Inverse Laplace transform problem

In summary, the conversation discusses using Mellin's inverse formula to calculate the Green's function of a PDE by setting the gamma parameter to 0, and emphasizes the importance of verifying the results and assumptions.
  • #1
xlines
96
0
Hello!

I am trying to solve a PDE for it's Green function, I found myself in a position to know G(is) where "i" is imaginary unit, G(s) is a Laplace transform of g(t) and g(t) is the Green's function I am after. So, I've been snooping around and found Mellin's inverse formula at

http://en.wikipedia.org/wiki/Inverse_Laplace_transform#Integral_form

Should I insert variable transform of s = -i*u over there and setting gamma to zero, I find that green's function g(t) is a Fourier transform of G(-iu) divided by a factor. If I can make sure I can set gamma to zero, is this procedure "legal"?

Any input or suggestion is appriciated!
 
Last edited:
Physics news on Phys.org
  • #2
Yes, this procedure is legal. The inverse Laplace transform formula can be used to calculate the inverse Fourier transform as well, by setting the gamma parameter to 0. Since you are trying to solve a PDE for its Green's function, it is important to make sure that your calculations are correct. You should check your results against known solutions and make sure that they match. Additionally, you should make sure that the assumptions you made in your calculations are valid for the problem at hand.
 

1. What is the Inverse Laplace Transform Problem?

The Inverse Laplace Transform Problem is a mathematical problem that involves finding the original function from its Laplace transform. In other words, it is the process of going from a complex frequency domain representation to a time domain representation.

2. How is the Inverse Laplace Transform Problem solved?

The Inverse Laplace Transform Problem can be solved using various techniques such as the Partial Fraction Decomposition, the Method of Residues, and the Bromwich Integral. These techniques involve breaking down the Laplace transform into simpler forms and then using inverse transforms to find the original function.

3. What are some applications of the Inverse Laplace Transform Problem?

The Inverse Laplace Transform Problem has various applications in engineering and physics, specifically in solving differential equations and analyzing dynamic systems. It is also used in signal processing, control systems, and circuit theory.

4. What are some challenges in solving the Inverse Laplace Transform Problem?

One of the main challenges in solving the Inverse Laplace Transform Problem is the possibility of multiple solutions or a lack of solutions. This can occur when the Laplace transform has singularities or when the Bromwich integral does not converge. Additionally, the complexity of the original function can also make the problem challenging to solve.

5. How is the Inverse Laplace Transform Problem related to the Laplace Transform?

The Inverse Laplace Transform Problem and the Laplace Transform are two sides of the same coin. The Laplace Transform converts a function from the time domain to the frequency domain, while the Inverse Laplace Transform converts it back from the frequency domain to the time domain. They are both widely used in mathematics and engineering to solve problems involving differential equations and dynamic systems.

Similar threads

I On a bijective Laplace transform
    • Differential Equations
Replies
1
Views
647
I On Laplace transform of derivative
    • Differential Equations
Replies
17
Views
849
I Laplace Transform of Sign() or sgn() functions
    • Differential Equations
Replies
1
Views
2K
I Laplace transform linearity problem
    • Differential Equations
Replies
2
Views
2K
A Inverse Laplace transform of a piecewise defined function
    • Differential Equations
Replies
5
Views
4K
What is the Inverse Laplace Transform of e^(-sx^2/2)?
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
I Solving PDE with Laplace Transforms & Inverse Lookup
    • Differential Equations
Replies
1
Views
1K
A Transforming a PDE with Laplace method
    • Differential Equations
Replies
2
Views
1K
Inverse Laplace transform where e^(st)F(s) is entire
    • Differential Equations
Replies
1
Views
1K
Engineering How would I solve this using Laplace transformation?
    • Engineering and Comp Sci Homework Help
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top