The Cauchy Problem for the Heat Equation

This study investigates analytical and numerical approaches to linear and nonlinear heat conduction equations based on Cauchy problem formulations. Modern approaches that extend the applicability of the classical Fourier heat conduction model, including phonon and ballistic transport models, thermal lag theories, thermodynamic formulations, and fractional-order differential equations, are briefly reviewed to provide background on advanced heat transfer models. These approaches are discussed only in the literature review and are not investigated numerically in the present study. The main objective of this study is to derive and analyze the analytical solution of the classical heat equation with an instantaneous point heat source and to investigate numerical solutions of linear and nonlinear heat conduction models. The exact solution of the one-dimensional problem is expressed in terms of the Green's function, and its key properties, including the Gaussian temperature distribution, conservation of thermal energy, and temporal evolution, are examined. Numerical simulations are performed using the method of lines implemented in Wolfram Mathematica for spatial grids with different resolutions. The considered models include the classical heat equation, equations with linear and nonlinear heat source terms, and a model with nonlinear thermal conductivity. The results demonstrate that for the linear problem, the numerical solutions obtained with and are in close agreement at . For the nonlinear-source model, the coarse-grid solution ( ) exhibits non-physical negative temperature values, which disappear upon grid refinement. The comparison between analytical and numerical results confirms the reliability of the computational approach and illustrates the influence of nonlinear terms in heat conduction equations and grid resolution on temperature evolution. The obtained analytical solution provides a benchmark for validating numerical methods, while the analysis of nonlinear heat conduction models and grid resolution effects offers practical recommendations for computational heat transfer simulations and further studies of advanced heat transfer problems in mathematical physics and engineering applications.