Solving a separable PDE with BC

In summary, the conversation discusses a question about interpreting boundary conditions for a partial differential equation (PDE) and specifically question 4 in an attached picture. The individual is unsure how to apply the given boundary conditions, as they do not equate to a constant value like in previous problems. They are able to solve for the PDE, but struggle with interpreting the boundaries in terms of the ordinary differential equations (ODEs) obtained. The conversation also includes a request to see the equations obtained when letting u(x,t) = X(x)T(t) and applying it to the two boundary conditions, with a preference for typed equations rather than an image.
  • #1
trap101
342
0
So I have a question in terms of interpreting the boundary conditions for a PDE. It is question 4 in the attached picture.

My question is that usually when I have encountered BC problems I have been given that my boundary conditions equal a given value, in terms of the diffusion equation:

U(0,t) = U(l,t) = 0,

but as yuo can see they don't equate it to a constant. I know how to solve through for the PDE, but now I'm having trouble interpreting those BC in order to apply them to the ODEs I have obtained. Is there a constant I should assume that they equal?
 

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  • #2
Show us what equations you get when you let ##u(x,t) = X(x)T(t)## and apply it to the two boundary conditions. And please type it here instead of giving an uneditable image.
 

Related to Solving a separable PDE with BC

1. How do I determine if a PDE is separable?

In order for a PDE to be separable, it must have the form u(x,y)=X(x)Y(y). This means that the equation can be separated into two independent equations, one involving only x and the other involving only y.

2. What are boundary conditions and why are they important in solving a separable PDE?

Boundary conditions are conditions that must be satisfied at the boundary of the domain in which the PDE is being solved. They are important because they provide necessary information for finding a unique solution to the PDE.

3. Can all separable PDEs be solved analytically?

No, not all separable PDEs can be solved analytically. Some may require numerical methods to find a solution.

4. How can I check if my solution to a separable PDE is correct?

You can check if your solution satisfies both the original PDE and the boundary conditions. Additionally, you can compare your solution to known solutions or use software to verify the solution.

5. Are there any common techniques for solving separable PDEs?

Yes, some common techniques for solving separable PDEs include separation of variables, method of characteristics, and integral transforms. The choice of technique may depend on the specific form of the PDE and boundary conditions.

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