j. mohapatra

The main focus of this work is to develop and implement an efficient lo-cal discontinuous Galerkin scheme for acquiring the numerical solution of the (1 + 1)-dimensional nonlinear Kolmogorov–Petrovskii–Piskunov equa-tion. The proposed framework employs a local discontinuous Galerkin discretization technique in the spatial direction and a higher-order totalvariation diminishing Runge–Kutta scheme in the temporal direction. The L2 stability of the local discontinuous Galerkin method, which is ensured by carefully selecting the interface numerical fluxes, is discussed in detail. The Kudryashov technique is also employed in this work to acquire the an-alytical traveling wave solution of the governing Kolmogorov–Petrovskii–Piskunov equation. Furthermore, the comparison between the obtained analytical and numerical solutions is demonstrated by computing the L2 and L∞ error norms. The accuracy and efficacy of the numerical local discontinuous Galerkin method solutions are validated by comparing them with analytical Kudryashov method solutions. For a more comprehensive understanding of the obtained analytical solutions, various graphical il-lustrations are presented in both two-dimensional and three-dimensional representations.

This paper presents a numerical solution for a time delay parabolic problem (reaction-diffusion) containing a small parameter. The numerical method combines the implicit Crank–Nicolson scheme for the time  derivative on the uniform mesh and the central difference scheme for the spatial derivative on the Shishkin type meshes. It is shown to be second-order uniformly convergent in time and space. Then Richardson extrapolation technique is applied to enhance the accuracy from second-order to fourth-order. The error analysis is carried out, and the method is proved to be uniformly convergent. These two methods are applied to two test examples in support of the theoretical results.

This article deals with an effcient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method.