High Accuracy Numerical Methods For Fractional Diffusion-wave Equations And Fractional Variational Problems

With the further research on complex systems in natural and social sciences,many phenomena in anomalous diffusion, viscoelasticity mechanics, soft material,electromagnetism, system control, biological medicine, economics and many other?elds have been described successfully by the mathematical models with fractional calculus(that is, calculus of integrals and derivatives of any arbitrary real or complex order) in recent years. The theory and applications of fractional calculus have gained considerable popularity and importance. Fractional differential equations provide an excellent instrument for the description of memory or nonlocal properties of various materials or processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models, in which such effects are in fact conventionally neglected. It is difficult or even impossible to achieve the analytical solutions in many cases, and there are usually these special functions with calculation troubles. These motivate greatly the researchers to engage in the numerical solution of fractional order systems.The current numerical methods for fractional diffusion-wave equations(FDWs), fractional variational problems(FVPs) and fractional optimal control problems(FOCPs) focus on the ?nite difference methods which usually mean a high storage requirement, large amount of calculation and low accuracy. Due to the exponential convergence when sinc functions and the Jacobi poly-fractonomials approximate many special functions, this thesis is devoted to utilizing this virtue and constructing the high-accuracy numerical methods for a class of the initial boundary value problems of FDWs, FVPs and FOCPs. The thesis consists of ?ve chapters and is organized as follows.The thesis begins in Chapter 1 with a brief historical review of the theory of fractional calculus and its applications. At the same time, the background and current situations about FDWs, FVPs, FOCPs and their numerical results are stated. Furthermore, the research motivation and the main contents are proposed brie?y.In Chapter 2, some special functions, the Jacobi orthogonal polynomials, three well known de?nitions and some properties in fractional calculus are stated. These results are needed in the later chapters.Chapter 3 develops an efficient numerical method for a class of the initial boundary value problems of FDWs with the Caputo fractional derivative of orderα(1 < α < 2). This approach is based on the ?nite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the original problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of 3- α order accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results. Next, the standard fractional diffusion equations(FDEs) describing subdiffusion are deduced from the master equation with integral form. It is proved that this master equation is equivalent with continuous time random walk(CTRW) model. Sinc-Chebyshev collocation method is used to solve the aforementioned FDEs and FDWs, and numerical examples con?rm the feasibility and efficiency of the method. In particular, when the double exponential transform is used in sinc collocation method, accuracy in space is improved greatly.An exponentially accurate Rayleigh-Ritz method is developed for solving FVPs in Chapter 4. The Jacobi poly-fractonomials based on fractional SturmLiouville eigen-problems are chosen as basis functions to approximate the exact solution, and the Rayleigh-Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature, and reduces request of the memory space. The fractional variational convergence is discussed. Numerical examples are given to illustrate the exponential convergence of the method.In Chapter 5, the numerical method for a class of generalized FOCPs is presented. The necessary optimality conditions and fractional Hamiltonian system are obtained by the fractional variational calculus. Based on the shifted Jacobi poly-fractonomials, the FOCPs with quadratic performance index and linear fractional dynamic constraint are solved. The variational convergence is analyzed, andnumerical examples con?rm the exponential convergence.