Numerical Solutions of Variable-Order Fractional Differential Equations

Abstract

This dissertation aims to develop numerical schemes to approximate solutions of variableorder fractional differential equations. The absence of fundamental properties in variableorder fractional operators, such as the semigroup property, left inverse property, and the fundamental theorem of fractional calculus, makes studying variable-order fractional differential equations analytically and numerically challenging. To address this gap in the literature, we present an alternative approach to deal with certain classes of variable-order fractional differential equations in this dissertation. This approach is applicable for numerous numerical techniques, including quadrature methods, spectral methods, wavelet methods, and many others. To illustrate its applicability, we utilize an operational matrix method based on fractional Legendre functions in this work. The method is applied to solve the linear and nonlinear variable-order fractional differential equations. We present three approaches: the differentiation matrix approach, the integration matrix approach, and the integral equation approach, along with numerical demonstrations to show the superior efficacy of the integral equation approach in achieving accurate results. The methodology extended to approximate the solution of nonlinear variable-order fractional differential equations, employing a quasilinearization technique. An error estimate for approximating the variable-order derivative of a function is derived, providing insights into the accuracy and reliability of the presented approach.