An accurate three spatial grid-point discretization of O(k2+h4) for the numerical solution of one-space dimensional unsteady quasi-linear biharmonic problem of second kind

摘要

In this article, using three spatial-grid points we propose two new two level implicit finite difference approximations of O(k2+h2) and O(k2+h4) in a coupled manner to the one-space dimensional unsteady quasi-linear biharmonic equation A(x,t,u,uxx)uxxxx+ut=f(x,t,u,ux,uxx,uxxx), 00 subject to the initial and boundary conditions u(x,0)=φ(x), u(0,t)=p0(t), uxx(0,t)=q0(t), u(1,t)=p1(t), uxx(1,t)=q1(t) are prescribed, where h>0 and k>0 are mesh sizes in x- and t-directions, respectively. The numerical solution of uxx is obtained as a by-product of the method and we do not require to discretize the boundary conditions. The methods are successfully tested on the problems having singularities. Numerical results are provided to demonstrate the convergence of new methods.