6.4.1: Fourier's Method
- Page ID
- 2183
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
Integrated by Justin Marshall.