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

Abstract The two dimensional diffusion equation of the form ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 1 D ∂ u ∂ t is considered in this paper. We try a bi-cubic spline function of the form ∑ i , j = 0 N , N C i , j ( t ) B i ( x ) B j ( y ) as its solution. The initial coefficients C i , j (0) are computed simply by applying a collocation method; C i , j  =  f ( x i ,  y j ) where f ( x ,  y ) =  u ( x ,  y , 0) is the given initial condition. Then the coefficients C i , j ( t ) are computed by X ( t ) =  e tQ X (0) where X ( t ) = ( C 0,1 ,  C 0,1 ,  C 0,2 , … ,  C 0, N ,  C 1,0 , … ,  C N , 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.