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Mathematical Notes

, Volume 103, Issue 1–2, pp 290–296 | Cite as

The Dirichlet Problem for an Ordinary Continuous Second-Order Differential Equation

  • B. I. Éfendiev
Article
  • 12 Downloads

Abstract

The extremum principle for an ordinary continuous second-order differential equation with variable coefficients is proved and this principle is used to establish the uniqueness of the solution of the Dirichlet problem. The problem under consideration is equivalently reduced to the Fredholm integral equation of the second kind and the unique solvability of this integral equation is proved.

Keywords

continuous differential equation fractional integro-differential operator Dirichlet problem extremum principle 

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References

  1. 1.
    A. M. Nakhushev, “Continuous differential equations and their difference analogues,” Dokl. Akad. Nauk SSSR 300 (4), 796–799 (1988) [SovietMath. Dokl. 37 (3), 729–732 (1988)].MathSciNetMATHGoogle Scholar
  2. 2.
    A. M. Nakhushev, “On the positivity of continuous and discrete differentiation and integration operators that are very important in fractional calculus and in the theory of equations of mixed type,” Differ. Uravn. 34 (1), 101–109 (1998) [Differ. Equations 34 (1), 103–112 (1998)].MathSciNetGoogle Scholar
  3. 3.
    A. V. Pskhu, “On the Theory of the Continual Integro-Differentiation Operator,” Differ. Uravn. 40 (1), 120–127 (2004) [Differ. Equations 40 (1), 128–136 (2004)].MathSciNetMATHGoogle Scholar
  4. 4.
    A. M. Nakhushev, Fractional Calculus and Its Application (Fizmatlit, Moscow, 2003) [in Russian].MATHGoogle Scholar
  5. 5.
    A. V. Pskhu, Partial Differential Equations of Fractional Order (Nauka, Moscow, 2005) [in Russian].MATHGoogle Scholar
  6. 6.
    A. M. Nakhushev, “The Sturm–Liouville problem for an ordinary second-order differential equation with fractional derivatives in the lowest terms,” Dokl. Akad. Nauk SSSR 234 (2), 308–311 (1977) [SovietMath. Dokl. 18 (2), 666–670 (1977)].MathSciNetMATHGoogle Scholar
  7. 7.
    L. Kh. Gadzova, “On the theory of boundary-value problems for a differential equation of fractional order with constant coefficients,” Dokl. Adygeisk. (Cherkessk.) Mezhdunar. Akad. Nauk 16 (2), 34–40 (2014).MATHGoogle Scholar
  8. 8.
    L. Kh. Gadzova, “Generalized Dirichlet problem for a fractional linear differential equation with constant coefficients,” Differ. Uravn. 50 (1), 121–125 (2014) [Differ. Equations 50 (1), 122–127 (2014)].MathSciNetMATHGoogle Scholar
  9. 9.
    B. I. Éfendiev, “Cauchy problemfor a second-order ordinary differential equation with a continual derivative,” Differ. Uravn. 47 (9), 1364–1368 (2011) [Differ. Equations 47 (9), 1378–1383 (2011)].MathSciNetGoogle Scholar
  10. 10.
    B. I. Éfendiev, “Steklov problemfor a second-order ordinary differential equation with a continual derivative,” Differ. Uravn. 49 (4), 469–475 (2013) [Differ. Equations 49 (4), 450–456 (2013)].Google Scholar
  11. 11.
    B. I. Éfendiev, “Dirichlet problem for second-order ordinary differential equations with segment-order derivative,” Mat. Zametki 97 (4), 620–628 (2015) [Math. Notes 97 (3–4), 632–640 (2015)].CrossRefMATHGoogle Scholar
  12. 12.
    B. I. Éfendiev, “Initial-value problem for a continuous second-order differential equation,” Differ. Uravn. 50 (4), 564–568 (2014) [Differ. Equations 50 (4), 562–567 (2014)].MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and AutomationNalchikRussia

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