Solving Parabolic Singularly Perturbed Problems by Collocation Using Tension Splines

Tension spline is a function that, for given partition x0 < x1 < . . . < xn, on each interval [xi, xi+1] satisfies differential equation (D − ρiD)u = 0, where ρi’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 form ε ∂u ∂x2 − c(x, t)u− p(x, t) ∂t = f(x, t) with Dirichlet boundary conditions. We tested our method on the time-uniform mesh with Nx × Nt elements. Numerical results show ε-uniformly convergence of the method.