A Two Level Implicit Difference Formula of O(k 2 +h 4 ) for the Numerical Solution of One Space Dimensional Unsteady Quasi-Linear Biharmonic Problem of First Kind

Abstract

In this piece of work, we discuss two new two-level implicit difference formulas of O(k²+h²)and O(k²+h⁴) in a coupled manner for the numerical solution of quasi-linear partial differential equation A(x,t,u,u_x,u_xx,u_xxx)u_xxxx+u_t=f(x,t,u,u_x,u_xx,u_xxx), 0 < x < 1, t > 0 subject to the initial condition u(x,0)=g₀(x), 0≤x≤1 and boundary conditions u(0,t)=p₀(t), u_x(0,t)=q₀(t), u(1,t)=p₁(t), u_x(1,t)=q₁(t), t≥0, where k > 0 and h > 0 are grid spacing in t- and x-directions, respectively. In both cases, we use only three spatial grid points and do not require to discretize the boundary conditions. The numerical solution of u_x is obtained as a by-product of the method. The stability analysis for a model problem is discussed. The methods are successfully applied to the problems in polar coordinates. Numerical examples are given to demonstrate the utility of new methods and their convergence.