An innovative approach to approximating solutions of fractional partial differential equations

Abstract

The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order \(\sigma\), we observe that as the value approaches an integer order, the solution converges towards the exact solution.