A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

Abstract

In this paper, by applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. It consists of two main parts. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequence $\left\{u^{\left(m\right)}\right\}$ based on a $N$ -order iterative scheme in case of $f\in C^N\left(\left[\mathrm{0,1}\right]\times \left[0,T^{\ast }\right]\times ℝ\right)\left(N\ge 2\right)$ , or a single-iterative scheme in case of $f\in C^1\left(\overline{\mathrm{\Omega }}\times \left[0,T^{\ast }\right]\times ℝ\right)$ . In Part 2, we begin with the construction of a difference scheme to approximate $u^{\left(m\right)}$ of the $N$ -order iterative scheme, with $N=2$ . Next, we present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.