Name
Title | ||
|---|---|---|
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 ∂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. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.1016/j.amc.2010.05.018 | Applied Mathematics and Computation |
Keywords | Field | DocType |
Diffusion equation,Galerkin’s method,Bi-cubic splines,Collocation method,Non-separable solution | Spline (mathematics),Mathematical optimization,Mathematical analysis,Galerkin method,Separable space,Square matrix,Initial value problem,Collocation method,Diffusion equation,Mathematics | Journal |
Volume | Issue | ISSN |
217 | 5 | 0096-3003 |
Citations | PageRank | References |
0 | 0.34 | 5 |
Authors | ||
7 | ||