Numerical Approximation Of Several Fractional Differential Equations And Stochastic Differential Delay Equations

It is well known that the differential equations with fractional order are gen-eralization of ordinary differential equations to non-integer order. they occur more frequently in different research areas and engineering, such as physics, control of dy-namical systems, chemistry etc. In addition, stochastic systems with delays also play an important role in the areas of science and engineering for a long time. This thesis mainly studies the numerical solutions of several fractional differential equations and stochastic differential delay equations.This Ph.D. thesis is divided into six chapters.In Chapter1, we introduce the historical background and recent research situa-tion of the numerical approximation of fractional differential equations and stochastic differential delay equations. Then, the main work of this thesis is also concerned.In Chapter2, the numerical solution of the nonlinear boundary value problems with Riemann-Liouville fractional derivative and deviating arguments is considered. By means of the monotone iterative technique and the method of lower and upper solutions, we introduce two well-defined monotone sequences of lower and upper solutions which converge uniformly to the actual solution of the problem, and then the existence result of solution for the boundary problems is established. Moreover, a numerical iterative scheme is introduced to obtain an accurate approximate solu-tion for the problem. In this chapter, we remove the results of existing papers, and establish the results under the weaker monotone conditions. Finally, an example is presented.In Chapter3, By means of the monotone iterative technique, we consider the numerical approximation of the nonlinear boundary value problems of Caputo frac-tional differential equation, and define the quasi-lower and quasi-upper solutions. Moreover, we also give an algorithm to construct two monotone sequences of quasi-lower and quasi-upper solutions. Moreover, the constructed sequences are proved to converge uniformly to the unique solution of the problem. After removing the results of existing papers, the theory in this chapter is proved under a weaker condition. In Chapter4, we first introduce the existence and uniqueness of the global so-lutions for stochastic delay differential equations with coefficients of the polynomial growth. Then, we prove the existence and uniqueness results for the global solu-tion of stochastic delay differential equations with Markovian switching under the polynomial growth. Moreover, we establish the convergence in probability of the Euler-Maruyama numerical solution to the exact solution of the problem. So, in this chapter, we extend the results from some paper to the nonlinear stochastic delay d-ifferential equations under the the polynomial growth condition instead of the linear growth condition.In chapter5, the main aim is to study the highly nonlinear stochastic differential equation with time-dependent delay. After providing the existence-and-uniqueness of the exact solution under the polynomial growth condition, we prove that the back-ward Euler-Maruyama numerical method can preserves boundedness of moments, and the implicit numerical approximation converges strongly to the true solution. Finally, a highly nonlinear example is considered to illustrate the main theory. In this chapter, which is natural extension of the previous chapter, we prove the strong convergence instead of the weak convergence.Chapter6is concerned with brief summary and Research prospects.