Inhomogeneous wave equation solution?

Best of luck to you! In summary, the conversation discusses ways to solve the inhomogeneous Klein-Gordon equation, including using the guess method and Green's functions. Suggestions are also given for potential initial and boundary conditions to consider.
  • #1
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inhomogeneous Klein-Gordon equation solution?

Homework Statement


Psi_xx - Psi_tt - 4Psi = exp(exp(3it))*dirac_delta(x)
DE valid for all x,t (no boundary conditions specified).

Homework Equations


Solve for Psi. If the DE is singular, then nontrivial solutions are okay.

The Attempt at a Solution


-I solved the homogeneous portion, Psi_homogeneous, of this equation via separation of variables but my solution is just for some random case of k^2 where: F''/F = G''/G + 4 = k^2. I chose the case where k^2 = 0 which gave me solutions for Psi_homogeneous like x*sin((4^0.5)*t) and x*cos((4^0.5)*t).

-With the guess method for Psi_particular, I have no idea what to guess on a general form of exp(exp(3it))*dirac_delta(x) to plug back into the PDE.

-I have read about Green's Functions but, man, I'm having a hard time understanding the guides I have seen because they skip so many of the intermediate steps. I understand that these Green's Functions can provide a general solution and that seems like what I'm looking for. I likely spent a lot of time for nothing on my first attempt with separation of variables for Psi_homogeneous and then looking for Psi_particular using the guess method...

-I'm curious if there is a general set of IC/BCs that I should be assuming as well?
 
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  • #2


First of all, great job on solving the homogeneous portion of the equation! That's a crucial step in finding the general solution.

To tackle the nonhomogeneous part of the equation, you're on the right track with the guess method. However, in this case, the guess might not be as straightforward as in other cases. One approach you could try is to use the method of variation of parameters. This involves finding a particular solution by varying the parameters in the general solution of the homogeneous equation. This method is often used for nonhomogeneous linear equations with constant coefficients, but it can also be applied to certain types of nonhomogeneous nonlinear equations, like the one you have here.

Another approach you could try is to use Green's functions, as you mentioned. Green's functions are useful for solving nonhomogeneous linear equations, and they can provide a general solution. However, they can be tricky to understand at first, so I suggest reviewing some of the guides and examples you've found and trying to apply them to your specific equation. As for IC/BCs, it might be helpful to assume some general ones, such as Psi(0,t) = 0 and Psi(x,0) = 0, to see if those lead to a solution.

I hope this helps. Good luck with your solution! Remember to always check your work and make sure it satisfies the original equation. And don't hesitate to ask for more help if you need it. Science is all about collaboration and learning from each other.
 

1. What is the inhomogeneous wave equation?

The inhomogeneous wave equation is a mathematical equation that describes the behavior of waves in a medium where there are external forces or sources present. It is a partial differential equation that relates the second time derivative of a wave to the second spatial derivative of the wave.

2. How is the inhomogeneous wave equation solved?

The inhomogeneous wave equation is typically solved using the method of separation of variables, which involves separating the variables of the equation into different parts and solving each part separately. This results in a general solution that can then be applied to specific boundary conditions to find the particular solution for a given problem.

3. What are the boundary conditions for solving the inhomogeneous wave equation?

The boundary conditions for solving the inhomogeneous wave equation depend on the specific problem at hand. They typically involve specifying the initial conditions of the wave, such as its initial position, velocity, and any external forces acting on it. These boundary conditions are necessary for finding the particular solution to the equation.

4. What is the difference between the inhomogeneous and homogeneous wave equation?

The inhomogeneous wave equation includes external forces or sources, while the homogeneous wave equation does not. This means that the solution to the inhomogeneous wave equation will include a particular solution for the external forces, while the solution to the homogeneous wave equation will only include the general solution for the wave itself.

5. What are some real-world applications of the inhomogeneous wave equation?

The inhomogeneous wave equation has many practical applications, such as describing the behavior of sound waves in a room with external noise sources, modeling the propagation of seismic waves in the Earth's crust, and predicting the behavior of electromagnetic waves in the presence of external electromagnetic fields. It is also used in fields such as acoustics, optics, and electromagnetism.

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