What are Bessel Functions and how can they help solve differential equations?

In summary, the conversation discusses Bessel functions and how they can be used to solve certain differential equations. It also mentions the use of graphs and provides a helpful pdf file for students.
  • #1
Ackbach
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MHB
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This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download.

Bessel Functions​

$$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$
is a Bessel function of the first kind of order $\nu$. The general solution of $x^2 \, y''+x \, y'+(x^2-\nu^2) \, y=0$ is $y=c_1 \, J_{\nu}(x)+c_2 \, J_{-\nu}(x)$. If $\nu=n$ is an integer, the general solution is $y=c_1 \, J_n(x)+c_2 \, Y_n(x)$ where $Y_n(x)$ is the Bessel function of the second kind of order $n$. Here, $Y_n(x)$ equals $\frac{2}{\pi} \, \ln\left(\frac{x}{s}\right)$ plus a power series.

The solutions of $x^2 \, y''+x \, y'+(-x^2-\nu^2) \, y=0$ are expressible in terms of modified Bessel functions of the first/second kind of order $\nu$, namely $I_{\nu}(x)$ and $K_{\nu}(x)$.

The graphs:

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You can use these graphs sometimes to work out initial conditions, particularly if any of them are zero.

Equations Solvable in Terms of Bessel Functions​

If $(1-a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the Cauchy-Euler equation $(x^2 y''+axy'+cy=0)$, the differential equation
$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p-1)x^p+b^2x^{2p}]y=0$$
has as general solution
$$y=x^{\alpha} \, e^{-\beta x^p} [C_1 \, J_{\nu}(\varepsilon x^q)+C_2 Y_{\nu}(\varepsilon x^q)]$$
where
$$\alpha=\frac{1-a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{|d|}}{q},\qquad \nu=
\frac{\sqrt{(1-a)^2-4c}}{2q}.$$
If $d<0$, then $J_{\nu}$ and $Y_{\nu}$ are to be replaced by $I_{\nu}$ and $K_{\nu}$, respectively. If $\nu$ is not an integer, then $Y_{\nu}$ and $K_{\nu}$ can be replaced by $J_{-\nu}$ and $I_{-\nu}$ if desired.

The following file is a pdf of the above.

View attachment 4760
 

Attachments

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  • I0Plot.pdf
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  • I1Plot.pdf
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  • K1Plot.pdf
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  • Bessel Function Cheat Sheet.pdf
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  • #2
That will be helpful to many students. Thanks for taking the time to type it up and for sharing.
 
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Related to What are Bessel Functions and how can they help solve differential equations?

1. What are Bessel functions?

Bessel functions are a special class of mathematical functions that are used to solve differential equations that arise in many areas of physics and engineering. They were first introduced by the mathematician Daniel Bernoulli and later studied by Friedrich Bessel, hence the name "Bessel functions".

2. What is the purpose of a Bessel functions cheat sheet?

A Bessel functions cheat sheet is a condensed reference guide that provides the most commonly used equations and properties of Bessel functions. It serves as a quick and handy tool for scientists and mathematicians who frequently use Bessel functions in their work.

3. How are Bessel functions used in real-world applications?

Bessel functions have a wide range of applications in physics, engineering, and other fields. They are used to solve problems related to heat transfer, fluid flow, electromagnetic fields, and many other physical phenomena. They also have applications in signal processing and image reconstruction techniques.

4. What are the different types of Bessel functions?

There are three main types of Bessel functions: the first kind (J), the second kind (Y), and the modified Bessel functions (I and K). Each type has its own set of properties and applications. The first kind is used for problems with circular or spherical symmetry, while the second kind is used for problems with cylindrical symmetry.

5. What are some key properties of Bessel functions?

Some important properties of Bessel functions include their recurrence relations, orthogonality, and generating functions. They also have special values at certain points, such as zeros and extrema, which are useful for solving boundary value problems. Additionally, Bessel functions have asymptotic behavior that can be used to approximate their values for large or small arguments.

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