B-Spline Wavelet Method For Solving Inverse Problem Of Diffusion Equation

Inverse problem widely exists in the field of natural science and engineering,which is a new and important research direction in current scientific research.Among them,the solution of diffusion equation inverse problem promotes the solution of many practical problems and becomes one of the fastest developing fields in computational mathematics.Therefore,how to quickly and efficiently solve the diffusion equation inverse problem has good scientific research value and practical significance.Based on thegood characteristics of B-spline function,and the variable scale ability of wavelet function in different resolution levels,in this paper,we use the cubic B-spline wavelet scaling functions to discretize the differential equation,and determine the unkown initial boundary value conditions of the 1-D and 2-D diffusion equation along with the Tikhonov regularization method,the messless collocation method and the quasi Newton method.For the inverse problem of one-dimensional diffusion equation,B-spline wavelet meshless method is used to determine the unknown initial conditions and the unknown boundary conditions.The B-spline wavelet collocation method is used to discretize the initial boundary conditions and additional condition of the diffusion equation,which can transform the inversion problem into a problem of solving a system of algebraic equations.Because the coefficient matrix of the algebraic equation system is ill-conditioned,A regularization B-spline wavelet method is proposed by in introducing the Tikhonov regularization method to alleviate the ill-posedness of the problem.The regularization B-spline wavelet method and the B-spline wavelet method are used to solve the numerical examples,respectively.,The numerical results show the superiority of the regularization B-spline wavelet method.For the inverse problem of two-dimensional diffusion equation,B-spline wavelet method is used to solve the direct problemf the two-dimensional diffusion equation.The numerical results for different time and time steps are analyzed to verify the effectiveness of forward problem.For the inversion of initial conditions,the unknown initial conditions are fitted with polynomials,and then the polynomials with parameters are substituted into the solution of forward problem.Based on the idea of solving extreme value,the inverse problem of two-dimensional diffusion equation can be transformed into a nonlinear optimization problem.Finally,the quasi Newton method is used to find the unknown parameters of the optimization problem.The numerical results of several examples demonstrate that the B-spline wavelet method is effective and feasible for solving the inverse problem of diffusion equation.