Robust Numerical Methods For Two Kinds Of Singularly Perturbed Problems And One Kind Of Nonlinear Fractional Differential Equation

In this thesis, we consider two kinds of singularly perturbed problems and one kind of nonlinear fractional differential equations. Singularly perturbed problems have important applications in fluid mechanics, elasticity, quantum mechanics, acoustics, optics, chemical reaction and optimum control etc. While partial differential equations with space fractional derivatives are usually used to model problems in mathematical finance, dislocation dynamics, gas detonation and anomalous diffusion in semiconduc-tor growth. Though these problems arise in many application areas, it is difficult to numerically resolve them. Therefore, it is meaningful to construct robust numerical schemes for these problems.The thesis consists of five chapters:In Chapter One, we describe the research background and progress for the sin-gularly perturbed problems and fractional differential equations and point out the problems we will consider and the main work of the thesis.In Chapter Two, we consider a singularly perturbed problem with two small parameters. The numerical scheme used is the simple upwind finite difference scheme. We first perform a maximum posteriori error analysis. Then, using the posteriori error estimate, we design an adaptive algorithm. Finally, we give some numerical results which confirm the theoretical analysis.In Chapter Three, we consider the same boundary value problem as that in Chap-ter two. Here, we use the weighted finite difference scheme which is a generalization of the numerical scheme used in Chapter two. First, with a detailed analysis, some maximum a posteriori error estimates are obtained. Then, based on the posteriori error estimates, we propose a monitor function and design an adaptive algorithm. Finally, some numerical results are given to confirm the theoretical analysis.In Chapter Four, we consider a singularly perturbed problem with just one s-mall parameter. The numerical scheme used is the four point upwind finite difference scheme. We first consider the stability of the numerical scheme and perform a maxi-mum posteriori error analysis. Then, an adaptive method is designed. The numerical results are given at the end of this chapter to confirm the theoretical analysis.In Chapter Five, we investigate the finite difference WENO scheme applied to a nonlinear fractional differential equation and some numerical simulations are given to confirm the effectiveness of the scheme.