Numerical solution of the first biharmonic problem by linear programming

Abstract

A novel method for solving linear boundary-value problems is outlined through the use of the linear-programming computational algorithm. The method is equivalent to obtaining an approximate solution by a Chebyshev fit wherein the maximum error is minimized. In this work the method is used to solve the first biharmonic problem ▽ 4 u = c in a rectangular two-dimensional region subject to u = 0 and δu/ δn = 0 on the boundary of the region. The advantages of the present method over the collocation method (wherein a set of linear equations are solved uniquely for a set of parameters) seem to be the following: (1) an estimate of the error in the approximate solution is given by the objective function of the linear program: (2) for a fixed number of parameters in the approximate solution, the present method gives the best value, in some sense, of these parameters.