Optimal approximation to a class of nonlinear evolution equations

In this paper, we have developed and discussed a generalized difference method (GDM) for the approximation of a class of nonlinear evolution equations with integral items in one space dimension. The main idea of the paper is to introduce a generalized Volterra projection operator and we design a generalized difference formulation with a new approximate value of initial function, which is of special importance for the error analysis. Then some theories of the generalized Volterra projection operator are analyzed. Finally, we deduce the optimal error estimates. Moreover, some conclusions are obtained and our further research work is proposed.