Skip to main content
SpringerLink
Log in

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

We aim to prove a unique solvability of a boundary-value problem with Dirichlet conditions for the Hallaire–Luikov moisture transfer equation involving generalized fractional derivative (Hilfer derivative) in time. The formal solution to the problem has been obtained in a series form using the method of spectral expansion. Imposing certain conditions on given functions and using certain properties of the multinomial Mittag–Leffler function, we prove a uniform convergence of corresponding infinite series. Moreover, a number of properties of the multinomial Mittag–Leffler function in some particular cases are also presented. Finally, an example solution is provided to illustrate the obtained results.

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

Fig. 1

Similar content being viewed by others

References

  • Ali M, Aziz S, Malik S (2020) Inverse problem for a multi-term fractional differential equation. Fract Calc Appl Anal 23(3):799–821

    Article  MathSciNet  MATH  Google Scholar 

  • Bulavatsky VM (2014) Closed form of the solutions of some boundary-value problems for anomalous diffusion equation with Hilfer’s generalized derivative. Cybern Syst Anal 50(4):570–577

    Article  MathSciNet  MATH  Google Scholar 

  • Chudnovsky AF (1976) Thermal physics of soils. Nauka, Moscow (in Russian)

    Google Scholar 

  • Furati KhM, Iyiola OS, Kirane M (2014) An inverse problem for a generalized fractional diffusion. Appl Math Comput 249:24–31

    MathSciNet  MATH  Google Scholar 

  • Gekkieva SK (2018) Nonlocal boundary-value problem for the generalized Aller–Lykov moisture transport equation. Vestnik KRAUNC. Fiz.-mat. nauki. 24(4):19–28

    MathSciNet  MATH  Google Scholar 

  • Gekkieva SKh, Kerefov MA (2019) Dirichlet boundary value problem for Aller–Lykov moisture transfer equation with fractional derivative in time. Ufa Math. J. 11(2):71–81

    Article  MathSciNet  MATH  Google Scholar 

  • Hallaire M (1963) Potential efficace de lB’eau dans le sol en regime de dessechement. Assemble generale de Berkeley (General Assembly of Berkeley), Publ. no. 62 (August 1963). Gentbrugge, pp 114B–122

  • Hilfer R (2000) Fractional time evolution. Applications of fractional calculus in physics. World Scientific, Singapore, pp 87–130

    Book  MATH  Google Scholar 

  • Ilyas A, Malik SA (2022) An inverse source problem for anomalous diffusion equation with generalized fractional derivative in time. Acta Appl Math 181:15

    Article  MathSciNet  MATH  Google Scholar 

  • Janssen H, Carmeliet J, Hens H (2002) The influence of soil moisture in the unsaturated zone on the heat loss from buildings via the ground. Int J Therm Envel Build Sci 25(4):275–296

    Article  Google Scholar 

  • Karimov E T, Kerbal S, Al-Salti N (2017) Inverse source problem for multi-term fractional mixed type equation. In: Ruzhansky M et al (eds) Advances in real and complex analysis with applications, trends in mathematics. pp 289–301

  • Kerefov MA, Gekkieva SK (2019) Second boundary value problem for the generalized Aller–Lykov equation. Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki (J.Samara state Tech. Univ. Ser. Phys. Math. Sci.) 23(4):607–621

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Kim M-H, Ri G-C, Hyong C (2014) Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives. Fract Calc Appl Anal 17(1):79–95

    Article  MathSciNet  MATH  Google Scholar 

  • Kulik VA (1972) Study of motion of soil water from the point of view of invariance of relatively continuous transformations groups Study of processes of energy and matter transfer in the system soil-plane-air. Nauka, Leningrad (in Russian)

    Google Scholar 

  • Li Z, Liu Y, Yamamoto M (2015) Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl Math Comput 257:381–397

    MathSciNet  MATH  Google Scholar 

  • Luchko Y (2021) General fractional integrals and derivatives with the Sonine Kernels. Mathematics 9:594

    Article  Google Scholar 

  • Luchko Y, Gorenflo R (1999) An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam 24(2):207–233

    MathSciNet  MATH  Google Scholar 

  • Luikov AV (1965) Application of the methods of thermodynamics of irreversible processes to the investigation of heat and mass transfer. J. Eng. Phys. Thermophys. 9(3):189–202

    Article  Google Scholar 

  • Luikov AV (1966) Heat and mass transfer in capillary-porous body. Pergamon Press, New York, p 523

    MATH  Google Scholar 

  • Makaova RKh (2016) The first boundary value problem for the non-homogeneous Hallaire equation. Vestnik KRAUNC Fiz.-mat. nauki. 16(4–1):45–49

    MathSciNet  MATH  Google Scholar 

  • Makaova RKh (2018) The Tricomi problem for a third order hyperbolic equation degenerating inside the domain. Vestnik KRAUNC Fiz.-mat. nauki. 23(3):67–75

    MathSciNet  MATH  Google Scholar 

  • Makaova RKh (2019) The boundary value problem for order equation of parabolic-hyperbolic type. Vestnik KRAUNC Fiz.-mat. nauki. 28(3):26–31

    MathSciNet  MATH  Google Scholar 

  • Nakhushev AM (2003) Fractional calculus and its application. Fizmatgiz, Moscow (in Russian)

    MATH  Google Scholar 

  • Tarasov VE (2011) Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (nonlinear physical science). Springer, Berlin, p 505

    Google Scholar 

  • Tarasov VE (2015) Non-linear fractional field equations: weak nonlinearity at power-law non-locality. Nonlinear Dyn 80:1665–1672

    Article  MATH  Google Scholar 

  • Tarasov VE (2022) Fractional dynamics with depreciation and obsolescence: equations with Prabhakar fractional derivatives. Mathematics 10:1540

    Article  Google Scholar 

  • Tarasov VE, Tarasova VV (2021) Economic dynamics with memory: fractional calculus approach. De Gruyter, Berlin, p 602

    Book  MATH  Google Scholar 

  • Titchmarsh E (1939) The theory of functions, 2nd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Wongsantisul Ph et al (2022) Hilfer fractional quantum derivative and boundary value problems. Mathematics 10(6):878

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Science, National University of Science and Technology, Muscat, Oman

    Nasser Al-Salti

  2. V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, Universitet street, 100174, Tashkent, Uzbekistan

    Erkinjon Karimov

  3. Fergana State University, 19 Murabbiylar street, 140100, Fergana, Uzbekistan

    Erkinjon Karimov

  4. Department of Mathematics, Sultan Qaboos University, Al-Khoudh 123, Muscat, Oman

    Sebti Kerbal

  5. FracDiff Research Group DR/RG/03, Sultan Qaboos University, Muscat, Oman

    Nasser Al-Salti, Erkinjon Karimov & Sebti Kerbal

Authors
  1. Nasser Al-Salti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erkinjon Karimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebti Kerbal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erkinjon Karimov.

Additional information

Communicated by Vasily E. Tarasov.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Al-Salti, N., Karimov, E. & Kerbal, S. A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative. Comp. Appl. Math. 42, 94 (2023). https://doi.org/10.1007/s40314-023-02231-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02231-y

Keywords

Mathematics Subject Classification

Search

Navigation