A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

摘要

The two dimensional diffusion equation of the form ∂2u∂x2+∂2u∂y2=1D∂u∂t is considered in this paper. We try a bi-cubic spline function of the form ∑i,j=0N,NCi,j(t)Bi(x)Bj(y) as its solution. The initial coefficients Ci,j(0) are computed simply by applying a collocation method; Ci,j = f(xi, yj) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients Ci,j(t) are computed by X(t) = etQX(0) where X(t) = (C0,1, C0,1, C0,2, … , C0,N, C1,0, … , CN,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.