1 Heat Equation
References: Stanford Notes
The (homogeneous) heat equation describes the diffusion of heat in general \(n\)-dimensional continuous medium.
Denoting the heat of material at location \(\boldsymbol{x}\) at time \(t\) by \(u(t,x)\) the heat equation (diffusion equation) gives the heat diffusion dynamics as \[ u_{t}-\Delta u=0, \] where \(\Delta\) denotes the Laplace Operator.
The non-homogeneous heat equation is \[ u_{t}-\Delta u=f, \] subject to appropriate initial boundary conditions. Considering the 1-dimensional case for intuition we have \[ u_{t}-ku_{xx}=0;~~k>0, \] where \(u(x, t)\) represents the temperature at point \(x\) and time \(t\) in a long, thin, perfectly insulated bar of material. Some of the elementary properties of the diffusion equation are:
- Linearity: If \(u_{1}\) and \(u_{2}\) are solutions, then so to is \(c_{1}u_{1}+c_{2}u_{2}\) for any constants \(c_{1}\) and \(c_{2}\);
- Second Order Equation;
- Parabolic Equation: Characteristics are given by \(\tau=\text{constant}\) thus information propagates along these lines in \((x,\tau)\) spaces, and if change is made to \(u\) at some point, its effect is felt instantaneously everywhere else;
- Analytic Function Solutions.
To derive the heat (diffusion) equation consider the diffusion of dye through a liquid, where the dye moves from high concentration to low concentration. Let \(u(x,t)\) be the concentration (mass per unit length) of th dye at position \(x\) in the pipe at time \(t\). The total mass of dye in the pipe from \(x_{0}\) to \(x_{1}\) at time \(t\) is given by \[ M(t)=\int_{x_{0}}^{x_{1}}{u(x,t)}~d{x} \implies \frac{\partial M}{\partial t}=\int_{x_{0}}^{x_{1}}{u_{t}(x,t)}~d{x}. \]