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Study on the Numerical Solution and Application of Fractional Partial Differential Equations

This paper mainly studies the numerical calculation methods of several types of time fractional partial differential equations and some applications of fractional calculus theory and numerical methods in science. First, for two-dimensional nonlinear fractional-order reaction subdiffusion equations, we propose two compact finite difference schemes, and use Fourier analysis to give a theoretical analysis of the stability and convergence of these two schemes. Secondly, for the first-order fractional Stokes problem of Cantonian second-order fluid under heating, we propose a numerical parameter estimation method to estimate the order of the Riemann-Liouville fractional derivative. Thirdly, for the two-dimensional fractional-order Cable equation, we propose a compact fourth-order finite-difference scheme in space, and use Fourier analysis to give a theoretical proof of stability and convergence. For the inverse problem, we propose a numerical parameter estimation method, and give the optimization of two fractional order derivatives. In the tumor hyperthermia experiment, we constructed a time-fractional heat wave model of the double-layered spherical tissue, and used the implicit difference method to give a numerical solution of the T model. For the inverse problem, with the help of thermal experimental data, we propose a nonlinear parameter estimation method that gives an optimal estimate of the unknown fractional derivative and relaxation time parameters. Finally, the transport process of steel ions across the intestinal wall, we established a spatial fractional order anomalous diffusion model under the action of concentration gradient and potential gradient, and obtained the numerical solution of the problem by finite difference method.