A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind

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Abstract

We present two new finite difference methods of order two and four in a coupled manner for the general one-dimensional nonlinear biharmonic equation yIV=f(x,y,y′,y″,y″′) subject to the boundary conditions y(a)=A0,y′(a)=A1,y(b)=B0,y′(b)=B1. In both cases, we use only three grid points and do not require to discretize the boundary conditions. First-order derivative of the solution is obtained as a by-product of the methods. The methods are successfully applied to the problems both in cartesian and polar coordinates. Numerical examples are given to illustrate the methods and their convergence.

MSC

G.1.7

Keywords

Finite difference method
Nonlinear biharmonic equation
Polar coordinate
NBSOR method
Maximum absolute error
Root-mean-square error

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