Numerical Methods and the Solution of Boundary Value Problems.

A study of several numerical methods for the solution of boundary value problems, in both one and two-dimensions, was conducted using the CDC 6600 computer. The method of finite differences was employed for solution of the equations in differential form. These numerical solutions were compared to those obtained by transforming the original differential equation into integral form and approximating their solution using numerical integration via the trapezoid rule. All numerical experiments were conducted using Dirichlet boundary conditions. In the one-dimensional cases studied it was found that both methods are equivalent, i.e., yield identical solutions when the integral representation had a linear weighted Greens function kernel. For the two-dimensional investigation the steady-state heat conduction equation was analyzed. Again, the method of finite differences in two-dimensions was compared to the integral approach, using cubic splines. The method of finite differences was found to be superior in calculating the internal temperature, at all nodal points, as compared to the integral-spline solution.