Numerical Solutions of Tempered Fractional Differential Equations and Fractional Darboux Problem

Abstract

The main objective of this dissertation is to formulate methods for numerically solving specific types of fractional differential equations, which include Caputo fractional order differential equations, tempered fractional differential equations, and the fractional Darboux problem. To enhance the accuracy of solution approximations, an approach known as the product integra tion technique is introduced. This technique is based on the Newton-Cotes quadrature formula and is established through the utilization of generalized Lagrange interpolation polynomials. A formula for numerically evaluating the fractional integrals is established using the product integration technique. This formula is then utilized to develop a two distinct schemes for solutions of three classes of fractional differential equation, that differ slightly. Numerical experiments are conducted to illustrate the applica bility and accuracy of all of the proposed schemes. Upper bound of error in the approximations for the proposed product integration technique are also worked out. After dealing with Caputo fractional differential equations, we proposed a technique to obtain a nu merical solutions of tempered fractional order differential equations. The product integration technique is used to develop a formula for the left and right-sided tempered fractional integrals. Next, by using these formulas, two different methods are developed for solving a general class of tempered fractional differential equations. Then, three distinct classes of initial value problems for tempered fractional dif ferential equations are taken into consideration and each class is illustrated by a numerical experiment to demonstrate the validity and effectiveness of the proposed methodology. Furthermore, we established a product integration technique for the numerical solution of terminal value problems for the general class of tempered fractional differential equations and then considered the simplest class, followed by an example. Moreover, upper bound of error in the numerical approximations are provided. Lastly, a numerical strategy is introduced for solving a class of linear fractional order Darboux prob lem that involves Caputo derivative. A formula is developed to numerically evaluate the two-dimensional fractional integrals using the product integration technique. Then, this formula is employed to derive a scheme for solving an initial value problem for a class of fractional order Darboux problem. The accuracy and efficiency of all the presented schemes is illustrated through numerical experiments. Fur thermore, the estimation for the upper bound of error in the numerical approximation is established.