# Solving Differential Equations in Terms of Bessel Functions

On this page you find some materials to the paper submited to the ISSAC conference 2008 by Wolfram Koepf, Mark van Hoeij and me. You will also find information about my master thesis which covers the same topic.

## The Algorithm

The algorithm was written by Mark van Hoeij and me during my visit at Florida State University, Tallahasse, FL, USA.

## ISSAC 2008

We submited a paper to the ISSAC conference, which will be held July 20--23, 2008 at the University of Linz, Austria. More information can be found on their website.

You will find two examples in the paper. The first one can almost be done by hand but some commands in Maple will help you. The latter can be found in more detail in my master thesis (example 3.29, see below). The operator given at the very end of the paper is an examples which cannot be solved by Maple 12. It can be solved by the following commands.

## Master Thesis

• Helper Functions (mpl)
• Transformation Algorithms (mws / mpl)

### Examples With Maple

You can either download a classic Maple worksheet (mws) created by Maple 9 or a plain text file (mpl) including only the input commands. The mpl files can be loaded into Maple with the read command.

#### Chapter 1: Preliminaries

• Example 1.4: Differential Operators (mws / mpl)
• Example 1.29: The gen_exp Command (mws / mpl)
• Example 1.30: Generalized Exponents of Bessel Operators (mws / mpl)

#### Chapter 2: Transformations

• Theorem 2.3: Transformation Algorithms (mws / mpl)
• Example 2.11: Singularities and Generalized Exponents After Transformations (mws / mpl)
• Example 2.22: Cyclic Vector Method (mws / mpl)
• Example 2.23: Reverse Gauge (mws / mpl)

#### Chapter 3: Solving in Terms of Bessel Functions

• Example 3.2: Finding Polar Parts (mws / mpl)
• Example 3.6: Finding Polar Parts II (mws / mpl)
• Example 3.12: Logarithmic Case (mws / mpl)
• Example 3.13: Logarithmic case, gauge transformation not needed (mws / mpl)
• Example 3.15: Half Integer Case nu=1/2 (mws / mpl)
• Example 3.17: Integer Case (mws / mpl)
• Example 3.18: Rational Case (mws / mpl)
• Example 3.20: Basefield Case (mws / mpl)
• Example 3.21: Basefield Case II (mws / mpl)
• Example 3.23: Problems With Constant Factor in Polar Parts (mws / mpl)
• Example 3.24: Finding the Constant Factor With Subfields (mws / mpl)
• Example 3.25: Algebraic Functions (mws / mpl)
• Example 3.26: Generalized Exponents of Whittaker Operators (mws / mpl)
• Example 3.28: Whittaker case (mws / mpl)
• Example 3.29: Example Everitt (mws / mpl)
• Example 3.30: Example Koepf/Foupouagnigni (mws / mpl)

#### Chapter 4: Conclusion

• Generalized Exponents in Other Cases (mws / mpl)

Last Update: July 17, 2008