Skip to main content

2020 | OriginalPaper | Buchkapitel

On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions

verfasst von : Merlin Mouafo Wouodjié

Erschienen in: Orthogonal Polynomials

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We present here an algorithm that combines change of variables, exp-product and gauge transformation to represent solutions of a given irreducible third-order linear differential operator L, with rational function coefficients and without Liouvillian solutions, in terms of functions \(S\in \left \{{{ }_1F_1}^2, ~{{ }_0F_2}, ~_1F_2, ~_2F_2\right \}\) where pF q with p ∈{0, 1, 2}, q ∈{1, 2}, is the generalized hypergeometric function. That means we find rational functions r, r 0, r 1, r 2, f such that the solution of L will be of the form
$$\displaystyle y=~ \exp \left (\int r \,dx \right )\left (r_0S(f(x))+r_1(S(f(x)))^{\prime }+r_2(S(f(x)))^{\prime \prime }\right ). $$
An implementation of this algorithm in Maple is available.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
1
To generate this differential equation, we use the hsum package from Wolfram Koepf (see [4]).
 
Literatur
1.
Zurück zum Zitat R. Debeerst, Solving differential equations in terms of Bessel functions. Master’s Thesis, Universität Kassel (2007) R. Debeerst, Solving differential equations in terms of Bessel functions. Master’s Thesis, Universität Kassel (2007)
2.
Zurück zum Zitat R. Debeerst, M. van Hoeij, W. Koepf, Solving differential equations in terms of Bessel functions, in Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation (ISSAC’08) (2008), pp. 39–46 R. Debeerst, M. van Hoeij, W. Koepf, Solving differential equations in terms of Bessel functions, in Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation (ISSAC’08) (2008), pp. 39–46
4.
Zurück zum Zitat W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities (Springer, Berlin, 2014)CrossRef W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities (Springer, Berlin, 2014)CrossRef
5.
Zurück zum Zitat J. Kovacic, An algorithm for solving second-order linear homogeneous equations. J. Symb. Comput. 2, 3–43 (1986)MathSciNetCrossRef J. Kovacic, An algorithm for solving second-order linear homogeneous equations. J. Symb. Comput. 2, 3–43 (1986)MathSciNetCrossRef
8.
Zurück zum Zitat M.F. Singer, Solving homogeneous linear differential equations in terms of second order linear differential equations. Am. J. Math. 107, 663–696 (1985)MathSciNetCrossRef M.F. Singer, Solving homogeneous linear differential equations in terms of second order linear differential equations. Am. J. Math. 107, 663–696 (1985)MathSciNetCrossRef
9.
Zurück zum Zitat M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations. Comprehensive Studies in Mathematics, vol. 328 (Springer, Berlin, 2003) M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations. Comprehensive Studies in Mathematics, vol. 328 (Springer, Berlin, 2003)
10.
Zurück zum Zitat M. van Hoeij, Factorization of linear differential operators. Ph.D. Thesis, Universitijt Nijmegen (1996) M. van Hoeij, Factorization of linear differential operators. Ph.D. Thesis, Universitijt Nijmegen (1996)
11.
Zurück zum Zitat M. van Hoeij, Rational solutions of the mixed differential equation and its application to factorization of differential operators, in Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation (ISSAC’96) (1996), pp. 219–225 M. van Hoeij, Rational solutions of the mixed differential equation and its application to factorization of differential operators, in Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation (ISSAC’96) (1996), pp. 219–225
12.
Zurück zum Zitat M. van Hoeij, Factorization of linear differential operators with rational functions coefficients. J. Symb. Comput. 24, 237–561 (1997) M. van Hoeij, Factorization of linear differential operators with rational functions coefficients. J. Symb. Comput. 24, 237–561 (1997)
13.
Zurück zum Zitat M. van Hoeij, Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients, in Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC’10) (2010), pp. 37–44 M. van Hoeij, Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients, in Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC’10) (2010), pp. 37–44
14.
Zurück zum Zitat Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients. Ph.D. Thesis, Florida State University (2012) Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients. Ph.D. Thesis, Florida State University (2012)
Metadaten
Titel
On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions
verfasst von
Merlin Mouafo Wouodjié
Copyright-Jahr
2020
DOI
https://doi.org/10.1007/978-3-030-36744-2_8