Numerical Methods For Nonlinear Reaction-diffusion Equations With Memory

In recent years, it is found that a class of reaction-diffusion phenomena possesses the ability of memory, that is, its standard diffusion term is replaced by a weighted convolution of diffusion in the past time. Many researchers have extensively discussed the solvability and attractors of the nonlinear reaction-diffusion equations with memory, but the current numerical methods seem only covering its linear counterpart. Thus, it is worth to study the numerical methods for such nonlinear reaction-diffusion equations with memory.There are two barriers in the numerical methods for such a kind of equations: one is the nonlinear term, another is the convolution of diffusion. The strategy in the thesis is as follows: firstly we take spatial derivatives of the original equations, and take the gradient of the unknown as the new unknown and frozen nonlinear term to carry out linearization, so as to establish a new equation; secondly use the Laplace transform to the convolution, and apply the six-point implicit scheme to set up the semi-discrete difference equation; finally make use of the numerical inversion of the Laplace transformation to get an approximation solution to the equation. In the thesis it also includes detail discussions on the numerical inversion of the Laplace transform, and some theoretical analysis on the above numerical method, and a numerical experiment shows that such a method is applicable and enjoys higher accuracy.