In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at left (or right) end of the domain. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of the algorithm is investigated. It is shown that proposed technique provides first order accuracy independent of the perturbation parameter. Several linear and nonlinear problems are solved by the proposed method and numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

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PurposeThe purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.Design/methodology/approachThe small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.FindingsThe proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.Research limitations/implicationsThe use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.Originality/valueTo validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.

This work focuses on the semi-analytical methods for obtaining the solutions of time fractional partial integro-differential equations. Adomian decomposition method (ADM) and homotopy perturbation method (HPM) are successfully applied. Further, the modified version of homotopy perturbation method is applied which is comparatively more accurate than the other two methods. These methods are shown to be efficient and converge rapidly to the exact solution.Graphs are plotted and tabular data are recorded which represents the accuracy of the proposed techniques.

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