Solutions of spatiotemporal fractional partial differential equations by residual power series methods

Abstract

The fractional residual power series is an analytic-numeric technique used to handle linear and nonlinear problems in multidisciplinary fields of engineering and science. This thesis presents an effective implementation of a technique based on the residual power series method (RPSM). Under Caputo sense, the method gives approximate solutions for classes of fractional partial differential equations (FPDEs) of fractional order 𝒶: 0 < 𝒶 ≤ 1. Time fractional Caputo derivatives are used to explain the fractional derivatives. The work started with the traditional RPSM being used to solve FPDE systems. The Laplace residual power series technique (LRPSM) is then used, which combines the RPSM and the Laplace transform (LT) by transforming the FPDEs into the Laplace space and using the RPSM to find the unknown coefficients. The suggested approach solves a wide range of linear and nonlinear FPDEs and allows us to obtain analytical and approximate solutions to the equations in a rapidly convergent power series with precisely computable structures, all without making any restrictive assumptions. After that, the LRPSM is then used to solve systems of FPDEs based on the generalized Taylor series expansion. Following that, the LRPSM is created and used to solve FPDEs of fractional order 2𝒶: 0.5 < 𝒶 ≤ 1 using the generalized Taylor series expansion. Finally, under Caputo's differentiability, an approximate solution based on the generalized Taylor series expansion for complex FPDEs including Schrodinger equations of fractional order 𝒶: 0 < 𝒶 ≤ 1 is suggested. Using illustrative examples, the suggested technique's efficiency, reliability, and performance are proven. The given algorithm is significant and simple for generating a fractional power series (FPS) solution without linearization, problem nature limitations, classification, or perturbation. Graphical representations for the approximate solutions with various values of 𝒶: 0 < 𝒶 ≤ 1 are provided to demonstrate the method's effectiveness. The findings show that the LRPSM is more straightforward, reliable, and easier to use compared to traditional RPSM. This is because LRPSM relies on the limit concept, whereas traditional RPSM uses the fractional derivative at every step to find the unknown coefficients. The approximation agrees well with the analytical solution. The approximate solutions' dynamical properties are described, and the profiles of several representative numeric solutions are illustrated. There are graphical justifications for the suggested method's dependability. As a result, the LRPSM is a systematic technique for obtaining an accurate solution that is entirely compatible with FPDEs as well as complex FPDEs. It is worth mentioning that all the results are computed using Mathematica 12 software package.