Solving Laplacian Operators and DEs

[Hertz]

Hi, lately I've been messing around a lot with the Laplacian operator and DE's including the Laplacian operator. Most recently, the equation below is the one I have been messing around with and trying to understand better.

[itex]\nabla^2 U(\vec{r})=C(\vec{r})U(\vec{r})[/itex]

This is pretty general though.. WAYY too general for me to tackle. So I've been starting with the 1D case, which I also can't seem to solve.

[itex]\frac{d^2}{dx^2}U(x)=C(x)U(x)[/itex]

My goal is to try to solve for U(x) in terms of C(x). Any ideas? Is there any way to know if such a solution exists? What about to the general equation above?

Thanks :)

 

[HallsofIvy]

Do you know the form of the Laplacian operator in spherical or polar coordinates

 

[Hertz]

No I don't but it wouldn't be too much of a hassle to figure it out. How could that help though?

 

[the_wolfman]

A couple of thoughts for progressing.

1) Try the case where C is constant. This actually gives you a Helmholtz equation.

2) For the more general case, it helps to assume that U or U dot n =0 at the boundary and C has a certain sign.
Then multiply by U and integrate over the domain, this will involve an integration by parts.

 

[blue_raver22]

Hello,

It's great to hear that you have been exploring the Laplacian operator and differential equations (DEs). These are important concepts in many areas of science and engineering. I understand that you are currently focusing on the 1D case, and are trying to solve for U(x) in terms of C(x).

First, let's clarify what the Laplacian operator represents. It is a differential operator that measures the curvature of a function at a given point. In other words, it tells us how much a function changes as we move away from a specific point. This is why it is often used in equations that involve physical quantities such as temperature, pressure, or electric potential.

Now, to solve for U(x) in terms of C(x), we need to use techniques from differential equations. In the 1D case, we can use separation of variables to rewrite the equation as two separate equations:

\frac{d^2 U}{dx^2} = \lambda U

and

\frac{d C}{dx} = \lambda C

where \lambda is a constant. These equations can then be solved to find the general solution for U(x). However, it is important to note that the solution may not exist for all values of C(x) and \lambda. This is where further analysis and boundary conditions come into play to determine the specific solution for a given problem.

For the general equation \nabla^2 U(\vec{r})=C(\vec{r})U(\vec{r}), the same principles apply. It can be solved using separation of variables, but the solution may be dependent on the specific boundary conditions and values of C(\vec{r}).

I hope this helps in your understanding of solving Laplacian operators and DEs. Keep exploring and experimenting, and don't hesitate to seek guidance from experts in the field. Good luck!