Non-hom heat eq. w/ Dirichlet + Neumann BC

[BangJensen]

I'm trying to find the analytical solution to the following equation:

c1*d2p/dz2-dp/dt = -c2*cos(omega*t)

where
- p is a function of spatial z and time t, p=p(z,t)
- d2p/dz2 is the second derivative of p wrt z
- dp/dt is the first derivative of p wrt t

c1, c2 and omega are constants.

Initial condition: p(z,0) = 0
Boundary condition 1: p(z,t) = 0 for z = 0
Boundary condition 2: dp/dt = 0 for z = d

Everywhere I have looked for solutions so far does not allow the combination of Dirichlet and Neumann boundary conditions or the spatial domain has to be infinite.

I hope someone can help here.

Thanks.

 

[Dr Transport]

Try Lebedev's book on applied math. Your problem can be done fairly easily.

 

[tacman]

It is understandable that you are having difficulty finding an analytical solution for this particular problem. The combination of Dirichlet and Neumann boundary conditions can be challenging to solve analytically, and it is not always possible to find a closed-form solution. Additionally, the finite spatial domain adds another layer of complexity to the problem.

However, there are some techniques that you can try to find a solution. One approach is to use separation of variables, where you assume that the solution can be written as a product of two functions, one depending only on z and the other only on t. This method has been successful in solving similar problems in the past, but it may require some trial and error to find the appropriate form of the functions.

Another approach is to use numerical methods, such as finite difference or finite element methods, to approximate the solution. These methods can handle different types of boundary conditions and finite domains, but they do require some programming skills.

Ultimately, it may not be possible to find an analytical solution for this problem. In that case, it is important to remember that numerical solutions can still provide accurate and useful results. Don't be discouraged if you are not able to find an analytical solution – there are other options available to you. Good luck with your research.