Abstract
This paper presents a numerical scheme for solving nonlinear time-fractional differential equations in the sense of Caputo. This method relies on the Laplace transform together with the modified Adomian method (LMADM), compared with the Laplace transform combined with the standard Adomian Method (LADM). Furthermore, for the comparison purpose, we applied LMADM and LADM for solving nonlinear time-fractional differential equations to identify the differences and similarities. Finally, we provided two examples regarding the nonlinear time-fractional differential equations, which showed that the convergence of the current scheme results in high accuracy and small frequency to solve this type of equations.
Highlights
Fractional calculus is a branch of mathematics that deals with real or complex number powers of the differential and integral operators
There are many applications for solving Fractional differential equations (FDEs), such as the fundamental function solutions to the fractional development advection-dispersion equation [1], the implicit difference approximation [2], the introduction of stable numerical schemes by replacing fractional quadrature rules to obtain the general solution for stability problems [3], the homotopy analysis method as an approximate technique to find Solitary wave solutions [4], and the Laplace and Fourier transforms [5]
In order to better understand the phenomenon described by a given nonlinear FDEs, the solutions of differential equations for the fractional order are largely involved
Summary
Fractional calculus is a branch of mathematics that deals with real or complex number powers of the differential and integral operators. Fractional differential equations (FDEs) are the generalization of the differential equations of integer order, studied through the theory of fractional calculus. Fractional derivatives provide more accurate models of real world problems than integer order derivatives. ( ) is the Caputo fractional derivative, ( ) is the source term, is the linear operator, and is the general nonlinear operator. The Caputo definition has the advantage of dealing properly with the initial value problems, in which the initial conditions are given in terms of the field variables and their integer-order, which is the case in most physical processes. The Caputo definition has the advantage of dealing properly with the initial value problems, in which the initial conditions are given in terms of the field variables and their integer-order, which is the case in most physical processes. 2.1 The fractional derivative of ( ) in the Caputo sense is defined as [1][9]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
