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

Document Type : Research Article

Authors

• A. Chand
• J. Mohapatra

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

10.22067/ijnao.2024.87965.1442

Abstract

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 total
variation 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.

Keywords

• The Kolmogorov–Petrovskii–Piskunov equation
• Kudryashov method
• Local discontinuous Galerkin method
• Total variation diminishing Runge–Kutta method
• Stability analysis

Main Subjects

• Computational Mathematics

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