6.4.1: Método de Fourier
- Page ID
- 118024
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La separación de variables ansatz\(c(x,t)=v(x)w(t)\) lleva al problema de los valores propios, véanse los argumentos de la Sección 4.5"
\ begin {eqnarray}
\ label {ewpar1}\ tag {6.4.1.1}
-\ triángulo v&=&\ lambda v\\\ mbox {in}\\ Omega\
\ etiqueta {ewpar2}\ tag {6.4.1.2}
\ frac {\ parcial v} {\ parcial n} &=&0\\\ mbox {on}\\ parcial\ Omega,
\ end {eqnqnarray
y a la ecuación diferencial ordinaria
\ begin {ecuación}\ tag {6.4.1.3}
\ label {ewpar3}
w' (t) +\ lambda Dw (t) =0.
\ end {ecuación}
Supongamos que\(\Omega\) es acotado y\(\partial\Omega\) suficientemente regular, entonces los valores propios de (\ ref {ewpar1}), (\ ref {ewpar2}) son contables y
$$0=\ lambda_0<\ lambda_1\ le\ lambda_2\ le\ ldots\ a\ infty.\]
Dejar\(v_j(x)\) ser un sistema completo de ortonormal (in\(L^2(\Omega)\)) funciones propias.
Las soluciones de (\ ref {ewpar3}) son
$$w_j (t) =C_je^ {-D\ lambda_jt},\]
donde\(C_j\) son constantes arbitrarias.
De acuerdo con el principio de superposición,
$$C_n (x, t) :=\ suma_ {j=0} ^N C_je^ {-D\ lambda_jt} v_j (x)\]
es una solución de la ecuación diferencial (\ ref {ewpar1}) y
$$c (x, t) :=\ suma_ {j=0} ^\ infty c_je^ {-D\ lambda_jt} v_j (x),\]
con
$$c_j=\ int_\ Omega\ c_0 (x) v_j (x)\ dx,\]
es una solución formal del problema del valor de límite inicial (6.4.1) - (6.4.3).
Difusión en un tubo
Considera una solución en un tubo, ver Figura 6.4.1.1.
Figura 6.4.1.1: Difusión en un tubo
Supongamos que la concentración inicial\(c_0(x_1,x_2,x_3)\) del sustrato en una solución es constante si se\(x_3=const.\) deduce de un resultado de unicidad por debajo de que la solución del problema del valor de límite inicial\(c(x_1,x_2,x_3,t)\) es independiente de\(x_1\) y\(x_2\).
Establezca\(z=x_3\), luego el problema del valor de límite inicial anterior se reduce a
\ begin {eqnarray*}
c_t&=&dc_ {zz}\\
c (z,0) &=&c_0 (z)\\
c_z&=&0,\\ z=0,\ z=l.
\ end {eqnarray*}
La solución (formal) es
$$c (z, t) =\ suma_ {n=0} ^\ infty C_ne^ {-D\ izquierda (\ frac {\ pi} {l} n\ derecha) ^2 t}\ cos\ izquierda (\ frac {\ pi} {l} nz\ derecha),\]
donde
\ 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\ izquierda (\ frac {\ pi} {l} nz\ derecha)\ dz,\\ n\ ge1.
\ end {eqnarray*}