Nonlinear Transient Heat Conduction Problems Research Articles

The Application of the Collocation Method Using Hermite Cubic Splines to Nonlinear Transient One-Dimensional Heat Conduction Problems

A collocation method for the solution of one-dimensional parabolic partial differential equations using Hermite splines as approximating functions and Gaussian quadrature points as collocation points is described. The method consists of expanding dependent variables in terms of piece-wise cubic Hermite splines in the space variable at each time step. The unknown coefficients in the expansion are obtained at every time step by requiring that the resultant differential equation be satisfied at a number of points (in particular at the Gaussian quadrature points) in the field equal to the number of unknown coefficients. This collocation procedure reduces the partial differential equation to a system of ordinary differential equations which is solved as an initial value problem using the steady-state solution as the initial condition. The method thus developed is applied to a two-region nonlinear transient heat conduction problem and compared with a finite-difference method. It is demonstrated that because of high-order accuracy only a small number of equations are needed to produce desirable accuracy. The method has the desirable characteristic of an analytical method in that it produces point values as against nodal values in the finite-difference scheme.