Novel analytical solutions to the \((3+1)\)-dimensional heat model using Lie symmetry method
Authors
U. Usman
- Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan.
- College of Electrical and Mechanical Engineering (CEME), National University of Sciences and Technology (NUST), H-12 Islamabad 44000, Pakistan.
A. Hussain
- Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan.
A. M. Zidan
- Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia.
J. Herrera
- Facultad de Ciencias Naturales e Ingenieria, Universidad de Bogota Jorge Tadeo Lozano, Bogota 110311, Colombia.
Abstract
This study focuses on applying the Lie symmetry method, to obtain exact solutions in multiple forms for the \((3+1)\)-dimensional (3D) heat model This equation is a well-known model frequently used to describe numerous complex physical phenomena. Initially, the geometric vector fields for the 3D heat-type equation are determined. Using Lie symmetry reduction, we report a wide array of exact analytical solutions that encompass trigonometric and hyperbolic solitons, Lambert functions, polynomials, exponential and inverse functions, hypergeometric forms, Bessel functions, logarithmic forms, rational forms, and solitary wave solutions. These solutions include many rational forms that uncover intricate physical structures that have not been previously reported. The solutions presented in this study are original and significantly distinct from previous findings. They have significant potential for application in diverse fields, including fiber optics, plasma physics, soliton dynamics, fluid dynamics, mathematical physics, and other applied sciences. The findings demonstrated that these mathematical techniques are efficient, straightforward, and robust, making them suitable for solving other types of nonlinear equations.
Share and Cite
ISRP Style
U. Usman, A. Hussain, A. M. Zidan, J. Herrera, Novel analytical solutions to the \((3+1)\)-dimensional heat model using Lie symmetry method, Journal of Mathematics and Computer Science, 41 (2026), no. 2, 132--149
AMA Style
Usman U., Hussain A., Zidan A. M., Herrera J., Novel analytical solutions to the \((3+1)\)-dimensional heat model using Lie symmetry method. J Math Comput SCI-JM. (2026); 41(2):132--149
Chicago/Turabian Style
Usman, U., Hussain, A., Zidan, A. M., Herrera, J.. "Novel analytical solutions to the \((3+1)\)-dimensional heat model using Lie symmetry method." Journal of Mathematics and Computer Science, 41, no. 2 (2026): 132--149
Keywords
- Heat-type equation
- Lie symmetry method
- invariant solutions
- geometric vector fields
- Lie algebra
- conservation laws
MSC
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