Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines

Aslam Khan; Abdul Ghafoor; Emel Khan; Kamal Shah; Thabet Abdeljawad; Aslam Khan; Abdul Ghafoor; Emel Khan; Kamal Shah; Thabet Abdeljawad

doi:10.3934/nhm.2024028

Networks and Heterogeneous Media

2024, Volume 19, Issue 2: 634-654. doi: 10.3934/nhm.2024028

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Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines

1.
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Mathematics, University of Malakand, Chakdara, Dir (L), KP, Pakistan
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefaco Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 06 February 2024 Revised: 24 May 2024 Accepted: 11 June 2024 Published: 24 June 2024

This paper aims to conduct the numerical solutions of the scalar reaction diffusion model with cubic non-linearity having constant and time-dependent coefficients. The proposed method is hybrid in nature in which Haar wavelets are used to approximate the spatial derivatives and the Runge-Kutta (RK) routines are used to solve the resultant system of ordinary differential equations. We illustrate the applicability and efficiency of the proposed method by computing $ L_2 $, $ L_{\infty} $, and $ L_{rms} $ error estimates for various test models. The numerical accuracy and stability of the Haar wavelet-based method of lines for solving the scaler reaction-diffusion model provides further insight into the use of this scheme for model equations across various disciplines.
- Haar wavelet and its integrals,
- Runge-Kutta method,
- generalized FitzHuge-Naguma reaction diffusion models
Citation: Aslam Khan, Abdul Ghafoor, Emel Khan, Kamal Shah, Thabet Abdeljawad. Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines[J]. Networks and Heterogeneous Media, 2024, 19(2): 634-654. doi: 10.3934/nhm.2024028

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Abstract

This paper aims to conduct the numerical solutions of the scalar reaction diffusion model with cubic non-linearity having constant and time-dependent coefficients. The proposed method is hybrid in nature in which Haar wavelets are used to approximate the spatial derivatives and the Runge-Kutta (RK) routines are used to solve the resultant system of ordinary differential equations. We illustrate the applicability and efficiency of the proposed method by computing $ L_2 $, $ L_{\infty} $, and $ L_{rms} $ error estimates for various test models. The numerical accuracy and stability of the Haar wavelet-based method of lines for solving the scaler reaction-diffusion model provides further insight into the use of this scheme for model equations across various disciplines.

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