An Alternative Explicit Expression of the Kernel of the One Dimensional Heat Equation with Dirichlet Conditions

TL;DR

This paper presents a novel explicit integral expression for the solution of the one-dimensional heat equation with Dirichlet conditions, avoiding infinite series by using Laplace transforms and Green's functions.

Contribution

It introduces a new finite integral kernel representation of the solution, differing from classical spectral series methods, using Laplace transform and Green's function construction.

Findings

Solution expressed as a sum of two finite integrals

Avoids infinite series in the solution representation

Provides a direct integral expression for the heat equation solution

Abstract

This paper is devoted to the study of the one dimensional non homogeneous heat equation coupled to Dirichlet Boundary Conditions. We obtain the explicit expression of the solution of the linear equation by means of a direct integral in an unbounded domain. The main novelty of this expression relies in the fact that the solution is not given as a series of infinity terms. On our expression the solution is given as a sum of two integrals with a finite number of terms on the kernel. The main novelty is that, on the contrary to the classical method, where the solutions are derived by a direct application of the separation of variables method, on the basis of the spectral theory and the Fourier Series expansion, the solution is obtained by means of the application of the Laplace Transform with respect to the time variable. As a consequence, for any $t \ge 0$ fixed, we must solve an Ordinary Differential Equation on the spatial variable, coupled to Dirichlet Boundary conditions. The solution of such a problem is given by the construction of the related Green's function.