Numerical Analysis of Time-fractional Parabolic Differential and Integro-differential Equations

Abstract

Over the last few decades, the subject of fractional calculus emerged in various areas of science and engineering. In this regard, fractional partial differential equations are used to describe anomalous diffusion. The main feature of the fractional differential equations is their nonlocal and heredity property, which makes their solution challenging. Moreover, the smooth initial data in a differential equation involving fractional derivatives may not provide a smooth solution. Due to such uncertain behavior, obtaining analytic solutions of the fractional partial differential equations is intricate or impossible in many cases. Consequently, efficient numerical techniques play a major role in evaluating an approximate solution of fractional partial differential equations and analyzing its asymptotic behavior. The present thesis intends to develop and analyze some efficient and stable numerical schemes for solving a class of time-fractional partial differential and integro-differential equations of parabolic type in one and two dimensions. The thesis starts with a brief history of fractional calculus, followed by some preliminaries of time-fractional differential equations and the approximation techniques to obtain their solutions. This thesis can be divided into two parts. The first part is devoted to constructing some layer-adaptive numerical techniques for different classes (including linear, semilinear, time-delayed and interface models) of time-fractional partial differential models of parabolic type. In general, the typical solution to such types of problems undergoes a sharp change at t = 0, namely, the interior layer region of the domain due to the presence of the weak singularity. The traditional numerical methods on uniform mesh fail to grasp such abrupt changes inside the layer region, and they degrade the convergence rate. The layer-adaptive graded mesh with the user-chosen grading parameter is used in the temporal direction to achieve optimal accuracy. The widely-used L1 technique is employed to discretize the fractional order differential operator in the temporal direction for most of the model problems. The corresponding proposed schemes achieve a superlinear rate of convergence for suitable choices of the mesh grading parameter. In some cases, the newly-proposed L1-2 and L2-1σ technique is used in the temporal direction to construct a higher order (up to second order) scheme. The Newton linearization technique alongside the Daftardar-Gejji and Jafari method tackle the nonlinear part of the problems. The second part develops numerical techniques for the time-fractional partial Volterra integro-differential equations of parabolic type, along with their possible extensions to problems with semilinearity. The typical solution for these problems is assumed to be sufficiently smooth, subject to the prescribed initial data. For all the model problems, the fractional derivative is discretized using the L1-2 technique on a uniform mesh in the temporal direction. The composite trapezoidal rule approximates the integral part, whereas the composite product trapezoidal formula is employed to approximate the integral involving a weakly singular kernel. The classical central difference formula and the cubic B-spline collocation method are used on a uniform mesh in the spatial direction. The operator-splitting idea in time is discussed for the two-dimensional problem. To solve the resulting system of algebraic equations, the well established Thomas’s algorithm is used. On a suitable norm, the stability and convergence for all the proposed schemes are performed thoroughly under sufficient regularity assumptions on the initial data and true solution of the considered model problem. The efficiency and applicability of the proposed schemes are tested through numerical experiments. Computational results are presented through several plots and tables to support the theoretical findings. At the end of this thesis, a brief summary of the findings and the scope of further enhancement of the proposed study are provided. The novelty of these schemes is their simplicity and efficiency compared to the existing methods.