Exact solution of system of nonlinear fractional partial differential equations by modified semi-separation of variables method

Abstract

A system of nonlinear fractional partial differential equations (FPDEs) is widely used in applied sciences, especially for modeling fluid dynamics and polymer-related problems. Given their importance, finding solutions to these systems is essential and a core property. Various methods have been developed to find a solution to a system of nonlinear FPDEs. However, these methods are difficult to implement and sometimes converge slowly. In the worst-case scenario, applying the differential transform method may produce a series that does not converge to the exact solution of a system of nonlinear FPDEs. The semi-separation of variables method (S-SVM) is a recent and reliable analytic method that has not been applied to obtain an exact solution to a system of nonlinear FPDEs. Furthermore, S-SVM has not been improved to observe faster convergence. In this paper, the S-SVM is used to obtain the exact solution to the system of nonlinear FPDEs. In addition, the S-SVM is further improved as a Modified S-SVM (MS-SVM), which is applied to find an exact solution to the system of nonlinear FPDEs. Also, numerical experiments using the S-SVM and the MS-SVM in both two and three dimensions are provided therein, along with a comparison of their solutions to those obtained from the Adomian Decomposition Method (ADM), the Laplace Variational Iteration Method (LVIM), and the Fractional Power Series Method (FPSM). The results show that the solutions obtained using S-SVM and MS-SVM converge faster than those from FPSM, ADM, and LVIM. Moreover, S-SVM and MS-SVM do not require the complex computation of Adomian polynomials.