An analytical and numerical approach for the $(1+1)$-dimensional nonlinear Kolmogorov-Petrovskii-Piskunov equation

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 08 مهر 1403

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.87965.1442

نویسندگان

Abhilash Chand؛ Jugal Mohapatra^*

Department of Mathematics, National Institute of Technology Rourkela, India.

چکیده

The main focus of this work is to develop and implement an efficient local discontinuous Galerkin scheme for acquiring the numerical solution of the $(1+1)$-dimensional nonlinear Kolmogorov-Petrovskii-Piskunov equation. The proposed framework employs a local discontinuous Galerkin discretization technique in the spatial direction and a higher-order total variation diminishing Runge-Kutta scheme in the temporal direction. The $ L_2 $ 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 analytical 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 $ L_2 $ and $ L_\infty $ 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 illustrations are presented in both two-dimensional and three-dimensional representations.

کلیدواژه‌ها

Kolmogorov-Petrovskii-Piskunov equation؛ Kudryashov method؛ Local discontinuous Galerkin method؛ Total variation diminishing Runge-Kutta method؛ Stability analysis