- #1
chimay
- 81
- 7
Hi,
I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones:
##
\left\{
\begin{aligned}
&C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\
&\Phi(0,r)=f_1(r) \\
&\Phi(L,r)=f_2(r)
\end{aligned}
\right.
##
##\epsilon## being the dielectric constant of the medium and ##C_{di}## a constant capacitance; ##f_1## and ##f_2## are two known function.
Forcing no ##\phi## dependence, and solving the equation by separation of variable:
##
\Phi(x,r)=(Ae^{\lambda x/r_0}+Be^{-\lambda x/r_0})J_0(\lambda r/r_0)
##
##\lambda## being the separation constant and ##J_0## the first type Bessel function of 0-order.
By applying the first boundary condition:
##
\lambda \frac{J_1(\lambda)}{J_0(\lambda)}=C_r
##
## C_r=r_0 C_{di}/\epsilon##
which can be solved to compute all the values of ##\lambda##.
By applying the remaining boundary conditions, I obtain the following set of equations:
##
\left\{
\begin{aligned}
&\sum_{m=1}^{\infty} (A_m+B_m)J_0(\lambda_m r/r_0)=f_1(r) \\
&\sum_{m=1}^{\infty} (A_m e^{\lambda_m L/r_0}+B_m e^{-\lambda_m L/r_0})J_0(\lambda_m r/r_0)=f_2(r) \\
\end{aligned}
\right.
##
which allow me to compute the values of all the coefficients, by exploiting the orthogonality between differently scaled Bessel functions:
##
\left\{
\begin{aligned}
&A_i=\frac{1}{(2\sinh{\lambda_i L/r_0})(\frac{1}{2}{J_1(\lambda_i)}^2)}(I_2-I_1 e^{-\lambda_i L/r_0}) \\
&B_i=\frac{1}{(2\sinh{\lambda_i L/r_0})(\frac{1}{2}{J_1(\lambda_i)}^2)}(I_1e^{\lambda_i L/r_0}-I_1) \\
\end{aligned}
\right.
##
##I=\int_0^1 \frac{r}{r_0} f(r) J_0(\lambda_i r/r_0) d(\frac{r}{r_0})##
Now the point that is driving me crazy is the following: how do you expect ##A_i## and ##B_i## to change as a function of the order ##i##? I expect them to decrease in modulus; indeed, this is the case for ##A_i##, but ##B_i## presents a strange behaviour, oscillating in sign and increasing in modulus. Can you see any mistake? If you want, I can provide you with all the mathematical passages that I omitted for brevity, the trends of ##A_i## and ##B_i## or show you whatever plot you may need. Any suggestion is really appreciated here.
Thank you all.
I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones:
##
\left\{
\begin{aligned}
&C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\
&\Phi(0,r)=f_1(r) \\
&\Phi(L,r)=f_2(r)
\end{aligned}
\right.
##
##\epsilon## being the dielectric constant of the medium and ##C_{di}## a constant capacitance; ##f_1## and ##f_2## are two known function.
Forcing no ##\phi## dependence, and solving the equation by separation of variable:
##
\Phi(x,r)=(Ae^{\lambda x/r_0}+Be^{-\lambda x/r_0})J_0(\lambda r/r_0)
##
##\lambda## being the separation constant and ##J_0## the first type Bessel function of 0-order.
By applying the first boundary condition:
##
\lambda \frac{J_1(\lambda)}{J_0(\lambda)}=C_r
##
## C_r=r_0 C_{di}/\epsilon##
which can be solved to compute all the values of ##\lambda##.
By applying the remaining boundary conditions, I obtain the following set of equations:
##
\left\{
\begin{aligned}
&\sum_{m=1}^{\infty} (A_m+B_m)J_0(\lambda_m r/r_0)=f_1(r) \\
&\sum_{m=1}^{\infty} (A_m e^{\lambda_m L/r_0}+B_m e^{-\lambda_m L/r_0})J_0(\lambda_m r/r_0)=f_2(r) \\
\end{aligned}
\right.
##
which allow me to compute the values of all the coefficients, by exploiting the orthogonality between differently scaled Bessel functions:
##
\left\{
\begin{aligned}
&A_i=\frac{1}{(2\sinh{\lambda_i L/r_0})(\frac{1}{2}{J_1(\lambda_i)}^2)}(I_2-I_1 e^{-\lambda_i L/r_0}) \\
&B_i=\frac{1}{(2\sinh{\lambda_i L/r_0})(\frac{1}{2}{J_1(\lambda_i)}^2)}(I_1e^{\lambda_i L/r_0}-I_1) \\
\end{aligned}
\right.
##
##I=\int_0^1 \frac{r}{r_0} f(r) J_0(\lambda_i r/r_0) d(\frac{r}{r_0})##
Now the point that is driving me crazy is the following: how do you expect ##A_i## and ##B_i## to change as a function of the order ##i##? I expect them to decrease in modulus; indeed, this is the case for ##A_i##, but ##B_i## presents a strange behaviour, oscillating in sign and increasing in modulus. Can you see any mistake? If you want, I can provide you with all the mathematical passages that I omitted for brevity, the trends of ##A_i## and ##B_i## or show you whatever plot you may need. Any suggestion is really appreciated here.
Thank you all.