High-precision Finite Difference Methods For Two-dimensional Parabolic Problems With The Nonlocal Boundary Conditions

In various fields,we can find all kinds of parabolic equations,which are sub-ject to nonlocal boundary conditions,and it is of great practical significance to solve the approximate solution of such problems.In this paper,the finite differ-ence schemes are derived for solving one-dimensional and two-dimensional nonlocal parabolic problems.Compared to other papers,the methods of this paper are simple and effective,and we give the proof procedure of the convergence seriously.The error obtained in the paper has the saturated order of convergence,which is O(? + h2),In addition,we give two numerical examples separately to confirm the validity and accuracy of the theoretical results.The main research work of the paper is described as follows:In chapter one,we review the research background and results of partial differ-ential equations with nonlocal boundary conditions.In chapter two,we give the basic lemmas and the proof process.In chapter three,for the one-dimensional nonlocal parabolic problem,we pro-duce the finite difference scheme,and give the proof procedure of the convergence by the method of discrete Fourier transform.In chapter four,for the two-dimensional nonlocal parabolic problem,we first make a transformation so that the problem can be translated into two other prob-lems.One is one-dimensional nonlocal problem,which we can solve by the method in chapter three.Another is two-dimensional parabolic problem,which is combined with two types of boundary conditions.We produce the finite difference scheme,and prove the convergence of the scheme by the discrete extremum principle.