9.1 Homogeneous Equation: Fourier–Poisson Formula

Consider the initial value problem (IVP) for the heat equation in space dimension n ⩾ 1:

The solution of the IVP (9.1.1) is given by the Fourier–Poisson formula:

for x ∈ ℝn and t > 0. The Fourier–Poisson formula (9.1.2) can be written as a convolution. For this purpose, we define the heat kernel or fundamental solution of the heat equation by

We can write the Fourier–Poisson integral (9.1.2) as the convolution of K and g as. Furthermore, the heat kernel K enjoys the following properties, which are frequently used:

9.1.1 Inhomogeneous Equation: Duhamel's Principle

We now consider the inhomogeneous heat equation:

Assume that the function f and its partial derivatives fxj are continuous in x ∈ ℝn, t > 0 and the function g is continuous and bounded. Owing to the linearity in the problem, the required solution can be written as some of two functions, namely the solution of the homogeneous equation with initial condition g and the solution of the inhomogeneous equation with zero initial condition. Thus, it suffices to consider equation (9.1.4) with g ≡ 0 therein. A solution of this problem can be obtained via the Duhamel's principle: Fix s ⩾ 0, and consider the IVP for the heat equation:

Denoting the solution by v(x, t; s), we get

using equation (9.1.2) and the change of variable t ↦ t − s. We can now write down the expression for a solution to equation (9.1.4) with g ≡ 0, as:

9.1.2 Heat Equation in a Finite Interval: Fourier Method

In this subsection, we consider initial boundary value problem for the one-dimensional heat equation and see how the Fourier method can be used to obtain the solution. Consider IVP for the heat equation in a finite interval [0, L] on the real line

The initial condition u(x, 0) = g(x) in equation (9.1.8) represents the initial temperature distribution at all points of the rod at the initial instant of time t = 0. At the end points, x = 0 and x = L, different boundary conditions may be given.