Introduction to the Heat Equation

An Introduction to the Heat Equation

Introduction

The heat equation is a cornerstone partial differential equation (PDE) in physics and mathematics. It elegantly describes how heat distributes or diffuses through a given region over time. Its principles stem from fundamental physical laws like energy conservation and Fourier's law of heat conduction. While its name suggests heat, the same equation models various diffusion phenomena, including particle movement and even aspects of financial markets.

Definition of the Heat Equation

Let $u$ represent the temperature at a spatial point $\mathbf{x}$ and time $t$. The heat equation is given by:

\frac{\partial u}{\partial t} = \alpha \nabla^2 u

Where:

$u(\mathbf{x}, t)$: Temperature (a function of space $\mathbf{x}$ and time $t$).
$\frac{\partial u}{\partial t}$: The rate of change of temperature at a fixed point.
$\alpha$: The thermal diffusivity, a positive material constant ($\alpha = k / (\rho c_p)$, with $k$=thermal conductivity, $\rho$=density, $c_p$=specific heat capacity). It indicates how quickly heat spreads.
$\nabla^2$: The Laplacian operator, representing the sum of second partial derivatives with respect to spatial coordinates.
- 1D: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2}$
- 2D: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$
- 3D: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}$

The equation essentially states that temperature change over time is proportional to the spatial curvature of the temperature field.

Boundary Conditions (BCs)

PDEs usually require additional conditions to pinpoint a unique, physically meaningful solution. For the heat equation, these often take the form of boundary conditions, specifying the thermal state on the edges ($\partial \Omega$) of the spatial domain ($\Omega$).

1. Dirichlet Boundary Condition (Type I): Prescribed Temperature

Concept: The temperature value is fixed on the boundary.

Form: $u(\mathbf{0}, t) = g$ and $u(\mathbf{L}, t) = g$ for $t > 0$.

Example: Imagine holding the ends of a rod at a constant $0^\circ C$. Here $g$ would be the constant function $0$ at the endpoints.

2. Neumann Boundary Condition (Type II): Prescribed Heat Flux

Concept: The rate of heat flow across the boundary is specified. This relates to the normal derivative of temperature via Fourier's Law.

Form: $\frac{\partial u}{\partial n}(\mathbf{x}, t) = \nabla u(\mathbf{x}, t) \cdot \mathbf{n} = h(\mathbf{x}, t)$ for $\mathbf{x} \in \partial \Omega$, $t > 0$, where $\mathbf{n}$ is the outward normal vector.

Example: An insulated boundary means no heat flows across, so $\frac{\partial u}{\partial n} = 0$.

3. Robin Boundary Condition (Type III): Mixed Condition

Concept: A linear combination of the temperature and its normal derivative is specified.

Form: $a(\mathbf{x}, t) u(\mathbf{x}, t) + b(\mathbf{x}, t) \frac{\partial u}{\partial n}(\mathbf{x}, t) = q(\mathbf{x}, t)$ for $\mathbf{x} \in \partial \Omega$, $t > 0$.

Example: Often models convective cooling/heating at the boundary, where heat transfer depends on the temperature difference with the surroundings (Newton's Law of Cooling).

Initial Condition (IC)

Because the heat equation describes how temperature *evolves* in time, we must also specify the starting temperature distribution throughout the domain.

Concept: The temperature profile at time $t=0$.

Form: $u(\mathbf{x}, 0) = f(\mathbf{x})$ for $\mathbf{x} \in \Omega$.

Explanation: $f(\mathbf{x})$ is a known function representing the initial temperature everywhere inside the domain.

A complete problem requires the PDE, boundary conditions on $\partial \Omega$, and an initial condition in $\Omega$.

Solution Terminology using Separation of Variables

Separation of variables is a powerful technique for solving linear, homogeneous PDEs with homogeneous boundary conditions on regular domains (like intervals, rectangles, circles). Let's look at the terminology using a 1D example:

PDE: $ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, \quad t > 0 $ BCs: $ u(0, t) = 0, \quad u(L, t) = 0, \quad t > 0 $ IC: $ u(x, 0) = f(x), \quad 0 \le x \le L $

The Process & Key Terminology:

Assume Product Solution: Postulate $u(x, t) = X(x) T(t)$.
Substitute & Separate: Plug into the PDE and rearrange to get functions of $x$ on one side and functions of $t$ on the other: $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda \quad (\text{Separation Constant})$ This introduces the Separation Constant, denoted here by $-\lambda$.
Form ODEs: This separation yields two ordinary differential equations (ODEs): $T'(t) + \alpha \lambda T(t) = 0 \quad \text{(Temporal ODE)}$ $X''(x) + \lambda X(x) = 0 \quad \text{(Spatial ODE)}$
Solve Spatial Problem (Sturm-Liouville): Apply the boundary conditions ($X(0)=0, X(L)=0$) to the spatial ODE. This forms an Eigenvalue Problem (specifically, a Sturm-Liouville problem).
- Eigenvalues ($\lambda_n$): Only specific values of $\lambda$ allow non-trivial solutions $X(x)$. For this setup, $\lambda_n = (\frac{n\pi}{L})^2$ for $n=1, 2, 3, \dots$.
- Eigenfunctions ($X_n(x)$): The corresponding solutions are the eigenfunctions. Here, $X_n(x) = \sin(\frac{n\pi x}{L})$.
Solve Temporal Problem: For each eigenvalue $\lambda_n$, solve the temporal ODE: $T_n(t) = C_n e^{-\alpha \lambda_n t} = C_n e^{-\alpha (n\pi/L)^2 t}$.
Form Product Solutions: Combine $X_n$ and $T_n$: $u_n(x, t) = C_n \sin(\frac{n\pi x}{L}) e^{-\alpha (n\pi/L)^2 t}$.
Apply Superposition Principle: Since the PDE/BCs are linear and homogeneous, sum the product solutions to form the general solution: $u(x, t) = \sum_{n=1}^{\infty} u_n(x, t) = \sum_{n=1}^{\infty} C_n \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha (n\pi/L)^2 t}$
Apply Initial Condition (Fourier Series): Use $u(x, 0) = f(x)$ to find the constants $C_n$. $u(x, 0) = \sum_{n=1}^{\infty} C_n \sin\left(\frac{n\pi x}{L}\right) = f(x)$ This requires expressing $f(x)$ as a Fourier Series (specifically, a Fourier Sine Series in this case). The Fourier Coefficients $C_n$ are found using orthogonality: $C_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx$

This process beautifully illustrates how concepts like eigenvalues, eigenfunctions, and Fourier series naturally arise when solving fundamental PDEs like the heat equation.