Spectral methods for solving fractional differential and integral equations

Abstract

Many problems in engineering, chemistry, physics and natural sciences yield nonlinear fractional differential and integral equations which need to be solved. Finding the exact solutions for such equations is usually not an easy mission. Because of that, finding an approximate solution becomes very important. The desired approximate solutions have to be of small error with low computational and time. The aim of the present thesis is to investigate the features of spectral methods for numerical solutions of fractional differential equations and fractional integro-differential equations subject to various kinds of non-local conditions. The speed of convergence is one of the great advantages of spectral methods. In addition to having exponential rates of convergence, spectral methods also have a high level of accuracy. In this thesis, firstly, we use a combination of shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau to convert non-smooth solution to smooth solution to solve one- and two-dimensional nonlinear variable-order fractional convection-diffusion equations. Different examples are used in comparisons with other methods of the same orders to check the efficiency of the proposed techniques. Next, we proposed shifted Jacobi-Gauss collocation method and fractional-order shifted Jacobi-Gauss collocation method for solving one- and two-dimensional mixed Volterra-Fredholm integro-differential equations, fractional Fredholm integral equations and variable-order fractional integro-differential equations with a weakly singular kernel. The Riemann-Liouville fractional derivative is considered. Several numerical examples in addition to two applications are tested to see the efficiency and applicability of the proposed technique. It is empirically shown that the proposed methods yield solutions with high accuracy and need less computational time than the other methods used in the comparisons. Next, we use shifted fractional Chebyshev-Gauss and fractional Legendre Gauss quadrature points to solve the non-linear variable-order fractional Bagley-Torvik differential equation and space- and time-variable-order fractional Fokker-Planck equation. Several numerical examples are selected to compare the proposed schemes with other schemes of the same order. Moreover, we adopt a transformed spectral scheme to solve nonlinear high-dimensional weakly singular integral equations with proportional delay admitting nonsmooth solutions. The modified spectral approach is built on multivariate Jacobi polynomials and smoothing transformations are adopted to circumvent the difficulty of singularity at the beginning of time. Furthermore, we investigate the convergence, existence, and uniqueness of approximation solutions. The proposed methods improve the accuracy of the obtained approximate solutions compared to other selected methods in the literature. In addition, the proposed methods reduce the needed CPU-time. These improvements are significant in the field of computational mathematics, chemistry, physics, and any applied sciences that produce nonlinear and nonsmooth problems.