The solution of a fourth-order partial differential equation using the method of lines

Abstract

The purpose of this paper is to investigate the solution of the fourth order boundary value problem with boundary conditions where α(s) is square integrable on 0≦s≦a, bounded from above by M and below by M >0 and 0≦z≦∞. This problem arises in certain areas of fluid dynamics. If [0, a] is divided into equal intervals of length h such that and ū is a vector with ith component u(si,z) the asssociated vector differential equation is . The eigenvalues of AB are shown to be nonnegative and the matrix of eigenveciors to have bounded condition number, which assures that the solution remains bounded as h→0, and hence the solution of the boundary value problem exists.