Skip to main content
SpringerLink
Log in

Solution of a Problem with Initial Conditions of the Cauchy Type for a Higher-Order Equation with a Hilfer Fractional Derivative

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

A Cauchy type problem for a higher-order equation with a Hilfer fractional derivative is considered. Existence and uniqueness theorems are proved for the solution of the problem in the class of bounded functions, the solution being constructed with the help of self-similar solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Hilfer, R., Fractional time evolution, in Applications of Fractional Calculus in Physics, Singapore: World Sci., 2000, pp. 87–130.

  2. Nakhushev, A.M., Uravneniya matematicheskoi biologii (Equations of Mathematical Biology), Moscow: Vyssh. Shkola, 1995.

    MATH  Google Scholar 

  3. Luchko, Yu. and Gorenflo, R., Scale-invariant solutions of a partial differential equation of fractional order, Fract. Calc. Appl. Anal., 1998, vol. 1, no. 1, pp. 63–78.

    MathSciNet  MATH  Google Scholar 

  4. Irgashev, B.Yu., Construction of singular particular solutions expressed via hypergeometric functions for an equation with multiple characteristics, Differ. Equations, 2020, vol. 56, no. 3, pp. 315–323.

    Article  MathSciNet  MATH  Google Scholar 

  5. Kim, M.-Ha., Chol-Ri, G., and Chol, O.H., Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives, Fract. Calc. Appl. Anal., 2014, vol. 17, no. 1, pp. 79–95.

    Article  MathSciNet  MATH  Google Scholar 

  6. Karimov, E.T., Tricomi type boundary value problem with integral conjugation condition for a mixed type equation with Hilfer fractional operator, Bull. Inst. Math., 2019, no. 1, pp. 19–26.

  7. Salakhitdinov, M.S. and Karimov, E.T., Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative, Uzbek Math. J., 2017, no. 4, pp. 140–149.

  8. Pskhu, A.V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka (Fractional Partial Differential Equations), Moscow: Nauka, 2005.

    Google Scholar 

  9. Wright, E.M., On the coefficients of power series having exponential singularities, J. London Math. Soc., 1933, vol. 8, no. 29, pp. 71–79.

    Article  MathSciNet  MATH  Google Scholar 

  10. Pskhu, A.V., Fundamental solutions and Cauchy problems for an odd-order partial differential equation with fractional derivative, Electron. J. Differ. Equations, 2019, vol. 2019, no. 21, pp. 1–13.

    MathSciNet  MATH  Google Scholar 

  11. Pskhu, A.V., The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 2009, vol. 73, no. 2, p. 351.

    Article  MathSciNet  MATH  Google Scholar 

  12. Karasheva, L.L., The Cauchy problem for a parabolic equation of high even order with a fractional derivative with respect to the time variable, Sib. Elektron. Mat. Izv., 2018, vol. 15, pp. 696–706.

    MathSciNet  MATH  Google Scholar 

  13. Voroshilov, A.A. and Kilbas, A.A., The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative, Differ. Equations, 2006, vol. 42, no. 5, pp. 638–649.

    Article  MathSciNet  MATH  Google Scholar 

  14. Voroshilov, A.A. and Kilbas, A.A., A Cauchy-type problem for the diffusion-wave equation with Riemann–Liouville partial derivative, Dokl. Math., 2006, vol. 73, no. 1, pp. 6–10.

    Article  MathSciNet  MATH  Google Scholar 

  15. Voroshilov, A.A. and Kilbas, A.A., Existence conditions for a classical solution of the Cauchy problem for the diffusion-wave equation with a partial Caputo derivative, Dokl. Math., 2007, vol. 75, no. 3, pp. 407–410.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kochubei, A.N., Diffusion of fractional order, Differ. Equations, 1990, vol. 26, no. 4, pp. 485–492.

    MathSciNet  MATH  Google Scholar 

  17. Wright, E.M., The generalized Bessel function of order greater than one, Q. J. Math. Oxford Ser., 1940, vol. 11, no. 1, pp. 36–48.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kadirkulov, B.J. and Jalilov, M.A., On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator, Bull. Karaganda Univ. Math. Ser., 2021, vol. 104, no. 4, pp. 89–102.

    Article  Google Scholar 

  19. Titchmarsh, E.C., Introduction to the Theory of Fourier Integrals, London: Clarendon Press, 1937. Translated under the title: Vvedenie v teoriyu integralov Fur’e, Moscow: Gos. Izd. Tekh.-Teor. Liter., 1948.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Namangan Institute of Engineering and Construction, Namangan, 160100, Uzbekistan

    B. Yu. Irgashev

  2. Romanovskii Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 100170, Uzbekistan

    B. Yu. Irgashev

Authors
  1. B. Yu. Irgashev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Yu. Irgashev.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Irgashev, B.Y. Solution of a Problem with Initial Conditions of the Cauchy Type for a Higher-Order Equation with a Hilfer Fractional Derivative. Diff Equat 58, 1195–1210 (2022). https://doi.org/10.1134/S001226612209004X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S001226612209004X

Search

Navigation