A method is proposed to approximate the solutions of a certain class of differential equations, linear or nonlinear, in two or three dimensions, provided that the boundary conditions are given on a rectangle or a parallelepiped, respectively. For other boundary shapes the co-ordinate system must be transformed to meet that requirement. As in illustration, an example is given for the solution of Poisson's equation in two dimensions with a constant heat source, giving the temperatures on the rectangle together with the heat-flow distribution along its edges. The basis of the method is a Taylor-series development around one point; the result is given in terms of as many partial derivatives in that point as is desired. A similar method has already been described by Small' for the 'heat equation' 38/& = a20/ax2. Compared with finite difference and finite element solutions, these methods have the advantage that the solution is continuous, whereas first and second derivatives such as heat fluxes are available at hardly any effort. The results are compared with those of the exact solution. Even if the size of the determinant is limited to 4 x 4, the accuracy is already better than 98.98 per cent. When more effort is spent to solve a system of 10 equations, the accuracy is better than 99.95 per cent.
METHOD
Consider a function OIx,,,z, which is continuous in all its derivatives. Then Writing for short
If the boundaries coincide with planes for x, y , z is constant respectively, they put relatively simple restrictions on the possible functions { p x , p,, p-, . . .}; the differential equation L(0, x , y , z) = can be worked out to relations between those functions.
As soon as sufficient equations are developed for the desired number of functions, they may-under the usual conditions-be solved. As soon as { p x , p , , p z , B } are known for all p x + p , + p z < r, the value of 8 at any x , y , z can be approximated by a Taylor series.
As a simple example, we consider a rectangle which extends from x = -1 to x = + 1 and