Laplace Transform of Bessel Diff Eq

In summary, the conversation discusses how to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2 and explores different methods to solve this problem. The conversation also mentions using the integral representation of J0(x) and the inverse Laplace transform formula to compute the Laplace transform.
  • #1
DMcG
6
0
Hello PF, maybe you can help with this one!

I need to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2.

My prof told me to start with the form:
x2y'' + x y' + (x2 + p2)y = 0, where p = 0 ITC.

What have I got so far?...

Doing the Laplace transform on both sides, where Y stands for L[y]:
x2(s2Y - sy(0) - y'(0)) + x(sY - y(0)) + x2Y = 0,

...then 3 lines of algebra...

Y = y(0)* (1 + xs)/(xs2 + s + x) + y'(0)* x/(xs2 + s + x).

Do I convolve these two terms on the RHS now? Do I factor the denominator first? Or do I use some recurrence relns for Bessel functions, which is what y(x) is?

Your help would be much appreciated!
 
Physics news on Phys.org
  • #2
DMcG said:
Hello PF, maybe you can help with this one!

I need to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2.

My prof told me to start with the form:
x2y'' + x y' + (x2 + p2)y = 0, where p = 0 ITC.

What have I got so far?...

Doing the Laplace transform on both sides, where Y stands for L[y]:
x2(s2Y - sy(0) - y'(0)) + x(sY - y(0)) + x2Y = 0,

...then 3 lines of algebra...

Something is not right here. I think.

[tex]L\{x^2y' ' \}=\frac{d^2}{ds^2} (s^2Y(s)- sy(0) - y'(0))[/tex]

http://en.wikipedia.org/wiki/Laplace_transform" [Broken]
 
Last edited by a moderator:
  • #3
I think it is easier to start with the integral representation

[tex]J_{0}(x) =\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left[i x\cos\left(\right\theta)\right]d\theta[/tex]

You can then directly computethe Laplace transorm by multiplying this by exp(-sx) and then integrate over x from zero to infinity after reversing the order of integration.

It also not difficult to apply the inverse Laplace transform formula

[tex]f(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\hat{f}(z)\exp(zx)dz[/tex]

to the formula (s^2 + a^2)^-1/2 and then show that you get the Bessel function.
 

1. What is the Laplace Transform of a Bessel Differential Equation?

The Laplace Transform of a Bessel Differential Equation is a mathematical tool used to solve problems involving Bessel functions. It transforms a function of a real variable t into a function of a complex variable s, which can then be easily solved using standard techniques.

2. What is the significance of using the Laplace Transform for solving Bessel Differential Equations?

The Laplace Transform is particularly useful for solving Bessel Differential Equations because it converts them into algebraic equations, which are much easier to solve. This makes it possible to find the solution to a Bessel equation in terms of simple algebraic functions.

3. How does the Laplace Transform work for Bessel Differential Equations?

The Laplace Transform of a Bessel Differential Equation involves taking the integral of the equation with respect to time and then applying the transform to both sides. This results in a new equation in terms of the complex variable s, which can then be solved using standard techniques.

4. Are there any limitations to using the Laplace Transform for Bessel Differential Equations?

One limitation of using the Laplace Transform for Bessel Differential Equations is that it only works for linear equations. Additionally, it may not be suitable for all types of boundary conditions and initial conditions.

5. How can the Laplace Transform of Bessel Differential Equations be applied in real-world situations?

The Laplace Transform of Bessel Differential Equations has many applications in fields such as engineering, physics, and mathematics. It can be used to solve problems involving heat flow, wave propagation, and vibration, among others. It is also used in signal processing and control systems to analyze and design systems with Bessel functions.

Similar threads

  • Differential Equations
Replies
4
Views
1K
  • Differential Equations
Replies
17
Views
808
  • Differential Equations
Replies
1
Views
591
  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
Replies
1
Views
3K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
710
Replies
13
Views
2K
  • Differential Equations
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top