Application of symmetry analysis and conservation laws to a fractional-order nonlinear conduction-diffusion model

A. Tomar; H. Kumar; M. Ali; H. Gandhi; D. Singh; G. Pathak; A. Tomar; H. Kumar; M. Ali; H. Gandhi; D. Singh; G. Pathak

doi:10.3934/math.2024833

AIMS Mathematics

2024, Volume 9, Issue 7: 17154-17170. doi: 10.3934/math.2024833

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Application of symmetry analysis and conservation laws to a fractional-order nonlinear conduction-diffusion model

A. Tomar ^{1
,
,},
H. Kumar ²,
M. Ali ³,
H. Gandhi ⁴,
D. Singh ⁵,
G. Pathak ⁶

1.
School of Computer Science Engineering and Technology, Bennett University, Greater Noida, India
2.
Government College Sector-9, Gurugram, Haryana, India
3.
Department of Basic Sciences, Preparatory Year, King Faisal University, Al Ahsa 31982, Saudi Arabia; mkasim@kfu.edu.sa
4.
State Institute of Advanced Studies in Teacher Education, Jhajjar & Gurugram, India
5.
Amity School of Applied Sciences, Amity University, Haryana, India
6.
GL Bajaj Institute of Technology and Management, Greater Noida, India

Received: 06 March 2024 Revised: 04 May 2024 Accepted: 10 May 2024 Published: 17 May 2024

In this paper, the Lie symmetry analysis was executed for the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order 'β'. In the study of groundwater flow and oil reservoir engineering where fluid flow through porous materials is crucial, BM played an important role. The Lie point infinitesimal generators and Lie algebra were constructed for the equation. The Lie symmetries were acquired for the ordinary fractional-order BM. The power series solution and its convergence were also analyzed with the application of the implicit theorem. Noether's theorem was employed to ensure the consistency of a system by deriving the conservation laws of its physical model.
- convection-diffusion equation,
- Buckmaster model,
- Riemann-Liouville derivatives,
- fractional differential equations,
- Erdelyi-Kober fractional operators
Citation: A. Tomar, H. Kumar, M. Ali, H. Gandhi, D. Singh, G. Pathak. Application of symmetry analysis and conservation laws to a fractional-order nonlinear conduction-diffusion model[J]. AIMS Mathematics, 2024, 9(7): 17154-17170. doi: 10.3934/math.2024833

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Abstract

In this paper, the Lie symmetry analysis was executed for the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order 'β'. In the study of groundwater flow and oil reservoir engineering where fluid flow through porous materials is crucial, BM played an important role. The Lie point infinitesimal generators and Lie algebra were constructed for the equation. The Lie symmetries were acquired for the ordinary fractional-order BM. The power series solution and its convergence were also analyzed with the application of the implicit theorem. Noether's theorem was employed to ensure the consistency of a system by deriving the conservation laws of its physical model.

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