Solução da equação de Laplace pelo método das linhas

Abstract

An introductory material on semi-analytical methods is presented, in which the Laplace equation is solved by the method of lines. It is an easy-to-read text, which is aimed at undergraduate physics students, presenting the derivations of the equations in a clear and detailed manner. An electrostatic problem is solved using three different methods. The problem consists of an infinitely long conductor with null potential in the boundaries, except in the top boundary, in which the potential is 1 volt. The rectangular section has a width of L = 3 and M = 2 high. The equation to be solved is the Laplace equation. First, the problem was solved by an analytical method, the separation of variables method. Then we used a semi-analytical method, the method of lines, which was implemented in two different ways. In the first approach of the method, the Laplace equation is transformed into a system of ordinary differential equations of the first order and in the second approach the Laplace equation is solved as a partial differential equation of second order.