Thi-Qar , Nasiriyah, Iraq


Daftardar-Jafari method, nonlinear Burger equation, heat-like equation, Caputo- Fabrizio fractional operator.


In this paper, exact and approximate solutions of the nonlinear Burgers equation, heat-like equation
and coupled nonlinear fractional Burger’s equations with Caputo-Fabrizio fractional operator by
using Daftardar-Jafari method (DJM) and Sumudu decomposition method (SDM) are presented and
discussed. The solutions of our equations are calculated in the form of rabidly convergent series
with easily computable components. Three illustrative applications are given to demonstrate the
effectiveness and the leverage of the present methods. Graphical results are utilized and discussed
quantitatively to illustrate the solution. The results reveal that the methods are very effective and
simple in determination of solution of the fractional partial differential equations.


G. Aguilar et al., Homotopy perturbation transform method for nonlinear differential equations

involving to fractional operator with exponential kernel, Advances in Difference Equations (2017)


M. Al-Refai, K. Pal, New Aspects of Caputo-Fabrizio Fractional Derivative, Progress in Fractional

Differentiation and Applications, 5(2) (2019) 157-166 .

D. Baleanu, et al., Approximate Analytical Solutions of Goursat Problem within Local Fractional

Operators, Journal of Nonlinear Science and Applications, 9 (2016) 4829-4837.

D. Baleanu, H. K. Jassim, A Modification Fractional Homotopy Perturbation Method for

Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets, Fractal and Fractional, 3(30)

(2019) 1-8.

D. Baleanu, et al., Solving Helmholtz Equation with Local Fractional Derivative Operators, Fractal

and Fractional, 3(43) (2019) 1-13.

D. Baleanu, et al., Approximate Solutions of the Damped Wave Equation and Dissipative Wave

Equation in Fractal Strings, Fractal and Fractional, 3(26) (2019) 1-12.

D. Baleanu, et al., A Modification Fractional Variational Iteration Method for solving Nonlinear Gas

Dynamic and Coupled KdV Equations Involving Local Fractional Operators, Thermal Science,

(1) (2018) S165-S175.

Belgacem, Fethi Bin Muhammed, and Ahmed Abdullatif Karaballi. Sumudu transform fundamental

properties investigations and applications.( International Journal of Stochastic Analysis 2006 (2006).

M. Caputo, M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel,

Progress in Fractional Differentiation and Applications, 1(2)(2015) 73-85.

V. Gill, K. Modi, and Y. Singh, Analytic solutions of fractional differential equation associated with

RLC electrical circuit, Journal of Statistics and Management Systems, 21(4) (2018) 575-582.

V. G. Gupta, P. Kumar, Approximate solutions of fractional linear and nonlinear differential

equations using Laplace homotopy analysis method. Int. J. Nonlinear Sci. 19(2)(2015) 113-

H. Jafari, M. Ghorbani, S. Ghasempour, A note on exact solutions for nonlinear integral equations

by a modified homotopy perturbation method. New Trends Math. Sci. 1(2), 22-26


H. Jafari, H. K. Jassim, J. Vahidi, Reduced Differential Transform and Variational Iteration

Methods for 3D Diffusion Model in Fractal Heat Transfer within Local Fractional Operators,

Thermal Science, 22 (2018) S301-S307.

H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan and F. Tchier, Reduced differential transform

method for partial differential equations within local fractional derivative operators, Advances in

Mechanical Engineering, 8 (2016) 1-6.

A. Shaikh et al. Analysis of differential equations involving Caputo-Fabrizio fractional operator and

its applications to reaction-diffusion equations Advances in Difference Equations

(2019) 2019:178

H. K. Jassim, M. A. Hussain, On approximate solutions for fractional system of

differential equations with Caputo-Fabrizio fractional operator, Journal of

Mathematics and Computer Science, 23 (2021), no. 1, 58--66