Learn Hankel Transform & Application to Laplace Eq.

[Clausius2]

Hey guys,

Do you have any advice of a place for learning about Hankel's Transform and its application to Laplace Equation?.

There are a couple of lines of a paper in which I am stuck on, I don't know how do they do this stuff:

Defining the operator $$L_m^2=\frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial}{\partial r}\right)-\frac{m^2}{r^2}+\frac{\partial^2}{\partial z^2}$$

then the solution of $$L_m^2 f=0$$ under the bipolar change of variables $$r=2\eta/(\eta^2+\xi^2)$$ and $$z=2\xi/(\eta^2+\xi^2)$$ is given by:

$$f=(\xi^2+\eta^2)^{1/2}\int_0^\infty(A(s)sinh(s\xi)+B(s)cosh(s\xi))J_m(s\eta)ds$$

I have tried to perform the change of variables in the differential operator, but it turns out to be the Hell when doing that for the second derivative. Any advice?

Thanks.

 

[gtoguy97]

To learn more about Hankel's Transform and its application to the Laplace equation, there are a number of online resources available. You can look into books such as "Integral Transforms in Science and Engineering" by Alexander K. Dym and Alexander V. Ryzhov or "The Hankel Transform and Its Applications" by Robert M. Corless, which provide detailed explanations of the topic. Additionally, there are a number of websites that offer tutorials on the subject. For example, Wolfram MathWorld provides an overview of the Hankel transform and its applications, and MathWorks provides a tutorial on using Hankel transforms to solve Laplace equations. There are also various YouTube videos that explain the concept in detail. As for your specific question about the change of variables, you may need to do a bit more research to get a detailed answer. It may be helpful to consult with a professor or tutor who has expertise in the field.

 

[tacman]

Hi there,

Thank you for reaching out. Learning about Hankel's Transform and its application to Laplace Equation can definitely be challenging, but there are some great resources available to help you understand it better. Here are a few suggestions:

1. Online courses: There are several online courses available on platforms like Coursera, Udemy, and edX that cover the topic of Hankel's Transform and its applications. These courses typically have video lectures, practice exercises, and quizzes to help you understand the concepts better. Some of these courses may even offer a certificate upon completion.

2. Textbooks: There are many textbooks available on the topic of Hankel's Transform and its applications, which you can purchase or access through your university library. Some recommended textbooks are "Integral Transforms and Their Applications" by Lokenath Debnath and Dambaru Bhatta, and "The Laplace Transform: Theory and Applications" by Joel L. Schiff.

3. Online resources: There are also many online resources available, such as lecture notes, tutorials, and practice problems, that can help you understand Hankel's Transform and its applications. You can search for them on websites like MathWorld and MathWorks.

As for your question about the paper, it seems like you are struggling with the change of variables in the differential operator. My advice would be to break down the process step by step and carefully perform the substitution. You can also try consulting a mathematics professor or tutor for further clarification.

I hope these suggestions help you in your learning journey. Good luck!