Green's function for homogeneous PDE

In summary, the speaker is seeking help with a challenging problem involving a linear parabolic PDE with a piecewise boundary condition. They have attempted to solve it using separation of variables and the method of characteristics, but are also considering using Green's functions. They are seeking literature and guidance on this specific case.
  • #1
nixs
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Hi there, could anyone help me on this particularly frustrating problem I am having... I have a linear parabolic homogeneous PDE in two variables with a boundary condition that is a piecewise function.

I can solve the pde (with a homogeneous BC) however trying to impose the actual BC makes it seem impossible. I think that using Green's functions will help - as I then have a convolution of the green's function with the BC - but I am finding it difficult to find any literature on this case, i.e. homogen pde & nonhomogen BCs.

Could anyone point me in the right direction? Below is the pde...

[tex]\frac{\partial u}{\partial t}+ a x \frac{\partial^2 u}{\partial x^2} + b \frac{\partial u}{\partial x}=0[/tex]

with BC:
[tex] u(x,T) = x-\rho \,\, \mbox{for}\,\, x>\rho[/tex]
[tex]\qquad =0 [/tex] otherwise

Thanks!
 
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  • #2


Hello, it sounds like you are dealing with a challenging problem. There are a few potential approaches you could take to solve this PDE with a piecewise boundary condition. One option is to use the method of separation of variables, where you assume a solution of the form u(x,t)=X(x)T(t) and then solve for the individual functions X and T using the given boundary conditions. Another approach is to use the method of characteristics, which involves transforming the PDE into a system of ODEs and then solving for the characteristic curves.

As you mentioned, using Green's functions could also be a useful approach. Green's functions are a powerful tool for solving inhomogeneous PDEs with nonhomogeneous boundary conditions. You can try searching for literature on "Green's functions for linear parabolic PDEs with piecewise boundary conditions" to find relevant resources. Additionally, consulting with a colleague or professor who specializes in PDEs may also provide helpful insights and guidance.

I hope this helps guide you in the right direction. Best of luck with your research!
 

1. What is the definition of a Green's function for homogeneous PDE?

A Green's function for homogeneous PDE is a mathematical function that satisfies a certain partial differential equation and its associated boundary conditions. It represents the response of a system to a delta function input, and can be used to solve the original PDE by convolution with a given forcing function.

2. What is the significance of Green's function in solving PDEs?

Green's function is a powerful tool in solving PDEs because it allows for the decomposition of a complex problem into simpler, more manageable parts. By using convolution with the Green's function, the solution to the original PDE can be obtained in a straightforward manner.

3. How is a Green's function derived for a given PDE?

The process of deriving a Green's function for a given PDE involves solving the PDE with a delta function input, and then applying the appropriate boundary conditions. This results in the Green's function, which can then be used to solve the original PDE with a given forcing function.

4. Can Green's function be used for non-homogeneous PDEs?

Yes, Green's function can also be used for non-homogeneous PDEs. In this case, the Green's function will depend on both the PDE and the non-homogeneous term. The solution to the non-homogeneous PDE can then be obtained by convolving the Green's function with the forcing function.

5. Are there any limitations or assumptions when using Green's function to solve PDEs?

One limitation of using Green's function is that it is only applicable for linear PDEs. Additionally, the PDE must have constant coefficients and the boundary conditions must be homogeneous. Assumptions must also be made about the smoothness of the solution and the behavior of the PDE at infinity.

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