Wavelet Method for the Solution of Caputo and Caputo-Hadamard Fractional Differential Equations

Abstract

In this thesis, main objective to develop reliable and effective numerical methods for solving fractional differential equations. In this work wavelets play a key role in developing numerical methods for solving linear fractional ordinary differential equations with boundary conditions, as well as fractional partial differential equations with both initial and boundary conditions. Also we develop a numerical scheme using operational matrices to solve Caputo and Caputo-Hadamard fractional differential equations. Furthermore, we reproduce a technique called Fast Evaluation, which handles these equations with initial conditions without using operational matrices. A Hadamard-Gegenbauer wavelet method is proposed for the solution of Caputo-Hadamard fractional differential equations. It is applied to solve linear Caputo-Hadamard fractional differential equations for both initial and boundary value problems. Using the Haar wavelet method, we solve fractional ordinary differential equations (ODEs) with initial and boundary value problems. We then present a novel numerical scheme to find approximate solutions for fractional partial differential equations (PDEs). This method combines Haar wavelets with a fast evaluation technique, known as the fast Haar wavelet method. To demonstrate the applicability and accuracy of the method, we solve several examples of Caputo fractional partial differential equations.