APPROXIMATE SOLUTION FOR TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

Abstract

This thesis investigates one-dimensional time-dependent partial differential equations, focusing on two types of fractional derivative definitions and their properties. The primary goal is to derive semianalytical approximate series solutions for the spatial variable ν within a bounded interval [a,b], where a and b are real numbers. Three powerful numerical methods are employed to obtain approximate analytical solutions for fractional order partial differential equations: the variational iteration method (VIM), the Adomian decomposition method (ADM), and the homotopy analysis method (HAM). These techniques balance the simplicity of analytical solutions with the accuracy of numerical approaches. The study includes a comprehensive convergence analysis of the approximate series solutions obtained from VIM, ADM, and HAM. The differential equation under investigation is derived from the traditional Fornberg-Whitham equation and the Helmholtz equation by replacing the integer order time derivative with noninteger derivatives of order μ in the range n - 1 < μ ≤ n, for n ∈ N, incorporating variable coefficients. Novel approaches are developed to compute the Laplace transform in the Atangana-Baleanu fractional derivative operator, enhancing the performance and accuracy of the semianalytical methods. The research extends to validate the effectiveness of fractional order methods. To demonstrate the applicability of these techniques, computational analyses of various test problems are provided, featuring two fractional derivatives and variable coefficients. Comparisons reveal that the absolute differences between the approximate solutions derived from VIM, ADM, and HAM decrease with the parameter μ approaches to the integer order. The findings indicate that the differences between ADM and HAM are consistently smaller than those involving VIM, signifying that while all methods yield similar results, ADM and HAM show closer alignment and potential excellence in specific scenarios. According to the results and graphical representation, it can be seen that the proposed methods are efficient in obtaining an analytical solution for time-fractional differential equations.