- #1
jadyliber
- 11
- 0
About serval differential equations where A, B, D, g, \chi, c are functions of r
\begin{eqnarray}
&-\frac{{\chi}'}{r}+\frac{c'}{c}\left(\frac{g'}{g} -{\chi}'\right)=\frac{e^{\chi}(q A B)^2}{r^2 g^2 c^2}& \\
&c c''+c c'\left(\frac{g'}{g}+\frac{2}{r} -\frac{{\chi}'}{2} \right)=-\frac{B'^2}{2 r^2}+\frac{e^{\chi}}{2 g^2 r^2}(q A B)^2 & \\
&-g'\left(\frac{1}{r}+\frac{c'}{2c}\right)- g \left(\frac{1}{r^2}+\frac{3c'}{c r}+\frac{c''}{c} \right)+3 = \frac{e^{\chi}}{4}A'^2+\frac{g B'^2}{4 r^2 c^2}+\frac{e^{\chi} (q A B)^2}{4 g r^2 c^2}+\frac{e^{\chi}}{4 }D'^2 & \\
&A'' +\left(\frac{2}{r}+\frac{\chi'}{2}+\frac{c'}{c} \right) A'-\frac{q^2B^2}{r^2 c^2 g } A=0& \\
&B'' +\left(\frac{g'}{g}-\frac{\chi'}{2}-\frac{c'}{c} \right) B'+\frac{e^{\chi}q^2A^2}{g } B=0& \\
&D'' +\left(\frac{2}{r}+\frac{\chi'}{2}+\frac{c'}{c} \right) D'=0&
\end{eqnarray}
The solution near r=0 is given by
\begin{eqnarray}
& A\sim A_0 e^{-\alpha/r}, ~~~B\sim B_0\left(1-\frac{e^{\chi_0}q^2A_0^2}{4\alpha^2}e^{-2\alpha/r}\right),
~~~c\sim c_0\left(1+\frac{e^{\chi_0}A_0^2}{8r^2}e^{-2\alpha/r}\right), & \nonumber\\
& \chi\sim \chi_0-\frac{e^{\chi_0}A_0^2\alpha}{2r^3}e^{-2\alpha/r},~~~g\sim r^2-\frac{e^{\chi_0}A_0^2\alpha}{2r}e^{-2\alpha/r}, ~~~D\sim -\frac{e^{\chi_0}A_0^2\alpha}{8r^3}e^{-2\alpha/r}. & \nonumber\\
\end{eqnarray}
where terms with subscript 0 are constant.
My question is how to get the whole range numerical solution of r using mathematica. Any code for similar equations are welcome.
\begin{eqnarray}
&-\frac{{\chi}'}{r}+\frac{c'}{c}\left(\frac{g'}{g} -{\chi}'\right)=\frac{e^{\chi}(q A B)^2}{r^2 g^2 c^2}& \\
&c c''+c c'\left(\frac{g'}{g}+\frac{2}{r} -\frac{{\chi}'}{2} \right)=-\frac{B'^2}{2 r^2}+\frac{e^{\chi}}{2 g^2 r^2}(q A B)^2 & \\
&-g'\left(\frac{1}{r}+\frac{c'}{2c}\right)- g \left(\frac{1}{r^2}+\frac{3c'}{c r}+\frac{c''}{c} \right)+3 = \frac{e^{\chi}}{4}A'^2+\frac{g B'^2}{4 r^2 c^2}+\frac{e^{\chi} (q A B)^2}{4 g r^2 c^2}+\frac{e^{\chi}}{4 }D'^2 & \\
&A'' +\left(\frac{2}{r}+\frac{\chi'}{2}+\frac{c'}{c} \right) A'-\frac{q^2B^2}{r^2 c^2 g } A=0& \\
&B'' +\left(\frac{g'}{g}-\frac{\chi'}{2}-\frac{c'}{c} \right) B'+\frac{e^{\chi}q^2A^2}{g } B=0& \\
&D'' +\left(\frac{2}{r}+\frac{\chi'}{2}+\frac{c'}{c} \right) D'=0&
\end{eqnarray}
The solution near r=0 is given by
\begin{eqnarray}
& A\sim A_0 e^{-\alpha/r}, ~~~B\sim B_0\left(1-\frac{e^{\chi_0}q^2A_0^2}{4\alpha^2}e^{-2\alpha/r}\right),
~~~c\sim c_0\left(1+\frac{e^{\chi_0}A_0^2}{8r^2}e^{-2\alpha/r}\right), & \nonumber\\
& \chi\sim \chi_0-\frac{e^{\chi_0}A_0^2\alpha}{2r^3}e^{-2\alpha/r},~~~g\sim r^2-\frac{e^{\chi_0}A_0^2\alpha}{2r}e^{-2\alpha/r}, ~~~D\sim -\frac{e^{\chi_0}A_0^2\alpha}{8r^3}e^{-2\alpha/r}. & \nonumber\\
\end{eqnarray}
where terms with subscript 0 are constant.
My question is how to get the whole range numerical solution of r using mathematica. Any code for similar equations are welcome.