# Laplace equation boundary conditions

• I
• chimay
In summary, the engineer needs to find four constants that will give the desired solution to the Laplace equation in cylindrical coordinates.
chimay
Hi,
I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##.
The boundary conditions are:
##
\left\{
\begin{aligned}
&\Phi(0,r)=V_B \\
&\Phi(L,r)=V_P \\
& -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\
&\frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_1}=0 \\
\end{aligned}
\right.
##
By separation of variable I obtain:
##
\Phi_(z,r)=(A e^{-\lambda z} + B e^{+\lambda z})(C J_0(\lambda r) + D Y_0(\lambda r))
##
##J_0## and ##Y_0## being zero order first type and second type Bessel functions.
The constants I have to determine are : ##A, B, C, D## and ##\lambda##; they are ##5## constants, but only ##4## equations are available coming from the behaviour of ##\Phi## at the boundaries of the domain, so I do not know how to proceed.
Is there any mistake in my reasoning?

You cannot find a product solution. The reason to do the variable separation is so that you can expand the solution in the eigenfunctions of Bessel's differential operator. The normalisation of your eigenfunctions is arbitrary.

Note that you really only have 4 constants as you can take an overall normalisation in one of the factors and absorb it in the other.

Is it the same like saying that I can fit my boundary conditions whatever the value of one constant among A, B, C and D?

Note what I said about the variable separation. In order to find the solution you will need a superposition of separated solutions. Your base functions in the radial direction will be composed of the Bessel functions and their normalisation constant will not matter (it will just change the normalisation of the expansion coefficients).

I have understood that. The first boundary condition gives:
##
\sum_m (A_m + B_m )(C_m J_0(\lambda_m r) + D_m Y_0(\lambda_m r))= V_B
##
and I can decide, for example, to choose ##A_m=1##; is this correct?

It would be more common to fix one of the constants in the radial (SL) part, but in essence yes.

Could you tell me why fixing ##C_m## and ##D_m## is more common?
Anyway, thank you for you help!

## 1. What is the Laplace equation in boundary value problems?

The Laplace equation is a second-order partial differential equation that describes the behavior of a physical quantity, such as temperature or electric potential, in a region where there are no sources or sinks. In boundary value problems, the equation is used to determine the values of the physical quantity at the boundaries of the region.

## 2. What are boundary conditions in the context of the Laplace equation?

Boundary conditions are the set of conditions that must be satisfied at the boundaries of the region in which the Laplace equation is being solved. These conditions provide information about the behavior of the physical quantity at the boundaries and are essential for finding a unique solution to the equation.

## 3. How are boundary conditions specified in the Laplace equation?

Boundary conditions can be specified in different ways, depending on the physical problem being studied. They can be specified as Dirichlet boundary conditions, which specify the value of the physical quantity at the boundary, or as Neumann boundary conditions, which specify the derivative of the physical quantity at the boundary.

## 4. What is the importance of boundary conditions in solving the Laplace equation?

Boundary conditions play a crucial role in solving the Laplace equation because they provide the necessary information to determine a unique solution. Without boundary conditions, the equation would have an infinite number of solutions, making it impossible to determine the behavior of the physical quantity in the region.

## 5. How do boundary conditions affect the behavior of the physical quantity in the region?

The type and values of the boundary conditions directly affect the behavior of the physical quantity in the region. Dirichlet boundary conditions specify fixed values at the boundaries, while Neumann boundary conditions specify the rate at which the physical quantity changes at the boundaries. These conditions influence the overall behavior of the physical quantity and can also affect the stability and convergence of the solution to the Laplace equation.

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