Advertisement

Nonlinear Oscillations

, Volume 11, Issue 3, pp 307–319 | Cite as

On one class of differential equations of fractional order

  • A. N. Vityuk
  • A. V. Mikhailenko
Article

We consider the Darboux problem for a differential equation of fractional order that contains a regularized mixed derivative. Sufficient conditions for the existence and uniqueness of a solution of this problem are obtained in the class of continuous functions. We also propose a method for finding an approximate solution of this problem and prove the convergence of this method.

Keywords

Differential Equation Integral Equation Approximate Solution Cauchy Problem Fractional Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Tekhnika, Minsk (1987).MATHGoogle Scholar
  2. 2.
    A. N. Kochubei, “Cauchy problem for evolution equations of fractional order,” Differents. Uravn., 25, No. 8, 1359–1367 (1989).MathSciNetGoogle Scholar
  3. 3.
    A. A. Kilbas and S. A. Marzan, “Nonlinear differential equations with fractional Caputo derivative in the space of continuously differentiable functions,” Differents. Uravn., 41, No. 1, 82–86 (2005).MathSciNetGoogle Scholar
  4. 4.
    A. A. Kilbas and S. A. Marzan, “Cauchy problem for differential equations with fractional Caputo derivative,” Dokl. Ros. Akad. Nauk, 339, No. 1, 7–11 (2004).MathSciNetGoogle Scholar
  5. 5.
    S. D. Éidel’man and A. A. Chikrii, “Dynamical game problems of approach for fractional-order equations,” Ukr. Mat. Zh., 52, No. 11, 1566–1583 (2000).MATHGoogle Scholar
  6. 6.
    S. Walczak, “Absolutely continuous functions of several variables and their application to differential equations,” Bull. Pol. Acad. Sci. Math., 35, No. 11–12, 733–744 (1987).MATHMathSciNetGoogle Scholar
  7. 7.
    G. E. Shilov and B. L. Gurevich, Integral, Measure, and Derivative [in Russian], Nauka, Moscow (1967).Google Scholar
  8. 8.
    A. F. Timan, Theory of Approximation of Functions of Real Variables [in Russian], Fizmatgiz, Moscow (1960).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics, Economics, and MechanicsOdessa National UniversityOdessaUkraine

Personalised recommendations