Numerical Solutions and their Convergence Analysis for Fractional Differential and Integro- Differential Equations Involving Weak Singularities

Abstract

An important aspect of fractional calculus is the study of fractional order integrals and derivatives and their applications. In the past few decades, the study on fractional differential equations (FDEs) and fractional order integro-differential equations (IDEs) has gained immense interest among many researchers due to its practical implementation in several fields of science and engineering. This helps in modeling many physical problems in control theory, cryptography, neural networks, fluid mechanics, financial market, viscoelasticity, electrodynamics, and bioengineering, etc. The analytical approximations of the FDEs and fractional order IDEs are not straightforward as the fractional differential operators are non-local defined by an integral over the entire region whereas the integer order derivatives are defined on a small neighbourhood of the given point. Further, given smooth data in the governing equation involving fractional order derivatives may not guarantee the smooth solution of the problem. Due to this uncertain behaviour, finding a solution for an FDE and fractional order IDE is not an easy task and there is need for semi-analytical/numerical methods. The major objective of the present thesis is to analyze FDEs as well as fractional IDEs in order to develop time-efficient, accurate, and computationally effective numerical methods such as the L1 scheme, the Adomian decomposition method, the homotopy perturbation method, and the modified Laplace decomposition method, etc., for solving them along with their stability and convergence analysis. This thesis is designed into two parts. The first part is devoted for the numerical simulation of differential equations (IVPs & IBVPs) involving fractional order derivatives, whereas the second part of the thesis shows the reliability of the present approach applying on fractional order IDEs as well as fractional order partial integro-differential equations (PIDEs) of Volterra type, Fredholm type and also of mixed Volterra-Fredholm type. Further, the thesis consists of a total of eleven chapters, out of which Chapter 1 describes the basic preliminaries about fractional calculus, which contains several definitions of fractional order derivatives including their various properties along with the objective and motivation to work on various fractional order models. Chapter 2 considers a one-term Riemann-Liouville fractional IVP for which a fully discrete finite difference scheme is constructed using L1 discretization on uniform mesh. Further, this scheme is applied to a more general multi-term fractional IVP in order to show its applicability in solving the fractional models. Then, in Chapter 3, we develop the idea and introduce the L1 discretization on a rectangular domain in order to solve a time fractional IBVP of mixed convection-diffusion-reaction type for which the L1 discretization is applied on a uniform mesh. Next, in Chapter 4, a time fractional Black-Scholes European option pricing model is considered in which the L1 discretization is introduced on a graded mesh in order to increase the order of accuracy. Chapter 5 presents the numerical solution of a time-space fractional Poisson’s equation by using the homotopy perturbation method. Then, Chapter 6 demonstrates the numerical investigation of a multi-term time fractional nonlinear KdV equation. A modified Laplace decomposition method is used to find an analytical approximate solution. The obtained results are compared with some existing methods. In case of fractional order IDEs, first we consider a fractional IVP with a Volterra integral operator in Chapter 7, where the L1 discretization is applied to approximate the fractional operator and a repeated quadrature rule is used to discretize the integral term. Then, in Chapter 8, we extend the present idea for solving a fractional order PIDE and the method is reconstructed on a nonuniform mesh to solve a more general multi-term time fractional PIDE. Next, Chapter 9 comprises the numerical simulation of a time fractional Black-Scholes model under jump-diffusion, where the model is converted into a time fractional PIDE with a Fredholm integral operator. Finally, a class of multi-term time fractional PIDEs of mixed Volterra-Fredholm type is considered in Chapter 10, where the Adomian decomposition method is applied for numerical convergence. At last, Chapter 11 highlights the concluding remarks obtained from various consequences of the works reported in this thesis followed by a few possible future directions of the present works. Several tests are performed on numerous extensive examples in numerical results and discussion section of each chapters to show the efficiency and accuracy of the proposed methods. Further, several comparison results are presented in terms of tables and figures in order to show the reliability of the present approach.