Finite difference schemes of the nonlinear pseudo-parabolic system

Abstract

In this paper, we deal with the finite difference method for the initial boundary value problem of the nonlinear pseudo-parabolic system

$$\begin{gathered} ( - 1)^M u_t + A(x,t,u,u_x , \cdots ,u_{x^{2M} } )u_{x^{2M} _t } = F(x,t,u,u_x , \cdots ,u_{x^{2M} } ), \hfill \\ u_{x^k } (0,t) = \psi _{0k} (t),u_{x^k } (L,t) = \psi _{1k} (t),k = 0,1, \cdots ,{\rm M} - 1, \hfill \\ u(x,0) = \phi (x), \hfill \\ \end{gathered} $$

in the rectangular domainQ _T =[0≤X≤L, 0≤t≤T], whereu(x,t)=(u ₁(x,t),u ₂(x,t), ...,u _m (x,t)),φ(x),ψ _0k (t),ψ _1k (t),F(x,t,u,u _x , ...,u _x 2m) arem-dimensional vector functions, andA(x,t,u,u _x , ...,u _x 2m is anm×m positive definite matrix. The existence and uniqueness of solution for the finite difference system are proved by the fixed-point theory. Stability, convergence and error estimates are derived.