J. Math. Computer Sci., 41 (2026), 132–149 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs Novel analytical solutions to the (3 + 1)-dimensional heat model using Lie symmetry method M. Usman a,b , Akhtar Hussain a,∗ , Ahmed M. Zidan c , Jorge Herrera d a Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan. b College of Electrical and Mechanical Engineering (CEME), National University of Sciences and Technology (NUST), H-12 Islamabad 44000, Pakistan. c Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia. d 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. Keywords: Heat-type equation, Lie symmetry method, invariant solutions, geometric vector fields, Lie algebra, conservation laws. 2020 MSC: 34C14, 34A26. ©2026 All rights reserved. 1. Introduction In recent decades, numerous robust nonlinear models have been employed to represent various real- world phenomena across diverse fields, including plasma physics [6, 24, 38, 41], optical fiber technology [21, 25, 43, 44], chemical physics, fluid dynamics, solid-state physics, and acoustics. Given their signif- icance, finding the exact solutions to these equations is of great importance. However, obtaining such solutions is often a challenging task because they are typically achievable only in specific cases. In re- cent years, considerable progress has been made in deriving exact explicit solutions for nonlinear partial ∗ Corresponding author Email addresses: musman.awan112@gmail.com (M. Usman), akhtarhussain21@sms.edu.pk (Akhtar Hussain), ahmoahmed@kku.edu.sa (Ahmed M. Zidan), jorgea.herrerac@utadeo.edu.co (Jorge Herrera) doi: 10.22436/jmcs.041.02.01 Received: 2025-03-04 Revised: 2025-03-24 Accepted: 2025-06-30