Application of the Method of Fractional Steps to Boundary Value Problems for Laplace’s and Poisson’s Equations

Abstract

Consider the Dirichlet problem in the rectangular region G

$$\frac{{{\partial ^2}u}}{{\partial x_1^2}} + \frac{{{\partial ^2}u}}{{\partial x_2^2}} = 0$$

(4.1.1)

$$u\left( {{x_1},{x_2}} \right) = f\left( {{x_1},{x_2}} \right),\,\left( {{x_1},{x_2}} \right) \in \gamma ,$$

(4.1.2)

where γ is the boundary G, G = {0 < x _i < π, i = 1, 2}. Along with Eq. (4.1.1) consider the unsteady problem

$$\frac{{\partial u}}{{\partial t}} = {a^2}\left( {\frac{{{\partial ^2}u}}{{\partial x_1^2}} = \frac{{{\partial ^2}u}}{{\partial x_2^2}}} \right),$$

(4.1.3)

$$u\left( {{x_1},{x_2},0} \right) = {u_0}\left( {{x_1},{x_2}} \right)$$

(4.1.4)

with the same steady boundary conditions (4.1.2).