Numerical Algorithms For Fractional Patrial Differential Equations And Its Applications In Mechanics

Recently, fractional calculus has been widely applied in various fields of science and engineering. Fractional calculus provides an excellent tool for the description of memory and hereditary properties of various materials and processes.In this paper, we mainly study the numerical method for some time frac-tional partial differential equations and some applications in mechanics. Firstly, we derive two compact finite difference schemes for the two-dimensional non-linear fractional reaction-subdiffusion equation, and analyze the stability and convergence of the two schemes by the Fourier method. Secondly, we propose a numerical method to estimate the unknown order of the Riemann-Liouville frac-tional derivative for the fractional Stokes’first problem for a heated generalized second grade fluid. Thirdly, we propose a fourth-order compact finite difference method for the two-dimensional fractional Cable equation, and discuss the sta-bility and convergence by means of the Fourier method. For the inverse problem, we propose an efficient numerical method to obtain the identification of the two fractional derivatives. Fourthly, we formulate a fractional thermal wave model for a bi-layered spherical tissue, and employ the implicit finite difference method to obtain the solution of the direct problem. Based on the experimental data, we propose an efficient numerical method to obtain the optimal estimation of the Caputo fractional derivative and the relaxation time parameters. Lastly, we present a fractional anomalous diffusion model to describe the uptake of sodium ions across the epithelium of gastrointestinal mucosa and their subsequent diffu-sion in the underlying blood capillaries using fractional Fick’s law. Concretely, it concludes:In Chapter 1, we first give a brief introduction to the history of the frac-tional calculus, and then we present some numerical methods for solving the time fractional partial differential equations and introduce the definitions of some frac- tional operators used in our paper. Finally, we present our main work in this paper.In Chapter 2, we consider the two-dimensional non-linear fractional reaction-subdiffusion equation where the non-linear source term g(u,x,y,t) has the second-order continuous partial derivative (?)2g(u,x,y,t)/(?)t2,and satisfies the Lipschitz condition with respect to u, that is, Firstly, to derive a novel fourth-order compact finite difference scheme, we in-tegrate both sides of the equation with respect to the time, and then use the Riemann-Liouville fractional integral definition to discretize the fractional term and the fourth-order compact difference scheme to approximate the second-order derivative. As for the integral of the non-linear source term, we here adopt the trapezoid formula to approximate it.Finally, we have obtained a novel com-pact difference method with fourth-order spatial accuracy. The stability and convergence of the compact difference method have been analyzed by the Fourier method. Secondly, we derive an improved compact difference scheme with second-order temporal accuracy and fourth-order spatial accuracy by the linear interpo-lation technique, and prove the stability and convergence by the Fourier method under some appropriate conditions. Finally, numerical examples have confirmed the effectiveness and accuracy of the proposed compact difference methods.In Chapter 3, we consider the fractional Stokes’ first problem for a heated generalized second grade fluid: We propose a numerical method to estimate the unknown order of the Riemann-Liouville fractional derivative. Implicit numerical method is employed to solve the direct problem. For the inverse problem, we first obtain the fractional sensitivity equation by means of the digamma function: where and then we introduce the Levenberg-Marquardt algorithm to estimate the un-known order of the Riemann-Liouville fractional derivative. In order to demon-strate the effectiveness of the proposed numerical method, two cases in which the measurement values contain random measurement error or not are con-sidered. The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of the Riemann-Liouville fractional derivative for the fractional Stokes’first problem for a heated generalized second grade fluid.In Chapter 4, we consider the two-dimensional fractional Cable equation:An efficient numerical method to obtain the identification of the fractional deriva-tives is investigated. Concerning the numerical treatment of the two-dimensional fractional Cable equation, a fourth-order compact finite difference method is pro-posed, and the stability and convergence of the compact difference method are discussed rigorously by means of the Fourier method. For the inverse problem of the identification of the fractional derivatives, Levenberg-Marquardt iterative method is employed, and the fractional sensitivity equation is obtained by means of the digamma function. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.In Chapter 5, we formulate a fractional thermal wave model for a bi-layered spherical tissue: Implicit finite difference method is employed to obtain the solution of the direct problem. The inverse analysis for simultaneously estimating the Caputo frac-tional derivative and the relaxation time parameters is implemented by means of the Levenberg-Marquart method. Compared with the experimental data, we can obviously find out that the estimated temperature increase values are in excellent-ly consistent with the measured temperature increase values in the experiment. We have also discussed the effect of the fractional derivative, the relaxation time parameters, the initial guess as well as the sensitivity problem. All the results show that the proposed fractional thermal wave model is efficient and accurate in modeling the heat transfer in the hyperthermia experiment, and the proposed numerical method for simultaneously estimating multiple parameters for the frac-tional thermal wave model in a spherical composite medium is effective.In Chapter 6, we present a fractional anomalous diffusion model to describe the uptake of sodium ions across the epithelium of gastrointestinal mucosa and their subsequent diffusion in the underlying blood capillaries using fractional Fick’s law. A heterogeneous two-phase model of the gastrointestinal mucosa is considered, consisting of a continuous extracellular phase and a dispersed cel-lular phase. The main mode of uptake is considered to be a fractional anomalous diffusion under concentration gradient and potential gradient. Appropriate par-tial differential equations describing the variation with time of concentrations of sodium ions in both the two phases across the intestinal wall are obtained using Riemann-Liouville space-fractional derivative and are solved by finite difference methods. The concentrations of sodium ions in the interstitial space and in the cells have been studied as a function of time, and the mean concentration of sodi-um ions available for absorption by the blood capillaries has also been studied. Finally, numerical results are presented graphically for various values of different parameters. This study demonstrates that fractional anomalous diffusion model is appropriate for describing the uptake of sodium ions across the epithelium of gastrointestinal mucosa.In Chapter 7, we give our conclusions and some possible research interest in our future work.