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

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.