Parameter-uniform cubic spline method for singularly perturbed parabolic differential equation with large negative shift and integral boundary condition

The singularly perturbed parabolic differential equations with integral boundary conditions and a large negative shift in the space variable are studied in this paper. The implicit Euler method for the temporal direction and the cubic spline method for the spatial direction on a piecewise uniform mesh (Shishkin mesh) are applied to formulate a parameter-uniform numerical method. To handle the integral boundary condition, the composite Simpson's rule is used. The proposed scheme has been shown to be uniformly convergent with order of convergence O & UDelta; t + N - 2 ln 2 N . The maximum absolute errors and rate of convergence for various perturbation parameters and mesh size values are tabulated for two model problems, which agrees with the theoretical estimates and the method is more accurate than the results of some methods existing in the literature.