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Collection of papers related to Kovacic algorithm for solving differential equations

Nasser M. Abbasi

November 1, 2022   Compiled on November 1, 2022 at 8:09pm

Contents

1 The papers
 1.1 An Algorithm for Solving Second Order Linear Homogeneous Differential Equations (1985 version)
 1.2 An Algorithm for Solving Second Order Linear Homogeneous Differential Equations (2005 version)
 1.3 An Implementation of Kovacic’s Algorithm for Solving Second Order Linear Homogeneous Differential Equations
 1.4 A DISCUSSION AND IMPLEMENTATION OF KOVACICS ALGORITHM FOR ORDINARY DIFFERENTIAL EQUATIONS
 1.5 Analysis and object oriented implementation of the Kovacic algorithm
 1.6 Kovacic’s Algorithm and Its Application to Some Families of Special Functions
 1.7 Note on Kovacic’s Algorithm
 1.8 Necessary Conditions for Liouvillian Solutions of (Third Order) Linear Differential Equations
 1.9 How to Solve Linear Differential Equations: An Outline
 1.10 Solving Second Order Linear Differential Equations with Klein’s Theorem
 1.11 Liouvillian solutions for second order linear differential equations with polynomial coefficients
 1.12 Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
 1.13 A rationality result for Kovacic’s algorithm
 1.14 NonLiouvillian Solutions for Second Order Linear ODEs
 1.15 Galois Groups of Second and Third Order Linear Differential Equations
 1.16 Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations
 1.17 Symbolic analysis of second-order ordinary differential equations with polynomial coefficients
 1.18 AN INTRODUCTION TO DIFFERENTIAL GALOIS THEORY
 1.19 Solving Second Order Linear Differential Equations with Klein’s Theorem
 1.20 Formal Solutions of Differential Equations
 1.21 ON SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ALGEBRAIC SOLUTIONS
 1.22 On second order homogeneous linear differential equations with Liouvillian solutions
 1.23 Solutions of linear ordinary differential equations in terms of special functions
 1.24 AN IMPLEMENTATION OF KOVACIC’S ALGORITHM FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS IN FORMAC
 1.25 Algorithm implementation in FriCAS
 1.26 Maxima implementation of Kovacic algorithm

  These are collection of papers related to Kovacic algorithm for solving differential equations. As I was studying the algorithm, I thought it will be good to have any related paper I find on the subject in one place for easy access. The HTML version has only the outlines. Only the PDF version has the actual papers shown.

Chapter 1
The papers

1.1 An Algorithm for Solving Second Order Linear Homogeneous Differential Equations (1985 version)

This is the original paper by JERALD J. KOVACIC.

1.2 An Algorithm for Solving Second Order Linear Homogeneous Differential Equations (2005 version)

By JERALD J. KOVACIC.

1.3 An Implementation of Kovacic’s Algorithm for Solving Second Order Linear Homogeneous Differential Equations

By B. David Saunders

1.4 A DISCUSSION AND IMPLEMENTATION OF KOVACICS ALGORITHM FOR ORDINARY DIFFERENTIAL EQUATIONS

By Carolyn J. Smith.

1.5 Analysis and object oriented implementation of the Kovacic algorithm

By Nasser M. Abbasi

1.6 Kovacic’s Algorithm and Its Application to Some Families of Special Functions

By Anne Duval, and Michele Loday-Richaud.

1.7 Note on Kovacic’s Algorithm

By FELIX ULMER and JACQUES-ARTHUR WEIL

1.8 Necessary Conditions for Liouvillian Solutions of (Third Order) Linear Differential Equations

By FELIX ULMER and Michael F. Singer.

1.9 How to Solve Linear Differential Equations: An Outline

By FELIX ULMER

1.10 Solving Second Order Linear Differential Equations with Klein’s Theorem

By Mark van Hoeij, Jacques-Arthur Weil and Thomas Cluzeau.

1.11 Liouvillian solutions for second order linear differential equations with polynomial coefficients

By Primitivo B. Acosta‑Humánez, David Blázquez‑Sanz, Henock Venegas‑Gómez

1.12 Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients

By MICHAEL F. SINGER. North Carolina State University. J. Symbolic Computation (1991) 11, 251-273.

1.13 A rationality result for Kovacic’s algorithm

By Marius van der Put, Peter A. Hendriks

1.14 NonLiouvillian Solutions for Second Order Linear ODEs

By L. Chan, E.S. Cheb-Terrab

1.15 Galois Groups of Second and Third Order Linear Differential Equations

By MICHAEL F. SINGER and FELIX ULMER. Journal of Symbolic Computation, 16, July 1993, 9 - 36

1.16 Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations

By MICHAEL F. SINGER and FELIX ULMER. Journal of Symbolic Computation, 16, July 1993, 37 - 74.

1.17 Symbolic analysis of second-order ordinary differential equations with polynomial coefficients

By Tolga Birkandan. arXiv:2010.01563v2, 28 June 2021.

1.18 AN INTRODUCTION TO DIFFERENTIAL GALOIS THEORY

By BRUCE SIMON. San Francisco State University. December 17, 2015.

1.19 Solving Second Order Linear Differential Equations with Klein’s Theorem

By M. van Hoeij and J.A. Weil. Florida State University and Universit´e de Limoges. ISSAC’05, July 24–27, 2005, Beijing, China.

1.20 Formal Solutions of Differential Equations

By MICHAEL F. SINGER. North Carolina State University. ISSAC ’89, Portland, Oregon, July 17, 1989.

1.21 ON SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ALGEBRAIC SOLUTIONS

By F. BALDASSARRI and B. DWORK. American Journal of Mathematics, Vol. 101, No. 1 (Feb., 1979), pp. 42-76

1.22 On second order homogeneous linear differential equations with Liouvillian solutions

By Winfried Fakler. Vniversitiit Karlsruhe. Karlsruhe, Germany. Elsevier Science. Theoretical Computer Science 187 (1997) 2748.

1.23 Solutions of linear ordinary differential equations in terms of special functions

By Manuel Bronstein and Sebastien Lafaille. INRIA – Projet. Proceedings of ISSAC’2002, Lille, ACM Press, 23-28

1.24 AN IMPLEMENTATION OF KOVACIC’S ALGORITHM FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS IN FORMAC

By A.Yu.Zherkov. Saratov State University, Saratov, USSR. 1987

1.25 Algorithm implementation in FriCAS

See https://github.com/fricas/fricas/blob/master/src/algebra/kovacic.spad

1.26 Maxima implementation of Kovacic algorithm

By Nijso Beishuizen.

See https://sourceforge.net/p/maxima/mailman/message/32164642/