High Order Algorithms And Theoretical Analysis For The Fractional Differential Equations

In recent decades, fractional differential equations have been playing more and more important roles, e.g., in mechanics(theory of viscoelasticity and viscoplasticity),(bio-)chemistry(modelling of polymers and proteins), electrical engineering(transmission of ultrasound waves), medicine(modelling of human tissue under mechanical loads). Because of the nonlocal properties of fractional operators, obtaining the analytical solutions of the fractional differential equations is more challenge or sometimes even impossible, effciently solving the fractional differential equations naturally becomes an urgent topic.The thesis consists of ?ve parts:In the ?rst chapter, we brie?y review the history of fractional operators, which are widely used in different ?elds.In Chapter 2, because of the nonlocal properties of fractional operators, higher order schemes play more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with ?rst order schemes but greatly improve the accuracy. The core object of this chapter is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference(WSLD) operators,for space fractional derivatives. Then we use the derived schemes to solve the space fractional diffusion equation with variable coeffcients in one-dimensional and twodimensional cases. And the unconditional stability and the convergence with the global truncation error O(τ2+ h4) are theoretically proved and numerically veri?ed.In Chapter 3, we discuss the properties and the numerical discretizations of the fractional substantial integral and the fractional substantial derivativewhere Ds =??x+ σ = D + σ, σ can be a constant or a function not related to x, sayσ(y); and m is the smallest integer that exceeds μ. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error O(hp)(p = 1, 2, 3, 4, 5) are theoretically proved and numerically veri?ed.In Chapter 4, the Feynman-Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function of Brownian functionals satis?es the Feynman-Kac formula, being a Schr¨odinger equation in imaginary time. The functionals of non-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kac equation, where the fractional substantial derivative is involved. Based on recently developed discretized schemes(in Chapter 3) for fractional substantial derivatives, this chapter focuses on providing algorithms for numerically solving the forward and backward fractional Feynman-Kac equations; since the fractional substantial derivative is non-local timespace coupled operator, new challenges are introduced compared with the ordinary fractional derivative. Two ways(?nite difference and ?nite element) of discretizing the space derivative are considered. For the backward fractional Feynman-Kac equation, the numerical stability and convergence of the algorithms with ?rst order accuracy are theoretically discussed. For all the provided schemes, including the ?rst order and high order ones, of both forward and backward Feynman-Kac equations,extensive numerical experiments are performed to show their effectiveness.In Chapter 5, the equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the L′evy?ights; and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the L′evy ?ights have divergent second order moments. However, in more practical problems, the physical domain is bounded and the involved observables have ?nite moments. Then the modi?ed equation can be derived by tempering the L′evy measure of the L′evy ?ights and the corresponding tempered space fractional derivative is introduced. This chapter focuses on providing the high order algorithms for the modi?ed equation, i.e., the equation with the time fractional substantial derivative and space tempered fractional derivative. More concretely, the contributions of this chapter are as follows: 1. the detailed numerical stability analysis and error estimates of the schemes with ?rst order accuracy in time and second order in space are given in complex space, which is necessary since the inverse Fourier transform needs to be made for getting the distribution of the functionals after solving the equation; 2. we further propose the schemes with high order accuracy in both time and space, and the techniques of treating the issue of keeping the high order accuracy of the schemes for nonhomogeneous boundary/initial conditions are introduced; 3. the multigrid methods are effectively used to solve the obtained algebraic equations which still have the Toeplitz structure; 4. we perform extensive numerical experiments, including verifying the high convergence orders and simulating the physical system which needs to numerically make the inverse Fourier transform to the numerical solutions of the equation.