Novel analytical solutions to the ( 3 + 1 ) -dimensional heat model using Lie symmetry method

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.