A Two Level Implicit Difference Formula of O(k2+h4) for the Numerical Solution of One Space Dimensional Unsteady Quasi-Linear Biharmonic Problem of First Kind
Affiliations: Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi – 110 007, India
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,ux,uxx,uxxx)uxxxx+ut=f(x,t,u,ux,uxx,uxxx), 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), ux(0,t)=q0(t), u(1,t)=p1(t), ux(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 ux 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.
