Numerical Methods For The Fractional Laplace Differential Equations

This work is devoted to constructing finite difference schemes for the fractional Laplace differ-ential equations and establishing the corresponding error estimates.In the first part,we construct a finite difference scheme for the one-dimensional fractional Laplace differential equation and establish the corresponding prior estimate.Based on the Laplace operator in integral form,we divide the integral region and get a small integral region containing singular points and another region containing no singular points.And we process the two regions respectively to obtain a difference scheme.Then we use the properties of symmetrically positive definite and strict diagonal dominance of the coefficient matrix to provide the convergence analysis of the difference scheme.At last,we present some numerical examples to verify the convergence order and validity of the scheme.In the second part,we study the finite difference method for one-dimensional evolutionary fractional Laplacian differential equation.Here we present the backward Euler scheme and the Crank-Nicolson scheme respectively,and give a priori estimation for the solution of difference scheme by using the maximum principle and the energy analysis method respectively.At last,we report the numerical experiment to check the convergence and effectiveness of the difference schemesIn the third part,we construct the finite difference scheme for the two-dimensional fractional Laplace differential equation and establish the corresponding prior estimate.The two-dimensional fractional Laplacian is still defined by a form of integral.The method of discretizing it is similar to one-dimensional problem.We divide the integral region and get a small rectangular region con-taining singular points and another region containing no singular points.And we deal with the two regions respectively to obtain a difference scheme.Different from the one-dimensional problem,the two-dimensional problem is more complicated,and the generation of the coefficient matrix is very challenging.With the same as that for the one-dimensional problem,the coefficient matrix is still symmetrically positive and strictly diagonally dominant,from which convergence of the difference scheme can be obtained.At last,we present some numerical examples to verify the convergence order and validity of the scheme.In the last part,we study the finite difference method for the two-dimensional evolutionary fractional Laplacian differential equation.Familiar with the second part,we also give the backward Euler scheme and the Crank-Nicolson scheme,and give a priori estimate with convergence proof for the solution of the difference schemes using the maximum principle and the energy analysis method,respectively.At last,we report the numerical experiment to check the convergence and effectiveness of the difference schemes.