8.6: Apéndice. Integración de las Ecuaciones
- Page ID
- 131892
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
La integración numérica de las ecuaciones 8.5.22-24 es directa (por regla de Simpson, por ejemplo) excepto cerca de perineme (\(x = 1\)) y aponeme (\(x = x_2\)), donde los integrandos se vuelven infinitos. Cerca de perineme, sin embargo, podemos sustituir\(x = 1 + \xi\) y cerca de aponeme podemos sustituir\(x = x_2(1 − \xi)\), y podemos expandir los integrands como series de potencia en\(\xi\) e integrar término por término. Aquí recojo los siguientes resultados para los intervalos\(x = 1\) hacia\(x = 1 + \epsilon \) y\(x = x_2 − \epsilon\) hacia\(x = x_2\), donde\(\epsilon\) deben elegirse para que sean suficientemente pequeños que\(\epsilon ^4\) sea menor que la precisión requerida.
\[I_1=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2}dx=M(1+\frac{1}{3}A_1\epsilon + \frac{1}{5}B_1\epsilon^2+\frac{1}{7}C_1\epsilon^3+...)\tag{8A.1}\]
\[I_2=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} \ln x \ dx=M(\frac{1}{3}\epsilon+\frac{1}{5}D_1\epsilon^2+\frac{1}{7}E_1\epsilon^3+...)\tag{8A.2}\]
\[I_3=\int^{1+\epsilon}_1[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} x^{-2} dx=M(1+\frac{1}{3}F_1\epsilon+\frac{1}{5}G_1\epsilon^2+\frac{1}{7}H_1\epsilon^3+...)\tag{8A.3}\]
\[I_4=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} dx=N[1+\frac{1}{3}A_2\epsilon/x_2+\frac{1}{5}B_2(\epsilon/x_2)^2+\frac{1}{7}C_2(\epsilon/x_2)^3+...)\tag{8A.4}\]
\[I_5=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} \ln x \ dx= I_4 \ln x_2 - N[\frac{1}{3}\epsilon/x_2 +\frac{1}{5}D_2(\epsilon/x_2)^2+\frac{1}{7}E_2(\epsilon/x_2)^3+...]\tag{8A.5}\]
\[I_6=\int^{x_2}_{x_2-\epsilon}[v_0^2(1-1/x^2)+2w_0 \ln x - (\ln x)^2]^{-1/2} x^{-2} dx= N[1+\frac{1}{3}F_2\epsilon/x_2+\frac{1}{5}G_2(\epsilon/x_2)^2+\frac{1}{7}H_2(\epsilon/x_2)^3+...]/x_2^2\tag{8A.6}\]
Las constantes se definen de la siguiente manera.
\[M = \left( \frac{2\epsilon}{v_0^2+w_0}\right)^{1/2}\tag{8A.7}\]
\[N= \left( \frac{2\epsilon x_2}{\ln x_2 - (v_0/x_2)^2-w_0} \right) ^{1/2}\tag{8A.8}\]
\[a_1=- \frac{3v_0^2+w_0+1}{2(v_0^2+w_0)}\tag{8A.9}\]
\[b_1= \frac{4v_0^2+\frac{2}{3}w_0+1}{2(v_0^2+w_0)}\tag{8A.10}\]
\[c_1= - \frac{5v_0^2+\frac{1}{2}w_0+\frac{11}{12}}{2(v_0^2+w_0)}\tag{8A.11}\]
\[a_2 = \frac{3(v_0/x_2)^2+w_0- \ln x_2 + 1}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.12}\]
\[b_2 = \frac{4(v_0/x_2)^2+\frac{2}{3}w_0-\ln x_2 +1}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.13}\]
\[c_2 = \frac{5(v_0/x_2)^2+\frac{1}{2}w_0- \frac{1}{2}\ln x_2 + \frac{11}{12}}{2 \left( (v_0/x_2)^2+w_0 - \ln x_2 \right)}\tag{8A.14}\]
\[A_n = -\frac{1}{2} a_n\tag{8A.15}\]
\[B_n = - \frac{1}{2}b_n + \frac{3}{8} a_n^2\tag{8A.16}\]
\[C_n = - \frac{1}{2}c_n + \frac{3}{4} a_n b_n - \frac{5}{16} a_n^3\tag{8A.17}\]
\[D_n = A_n +\frac{1}{2} (-1)^n\tag{8A.18}\]
\[E_n = B_n + \frac{1}{2}(-1)^n A_n + \frac{1}{3}\tag{8A.19}\]
\[F_n = A_n +2(-1)^n\tag{8A.20}\]
\[G_n = B_n + 2(-1)^n A_n + 3\tag{8A.21}\]
\[H_n = C_n + 2(-1)^n B_n + 3A_n + 4(-1)^n\tag{8A.22}\]
\[\nonumber n= 1,2\]