How Do You Solve This PDE Using Separation of Variables?

[Inigma]

I have a battle with the following direct partial integration and separation of variables toffee:

I have to solve,
$u(x,y)=\sum_{n=1}^{∞}A_n sin\lambda x sinh \lambda (b-y)$

If there were no boundary or initial conditions given, do I assume that λ is $\frac{n\pi}{L}$ and do I then solve $A_n$? If I am going in the wrong direction here, please point me in the right direction... thanks!

 

[meldraft]

As far as I know, there is no way to solve this further without boundary conditions. You need a condition of the type $u(x_0,y)=g(y)$ or $u(x,y_0)=g(x)$. By evaluating the equation with the boundary condition, you can use Fourier series to find the coefficient $A_n$.

 

[Inigma]

Meldraft:
Thank you for your contribution.
I agree and am also expecting boundary conditions... I want to carry on using the Fourier approach to solve An but due to the lack of boundary conditions, I find it impossible unless I miss something. I am not sure if the Eigen separation constant (λ) may lead to solving the 'boundary conditions" in some degree based on the arrangement of the equation (hyperbolic)... I am stumped...

 

[meldraft]

I am not sure that I understand what your question is. You have a general solution. If you had boundary conditions you could define the constant, but as it is, there is really nothing more you can do. Are you trying to solve it without applying a boundary condition? If so, PDE theory suggests that there are infinite possible solutions, so you cannot possibly get a unique value for your constant, without applying a boundary condition!

 

[blue_raver22]

First of all, great job on attempting to solve this partial differential equation (PDE) using direct partial integration and separation of variables. This is a common method used to solve PDEs, so you are on the right track.

To answer your question, it is important to note that the value of λ is not determined solely by the given boundary or initial conditions. In fact, the value of λ is often chosen based on the form of the PDE and the boundary conditions. So, assuming λ = \frac{n\pi}{L} may not always be the correct approach.

Instead, you can try substituting the given form of u(x,y) into the PDE and see if it satisfies the equation. If it does, then you can use the method of separation of variables to determine the values of A_n and λ. If it doesn't satisfy the equation, then you may need to try a different form for u(x,y) or use a different method to solve the PDE.

It is also important to keep in mind that solving PDEs can be a complex and iterative process. So don't get discouraged if you don't find the solution immediately. Keep trying different approaches and methods, and don't hesitate to seek help from other resources or colleagues if needed.

In summary, solving PDEs requires careful consideration of the given equation and boundary conditions, as well as using appropriate methods and techniques. Keep exploring and experimenting, and you will eventually find the solution. Good luck!