Solving Parabolic Singularly Perturbed Problems By Collocation Using Tension Splines

Tension spline is a function that, for given partition x(0) < x(1) < ... < x(n), on each interval [x(i), Xi+1] satisfies differential equation (D-4 - p(i)(2)D(2))u = 0, where pi's are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions. In this paper, we adapt collocation method to solve a time dependent reaction-diffusion problem of the formepsilon(2) partial derivative(2)u/partial derivative x(2) - c(x,t)u - P(x,t) partial derivative u/partial derivative t = f(x,t)with Dirichlet boundary conditions. We tested our method on the time-uniform mesh with N-x x N-t elements. Numerical results show epsilon-uniformly convergence of the method.