On the Numerical Solutions of Non-Local Problems of Fractional Order Differential Equations

Abstract

The thesis focuses on the study of some problems related to the second order boundary value problem (BVP). Four generalization styles are discussed. In the first, the Non-local conditions of the Dirichlet and Neumann types are studied. In the second, the fractional order BVP is considered. In the third, the singular perturbed problem is treated. In the fourth, the non-linear problem with exponential nonlinearity is considered. The finite difference representation of the BVP is employed to reduce the problem into a system of algebraic equations, in addition, the implicit- explicit treatment to the nonlocal problems is introduced. In the implicit- explicit treatment the computational work is reduced considerably. Two methods of grid labelling (natural and red-black) are studied. The effect of grid labelling on the construction of the algebraic systems as well as on the convergence rates of relaxation and modified relaxation methods are considered.