Reproducing Kernel Spline Method For Time-delay Regular Diffusion Equation

Image recovery is the basic problem of image processing and computer vision.Partial di?erential equations(PDEs)is one of the important tools for image process-ing.Among them,di?usion equation has attracted the attention of many scholars because its complete mathematical theory and important physical significance.In this paper,a system of time delay regular di?usion equations is proposed,in which the di?usion coe cient in the di?usion equation depends on the coupled time delay regular equation.The source term in the di?usion equation can change the original di?usion mode,from the physical process of cold and hot neutralization to cold and hot.Compared with the classical di?usion equation,the time delay regularization of the time delay regular di?usion equation is very important for restoring the image structure.It can consider both the di?usion information in the current space and the past time information.This kind of equation has a wide application prospect in image problems.In this paper,the time delay regular di?usion equations are analyzed theoreti-cally and numerically.Theoretically,the existence and uniqueness of weak solutions of the equations are proved by Rothe method.Rothe's method is outstanding in proving the existence and uniqueness of solutions of parabolic partial di?erential equations.The e ciency of Rothe method is that it divides the time direction into di?erent time layers,and connects the time layers of each layer by using fi-nite segment polyline function.After the time layers are separated,the problem of parabolic equation is transformed into the problem of elliptic equation.Numerically,the backward implicit Euler method is adopted to ensure the sta-bility of the numerical scheme.There are both time derivative and space derivative in regard to the di?usion equation.For the derivative in the time direction,the finite di?erence scheme is used.For the approximation of spatial direction,a new set of bases is constructed by combining the reproducing kernel functions and the spline unctions with the collocation method to solve the system of equations in the reproducing kernel function space W~3[0,1].In addition,the convergence analysis and error estimation of reproducing kernel spline method are given.Numerical examples show the e?ectiveness of the method.