Can Laplace Transforms be Applied to Finite Intervals?

In summary, the conversation discusses solving a problem involving a string of length L that is connected to supports and subjected to a load. The goal is to find the displacement u, but the proposed solutions attempted a Laplace transform on a finite interval, which is not possible. The correct approach would involve finding g using the delta dirac and then using it to solve the problem.
  • #1
LCSphysicist
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Homework Statement
I don't know why, but my answer is wrong.
Relevant Equations
.
"Consider a string of length L that is connected at both ends to supports and is subjected to a load (external force per unit length) of f(x). Find the displcament u"
https://i.stack.imgur.com/yVIDG.png

We need to solve this:
$$Tu_{xx} = f(x)$$ subject to $$u(0)=u(L)=0$$

But i don't understand what is the problem in my solutions:
$$Tu_{xx} = f(x) = > Tu_{xx} = \delta({x-\epsilon})\\T(Us^2-su(0)-u_{x}(0)) = e^{-s \epsilon}\\U = (e^{-s \epsilon}/T + u_{x}(0))/s^2$$

So, we know that the inverse of $$e^{-s \epsilon}/(Ts^2) = H(x-\epsilon)x/T$$ and inverse of $$(u_{x}(0))/s^2 = u_{x}(0)x$$

So, shouldn't the final answer be $$u = \int_{0}^{L} f(\epsilon) (H(x-\epsilon)x/T + u_{x}(0)x) d \epsilon$$

Why is it wrong?
 
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  • #2
You have attempted a Laplace transform on a finite interval.
 
  • #3
Orodruin said:
You have attempted a Laplace transform on a finite interval.
Hello. Thank you. But since we are restrict to a finite interval in x not infinite, this method is not possible? I mean, the method of find g using the delta dirac.
Or do i applied wrong? Could you give a tip to go on?
 

1. What is a Green function?

A Green function is a mathematical tool used in physics and engineering to solve differential equations. It represents the response of a system to an impulse or point source input.

2. How are Green functions used in string theory?

In string theory, Green functions are used to calculate the propagation of strings in spacetime. They describe the interactions between strings and other objects, such as particles and branes.

3. What is the significance of Green functions in quantum field theory?

In quantum field theory, Green functions play a crucial role in calculating the scattering amplitudes of particles. They also provide a way to calculate the vacuum expectation values of fields, which are important for understanding the behavior of quantum systems.

4. Can Green functions be used to solve boundary value problems?

Yes, Green functions can be used to solve boundary value problems in various fields, including electromagnetism, fluid dynamics, and elasticity. They provide a powerful method for finding solutions to differential equations with specified boundary conditions.

5. Are there any limitations to using Green functions?

While Green functions are a useful tool, they have some limitations. They can only be used for linear systems, and their use may be limited in systems with complicated boundary conditions. Additionally, they may not always provide an exact solution and may require approximations for practical applications.

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