6.4.1: Fourier's Method

Separation of variables ansatz $c(x,t)=v(x)w(t)$ leads to the eigenvalue problem, see the arguments of Section 4.5,

$$\begin{eqnarray} \label{ewpar1}\tag{6.4.1.1} -\triangle v&=&\lambda v\ \ \mbox{in}\ \Omega\\ \label{ewpar2} \tag{6.4.1.2} \frac{\partial v}{\partial n}&=&0\ \ \mbox{on}\ \partial\Omega, \end{eqnarray}$$

and to the ordinary differential equation

$$\begin{equation} \tag{6.4.1.3} \label{ewpar3} w'(t)+\lambda Dw(t)=0. \end{equation}$$

Assume $\Omega$ is bounded and $\partial\Omega$ sufficiently regular, then the eigenvalues of ($\ref{ewpar1}$), ($\ref{ewpar2}$) are countable and

$$0=\lambda_0<\lambda_1\le\lambda_2\le\ldots\to\infty.\]

Let $v_j(x)$ be a complete system of orthonormal (in $L^2(\Omega)$) eigenfunctions.
Solutions of ($\ref{ewpar3}$) are

$$w_j(t)=C_je^{-D\lambda_jt},\]

where $C_j$ are arbitrary constants.

According to the superposition principle,

$$c_N(x,t):=\sum_{j=0}^N C_je^{-D\lambda_jt}v_j(x)\]

is a solution of the differential equation ($\ref{ewpar1}$) and

$$c(x,t):=\sum_{j=0}^\infty C_je^{-D\lambda_jt}v_j(x),\]

with

$$C_j=\int_\Omega\ c_0(x)v_j(x)\ dx,\]

is a formal solution of the initial-boundary value problem (6.4.1)-(6.4.3).

Diffusion in a tube

Consider a solution in a tube, see Figure 6.4.1.1.

Figure 6.4.1.1: Diffusion in a tube

Assume the initial concentration $c_0(x_1,x_2,x_3)$ of the substrate in a solution is constant if $x_3=const.$ It follows from a uniqueness result below that the solution of the initial-boundary value problem $c(x_1,x_2,x_3,t)$ is independent of $x_1$ and $x_2$.

Set $z=x_3$, then the above initial-boundary value problem reduces to

$$\begin{eqnarray*} c_t&=&Dc_{zz}\\ c(z,0)&=&c_0(z)\\ c_z&=&0,\ \ z=0,\ z=l. \end{eqnarray*}$$

The (formal) solution is

$$c(z,t)=\sum_{n=0}^\infty C_ne^{-D\left(\frac{\pi}{l}n\right)^2 t}\cos\left(\frac{\pi}{l}nz\right),\]

where

$$\begin{eqnarray*} C_0&=&\frac{1}{l}\int_0^l\ c_0(z)\ dz\\ C_n&=&\frac{2}{l}\int_0^l\ c_0(z)\cos\left(\frac{\pi}{l}nz\right)\ dz,\ \ n\ge1. \end{eqnarray*}$$

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.