Solution of Multiple Point, Nonlinear Boundary Value Problems by Method of Weighted Residuals

A new numerical method for solving non-linear boundary value problems with the boundary values specified at multiple points is presented. The present paper is an extension of an earlier work where boundary conditions were specified only at the end. The method proceeds with first linearizing the problem by an initial guess for the nonlinear terms. Next the method of weighted residuals is applied to compute all boundary quantities for the approximate solution corresponding to the linearized version. This converts the boundary value problem to an initial value problem which is solved by a Runge-Kutta scheme. The resulting solution is used as an improved guess for the next iteration. The process is repeated until convergence to a prescribed tolerance is achieved. Illustrative applications from bending of sandwich beams and outflow of an incompressible fluid from a narrow two dimensional slit are included.