How to solve the wave equation with Dirac delta function initial conditions?

In summary, the conversation discusses solving the IVP for the wave equation, with given initial conditions and using D'Almbert's solution. The individual is unsure how to handle the dirac function inside the integral and is seeking guidance. They have already researched commonly available information on the dirac delta function.
  • #1
FrattyMathMan
2
0

Homework Statement


Solve the IVP for the wave equation:

Utt-Uxx=0 for t>0
U=0 for t=0
Ut=[dirac(x+1)-dirac(x-1)] for t=0


2. The attempt at a solution

By D' Almbert's solution:

1/2 integral [dirac(x+1)-dirac(x-1)] dx from (x-t) to (x+t)

I apologize for not using Latex- my browser does not seem to agree with the reference.

My issue is what am I to do with the dirac function inside the integral? Can anyone point me in the right direction? I've already read all common literature on the dirac delta function.

Thank you for any help!
 
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  • #2
Any help? Seriously drawing a blank!

Am I supposed to just use the heavyside function?
 

1. What is an initial value problem (IVP) for a partial differential equation (PDE) wave equation?

An initial value problem for a PDE wave equation involves finding a solution to the wave equation that satisfies both the equation itself and certain initial conditions. These initial conditions typically involve specifying the initial position and velocity of the wave at a given time.

2. What is the difference between an IVP and a boundary value problem (BVP) for a PDE wave equation?

An IVP involves finding a solution to the wave equation at a specific point in time, while a BVP involves finding a solution over a specific spatial interval. In an IVP, the initial conditions are given at a fixed time, while in a BVP, the boundary conditions are given at fixed spatial boundaries.

3. How is an IVP for a PDE wave equation solved?

The most common method for solving an IVP for a PDE wave equation is through the use of separation of variables. This involves assuming a solution in the form of a product of functions of time and space, and then solving for the individual functions using the given initial conditions.

4. What are the applications of IVP for PDE wave equations?

IVP for PDE wave equations have numerous applications in physics and engineering, including studying the behavior of waves in different mediums, such as sound waves in air or electromagnetic waves in space. They can also be used to model and analyze phenomena such as earthquakes, vibrations in structures, and fluid dynamics.

5. What are the challenges in solving IVP for PDE wave equations?

One of the main challenges in solving IVP for PDE wave equations is the mathematical complexity involved. The equations are often nonlinear and have multiple variables, making them difficult to solve analytically. This has led to the development of numerical methods, such as finite difference or finite element methods, to approximate solutions to these equations.

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