A robust technique of cubic Hermite splines to study the non-linear reaction-diffusion equation with variable coefficients

Abdul-Majeed Ayebire; Inderpreet Kaur; Dereje Alemu Alemar; Mukhdeep Singh Manshahia; Shelly Arora; Abdul-Majeed Ayebire; Inderpreet Kaur; Dereje Alemu Alemar; Mukhdeep Singh Manshahia; Shelly Arora

doi:10.3934/math.2024398

AIMS Mathematics

2024, Volume 9, Issue 4: 8192-8213. doi: 10.3934/math.2024398

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Research article

A robust technique of cubic Hermite splines to study the non-linear reaction-diffusion equation with variable coefficients

1.
Department of Mathematics, Punjabi University, Patiala, Punjab-147002, India
2.
Department of Statistics, Bolgatanga Technical University, Bolgatanga, Ghana
3.
Department of Applied Sciences, Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India
4.
Department of Mathematics, College of Natural and Computational Science, Jigjiga University, Jigjiga-1020, Ethiopia

Received: 06 December 2023 Revised: 07 February 2024 Accepted: 21 February 2024 Published: 27 February 2024
MSC : 35K51, 35K55, 65M06, 65M12

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $ h^4 $, and the CN scheme is convergent of order $ \Delta t^2 $. The technique is found to be convergent of order $ O(h^{2}\big(\gamma_2 \varepsilon_j\Delta t + \gamma_0(1+\bar{\alpha})h^2\big)+\Delta t^2) $. The step size in the space direction is taken to be $ h $, and the step size in the time direction is $ \Delta t $. Stability of the proposed scheme was studied using the $ L_2 $ and $ L_{\infty} $ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.
- reaction-diffusion equation,
- orthogonal collocation,
- Hermite splines,
- Crank Nicolson scheme,
- stability analysis
Citation: Abdul-Majeed Ayebire, Inderpreet Kaur, Dereje Alemu Alemar, Mukhdeep Singh Manshahia, Shelly Arora. A robust technique of cubic Hermite splines to study the non-linear reaction-diffusion equation with variable coefficients[J]. AIMS Mathematics, 2024, 9(4): 8192-8213. doi: 10.3934/math.2024398

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Abstract

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $ h^4 $, and the CN scheme is convergent of order $ \Delta t^2 $. The technique is found to be convergent of order $ O(h^{2}\big(\gamma_2 \varepsilon_j\Delta t + \gamma_0(1+\bar{\alpha})h^2\big)+\Delta t^2) $. The step size in the space direction is taken to be $ h $, and the step size in the time direction is $ \Delta t $. Stability of the proposed scheme was studied using the $ L_2 $ and $ L_{\infty} $ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.

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