Stability and Error Analysis of Numerical Schemes for 1D and 2D Fractional Differential Equations

Abstract

"The aim of this thesis is to construct and analyze some simple, yet very efficient numerical methods to approximate solutions to fractional differential equations (FDEs) which find wide-ranging applications across numerous fields. From understanding fluid dynamics in engineering to modeling chemical reactions in chemistry, and from analyzing electrical networks in physics to optimizing control systems in robotics, these mathematical models underpin crucial aspects of modern technology and scientific inquiry. In FDEs, weakly singular kernels play a significant role. These kernels have singularities that are milder compared to classical calculus. Since most FDEs lack analytical solutions, we look towards different numerical methods as the optional way. However, when dealing with FDEs with weakly singular kernels, standard numerical techniques may not suffice, and thus specialized techniques are needed to ensure accurate and efficient computations. The non-uniform mesh generation strategies help us to get rid of this issue of singularities by distributing a sufficient number of mesh points near the singular point."