Researches On Finite Difference Approximation For Two Types Of Space Fractional Partial Differential Equation

In recent decades, fractional calculus theory has been widely applied to simulate complex phenomena in the mechanics and engineering. In comparison to the mathematical model utilising classical theory of Newton-Leibnitz calculus, the model involving fractional calculus provides a more adequate and accurate way to interpret the self-similar and power-law phenomena. Due to the difficulty and complexity of obtaining the solution of space-fractional model, this thesis aim to discuss the finite difference approximation for two types of space-fractional partial differential model and some important related issues. Specifically, this dissertation including:In chapter 1, we give a brief introduction to the history of the fractional calculus and present some common used definitions of fractional derivative. Then we have a short talk on the background of this dissertation. Finally, we present the main structure of this thesis.In chapter 2, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law: (?) a≤x≤b,0<α<1,t>0, where aDxα, xDbα are left and right Riemann-Liouville fractional differential op-erator of order a. For given initial and boundary conditions, a semi-implicit difference method (SIDM) for this equation is proposed. The consistency, u-nique solvability, stability, and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like ma-trices. This fast iterative method significantly reduces the storage requirement of O(n2) and computational cost of O(n3) down to O(n) and O(nlogn), where n is the number of grid points. The method retains the same accuracy as the un-derlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.In chapter 3, we consider an inverse problem for identifying the fraction-al derivative indices in a two-dimensional space-fractional diffusion model with variable diffusivity coefficients: (?) (x, y)∈Ω, 0 <α<1,0<β<1, t > 0. Firstly, we derive an implicit difference method (IDM) for the direct problem and establish the stability and convergence of the IDM. For the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) which is superior in performance to Gaussian elimination but attains the same accuracy. Secondly, we utilize the Levenberg-Marquardt(LM) regular-ization technique combined with the Armijo rule to solve the modified nonlinear squares model associated with the parameter identification. Finally, we carry out numerical test to verify the accuracy and efficiency of the IDM. Numerical inversions are performed with both the accurate data and noisy data to check the effectiveness of the LMT regularization method. The convergence behaviors of the LMT for the inverse problem in space-fractional diffusion model are shown graphically. In chapter 4, a class of unconditionally stable difference schemes based onthe Pade approximation are presented for the Riesz space-fractional telegraph equation: (?) a≤x≤<b,0≤t≤T, 1<γ≤2, where RDzγ, is the Riesz fractional differential operator of order 7. Firstly, we introduce a new variable to transform the original differential equation to an e-quivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Final-ly, we use (1,1), (2,2) and (3,3) Pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equation-s. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes are accurate and efficient for solving a space-fractional hyperbolic equation.In chapter 5, we derive a fast unconditionally stable finite difference scheme of high order for the two-dimensional Riesz space-fractional model: (?) +f(x,y,t), (x,y)∈Ω,0<t≤T, Similar to what have done for the model discussed in the previous chapter, with the help of Kronecker tensor product, we obtain the semi-discrete scheme. For the discretization in the temporal direction, a cubic spline interpolation method is used to derive the fully discrete scheme with order of 0(hx2+hy2+τ4). Con-sidering the computational effort and consumed CPU time, we apply the fast bi-conjugate gradient stabilized method to the two-dimensional case. The cor-responding complexity is analyzed, numerical examples have been run to testify the theoretical analysis.In chapter 6, we present the summary of this thesis and give a brief talk on the possible research interest for the future.