Computational Physics Lectures: Partial differential equations

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Diffusion equation, dimensionless form

We note that the dimension of

$D$ is time/length$^2$. Introducing the dimensionless variables

$\alpha\hat{x}=x$ we get

$\begin{equation*} \frac{\partial^2 T(x,t)}{\alpha^2\partial \hat{x}^2}= D\frac{\partial T(x,t)}{\partial t}, \end{equation*}$ and since

$\alpha$ is just a constant we could define

$\alpha^2D= 1$ or use the last expression to define a dimensionless time-variable

$\hat{t}$ . This yields a simplified diffusion equation

$\begin{equation*} \frac{\partial^2 T(\hat{x},\hat{t})}{\partial \hat{x}^2}= \frac{\partial T(\hat{x},\hat{t})}{\partial \hat{t}}. \end{equation*}$ It is now a partial differential equation in terms of dimensionless variables. In the discussion below, we will however, for the sake of notational simplicity replace

$\hat{x}\rightarrow x$ and

$\hat{t}\rightarrow t$ . Moreover, the solution to the

$1+1$ -dimensional partial differential equation is replaced by

$T(\hat{x},\hat{t})\rightarrow u(x,t)$ .