COMBINATION OF LAPLACE TRANSFORM AND MODIFIED ADOMIAN DECOMPOSITION METHOD FOR SOLVING NONLINEAR TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Abstract

A nonlinear time-fractional partial differential equation is used for accurate modeling of complex real-world phenomena. Solving these fractional problems is one of the most challenging tasks in various fields of science because of the nonlinear terms and fractional operators in the problems. For this reason, the development of appropriate numerical methods for solving nonlinear time-fractional partial differential equations is an active area of research. In this study, we combine the Laplace transform with the modified Adomain decomposition method for solving nonlinear time-fractional partial differential equations. The fractional derivatives are described in terms of the Caputo sense. In this approach, we used the Laplace transform to change the time-fractional partial differential equations into algebraic differantial equation. Then, the inverse Laplace transform is applied by using the prescribed initial conditions. Hereafter, the modified Adomian decomposition method was applied to decompose the nonlinear term and generate a series solution of the proposed method. For the proposed method, the stability was confirmed using the T-stable approach, and the convergence of the method was also analyzed in the Banach space. The solution of the suggested technique is represented in a series of modified Adomian components, which is convergent to the exact solution of the given problems after some iteration. Furthermore, the effectiveness of the LTMADM is demonstrated by solving four test problems. The numerical results of the proposed method were compared with LTDM,ATHPM andATHPM,LRPSM and ETDM,LRDTM and LADM in the literature in terms of their absolute error. The findings show the proposed method provides a more accurate solution than other methods. Based on the obtained numerical results, we realize that the proposed method is well suited for solving nonlinear time-fractional partial differential equations and enables us to obtain more accurate solutions than the other methods. In addition, the obtained results are illustrated in tables and figures for different values of fractional order and compared with the exact solution at . The plotted graphs illustrate the behavior of the solution for different values of order . The results in each table and figure show that the results of the proposed method are in good agreement with the exact solution. As a conclusion, the proposed method is straightforward and enables us to obtain more accurate solutions after a few iteration