Research On Inverse Problem In Fractional Biological System

In population ecology,many problems can be described by nonlinear fractional equations,and most of these nonlinear fractional equations with practical application background cannot be solved accurately,or the expressions of the solutions are very complex,so the use of numerical methods to solve practical problems is effective and widely used.In this paper,the finite difference method and the optimal perturbation algorithm are used to study the initial and boundary value problems of the fractional logistic nonlinear diffusion model and the fractional Lotka-Volterra competition model.The main contents of the paper are as follows:In chapter 1,the research significance of the topic is introduced,and the research status and development trend at home and abroad are briefly described.Then,the concept and properties of Caputo fractional derivative are introduced.Finally,the main work of this paper is given.In chapter 2,we mainly study the forward problem of the fractional logistic nonlinear diffusion model.Taylor expansion is carried out for the nonlinear terms,and an implicit difference scheme is established to solve the forward problem.The L~2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality,and numerical examples are presented to support the theoretical analysis.In chapter 3,we mainly study the forward problem of fractional Lotka-Volterra competition model,and establish a semi-implicit difference scheme to solve the problem.The L~? stability and convergence of the scheme are proved by the combination of the Lipschitz condition of nonlinear terms and the mathematical induction method,and numerical examples are presented to support the theoretical analysis.In chapter 4,we mainly consider the fractional logistic nonlinear diffusion model and the fractional Lotka-Volterra competition model to determine the inverse problem of multiple model parameters,and introduce the method of solving the inverse problem—the optimal perturbation regularization algorithm.Using the observation data of internal points in space as additional data,applying the optimal perturbation regularization algorithm to perform numerical inversion,the calculation results show the numerical stability of the inverse problem.In chapter 5,we give summary and the main results of this paper,and put forward the some problems in the research and the research direction in the future.