Solving the Poisson equation with spherically symmetric functions

In summary, the conversation discusses the solution to the Poisson equation and how it can be approached using different formulas. The formula from the lecture involves a fundamental solution while the other formula considers spherically symmetric functions. The conversation also includes a question about the distinction between ||x|| and r in the formula and suggests using the lecture formula for a simpler solution.
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docnet
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Homework Statement
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I tried to follow the method outlined in lectures, and ended up with an incorrect solution. My understanding of PDEs is a bit shaky so I thank anyone for constructive feedback or information. :bow:

The solution to the Poisson equation
\begin{equation}
-\Delta u(x)=\frac{q}{\pi a^3}e^{-\frac{2||x||}{a}}
\end{equation}
is given by
\begin{equation}
u(x)=\int_{R^3}\Phi(x-y)f(y)dy
\end{equation}
in ##R^3## and spherically symmetric ##f(y)## this is
\begin{equation}
u(x)=\frac{1}{2||x||}\int_0^\infty r\tilde{f}(r)\Big(||x||+r-\Big|||x||-r\Big|\Big)dr
\end{equation}
\begin{equation}
\frac{1}{2||x||}\int_0^\infty r\frac{q}{\pi a^3}e^{-\frac{2||x||}{a}}\Big(||x||+r-\Big|||x||-r\Big|\Big)dr
\end{equation}
\begin{equation}
\frac{1}{||x||}\frac{q}{\pi a^3}\int_0^\infty e^{-\frac{2r}{a}}r^2dr
\end{equation}
after integration by parts
\begin{equation}
u(x)=\frac{1}{||x||}\frac{q}{4\pi}
\end{equation}
 
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docnet said:
My understanding of PDEs
Shouldn't be a problem :smile: in this case: no PDE involved !
First a question: it says
1614514885696.png
And what would that be ? My telepathic capabilities fail me.

You have (from e.g. here) $$ \Delta u(r) = {1\over r^2}{\partial \over \partial r} \Bigl ( r^2 {\partial u\over \partial r} \Bigr ) = - {q\over 4\pi a^3}\; e^{\displaystyle {-{2r\over a}}}$$ which is no longer a PDE, so we can replace all ##\ \partial \ ## by ##\ ## d ##\ ## !

But the ##\ formula\ from\ the \ lecture\ ## might offer a more comfortable path to the solution ?##\ ##
 
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  • #3
BvU said:
Shouldn't be a problem :smile: in this case: no PDE involved !
First a question: it says
And what would that be ? My telepathic capabilities fail me.

You have (from e.g. here) $$ \Delta u(r) = {1\over r^2}{\partial \over \partial r} \Bigl ( r^2 {\partial u\over \partial r} \Bigr ) = - {q\over 4\pi a^3}\; e^{\displaystyle {-{2r\over a}}}$$ which is no longer a PDE, so we can replace all ##\ \partial \ ## by ##\ ## d ##\ ## !

But the ##\ formula\ from\ the \ lecture\ ## might offer a more comfortable path to the solution ?##\ ##

Thanks for the reply. The formula for the lecture goes like

##-\Delta u=f## has a fundamental solution ##\int_{R^n}\Phi(x-y)f(y)dy##

where ##\Phi(x)=\frac{1}{n(n-2)w_n||x||^{n-2}}## and ##w_n=\frac{4\pi}{3}## for ##R^3##.

For spherically symmetric ##f##, the formula becomes $$\frac{1}{2||x||}\int^\infty_0rf(r)\Big(||x||+r-\Big|||x||-r\Big|\Big)dr$$

One thing I am confused about is why the formula distinguishes between ##||x||## and ##r## since they are the same parameters?
 
Last edited:

Related to Solving the Poisson equation with spherically symmetric functions

1. What is the Poisson equation?

The Poisson equation is a partial differential equation that describes the relationship between a function and its sources or sinks. It is commonly used in physics and engineering to model phenomena such as heat transfer, electrostatics, and fluid flow.

2. How is the Poisson equation solved?

The Poisson equation is typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the domain into smaller elements and solving for the function at discrete points within each element. The solutions from each element are then combined to obtain an overall solution for the entire domain.

3. What are the applications of solving the Poisson equation?

The Poisson equation has a wide range of applications in various fields, including physics, engineering, and mathematics. It is used to model phenomena such as heat transfer, electrostatics, fluid flow, and diffusion. It is also used in image processing, computer graphics, and finance.

4. What are the challenges in solving the Poisson equation?

One of the main challenges in solving the Poisson equation is the need for accurate and efficient numerical methods. As the equation involves derivatives, small errors in the discretization can lead to significant errors in the solution. Additionally, the size and complexity of the domain can also affect the accuracy and efficiency of the solution.

5. Can the Poisson equation be solved analytically?

In some cases, the Poisson equation can be solved analytically using techniques such as separation of variables or the method of eigenfunction expansion. However, these analytical solutions are limited to simple geometries and boundary conditions. In most cases, numerical methods are required to obtain a solution.

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