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Mat. Sb., 2011, Volume 202, Number 4, Pages 111–122 (Mi msb7645)  

This article is cited in 41 scientific papers (total in 41 papers)

Initial-value problem for a linear ordinary differential equation of noninteger order

A. V. Pskhu

Scientific Research Institute of Applied Mathematics and Automation, Kabardino-Balkar Scientific Centre of the Russian Academy of Sciences

Abstract: An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the equation; these conditions are necessary for the equation under consideration. The problem is reduced to an integral equation; an explicit representation of the solution in terms of the Wright function is constructed. As a consequence of these results, necessary and sufficient conditions for the solvability of the Cauchy problem are obtained.
Bibliography: 7 titles.

Keywords: fractional order derivative, Cauchy problem, differential equation of fractional order, Wright function, Hille-Tamarkin formula.

DOI: https://doi.org/10.4213/sm7645

Full text: PDF file (476 kB)
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English version:
Sbornik: Mathematics, 2011, 202:4, 571–582

Bibliographic databases:

UDC: 517.926
MSC: 34A99, 26A33
Received: 29.10.2009

Citation: A. V. Pskhu, “Initial-value problem for a linear ordinary differential equation of noninteger order”, Mat. Sb., 202:4 (2011), 111–122; Sb. Math., 202:4 (2011), 571–582

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Pskhu A.V., “Zadacha Koshi dlya lineinogo obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 13:1 (2011), 107–108  elib
    2. Gadzova L.Kh., “Kraevaya zadacha dlya lineinogo differentsialnogo uravneniya drobnogo poryadka s postoyannymi koeffitsientami”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 13:1 (2011), 47–49  elib
    3. Mamchuev M.O., “Obschee reshenie nagruzhennogo parabolicheskogo uravneniya vtorogo poryadka s postoyannymi koeffitsientami”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 14:1 (2012), 46–50  elib
    4. Mamchuev M.O., Kraevye zadachi dlya uravnenii i sistem uravnenii s chastnymi proizvodnymi drobnogo poryadka, Izdatelstvo KBNTs RAN, Nalchik, 2013  elib
    5. Gadzova L.Kh., “Zadacha Shturma-Liuvillya dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Materialy XI shkoly molodykh uchenykh «Nelokalnye kraevye zadachi i rodstvennye problemy sovremennogo analiza i informatiki», Izd-vo KBNTs RAN, 2013, 20–21  elib
    6. Gadzova L.Kh., “Zadacha dirikhle dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 15:2 (2013), 36–39  elib
    7. L. Kh. Gadzova, “Generalized Dirichlet problem for a fractional linear differential equation with constant coefficients”, Differ. Equ., 50:1 (2014), 122–127  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. B. I. Efendiev, “Initial-value problem for a continuous second-order differential equation”, Differ. Equ., 50:4 (2014), 562–567  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    9. L. Kh. Gadzova, “Zadacha Shturma-Liuvillya dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka s postoyannymi koeffitsientami”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 16:3 (2014), 12–16  elib
    10. Zh. A. Balkizov, “Analog zadachi Trikomi dlya uravneniya parabolo-giperbolicheskogo tipa tretego poryadka s operatorom Gellerstedta v oblasti giperbolichnosti”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 16:2 (2014), 20–27  elib
    11. L. Kh. Gadzova, “K teorii kraevykh zadach dlya differentsialnogo uravneniya drobnogo poryadka s postoyannymi koeffitsientami”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 16:2 (2014), 34–40  elib
    12. B. I. Efendiev, “Dirichlet Problem for Second-Order Ordinary Differential Equations with Segment-Order Derivative”, Math. Notes, 97:4 (2015), 632–640  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. V. Kolokoltsov, “On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations”, Fract. Calc. Appl. Anal., 18:4 (2015), 1039–1073  crossref  mathscinet  zmath  isi  elib  scopus
    14. Gadzova L.Kh., “Dirichlet and Neumann problems for a fractional ordinary differential equation with constant coefficients”, Differ. Equ., 51:12 (2015), 1556–1562  crossref  mathscinet  zmath  isi  scopus
    15. L. H. Gadzova, “On the asymptotics for the fundamental solution of the ordinary fractional order differential equation with constant coefficients”, Bulletin KRASEC. Phys. & Math. Sci., 13:2 (2016), 5–9  mathnet  crossref  crossref  mathscinet  elib
    16. L. Kh. Gadzova, “Zadacha Neimana dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Vladikavk. matem. zhurn., 18:3 (2016), 22–30  mathnet
    17. F.T. Bogatyreva, “Nachalnaya zadacha dlya uravneniya drobnogo poryadka s postoyannymi koeffitsientami”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2016, no. 4-1(16), 21–26  mathnet  crossref  mathscinet  elib
    18. R. Ashurov, A. Cabada, B. Turmetov, “Operator method for construction of solutions of linear fractional differential equations with constant coefficients”, Fract. Calc. Appl. Anal., 19:1 (2016), 229–252  crossref  mathscinet  zmath  isi  scopus
    19. M. B. Borikhanov, B. Kh. Turmetov, “On construction of solutions of linear fractional differential equations with constant coefficients”, International Conference on Analysis and Applied Mathematics (ICAAM 2016) (Almaty, Kazakhstan, 7–10 September 2016), AIP Conference Proceedings, 1759, eds. A. Ashyralyev, A. Lukashov, Amer. Inst. Physics, 2016, 020151  crossref  isi  scopus
    20. A. V. Pskhu, “Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator”, Differ. Equ., 52:12 (2016), 1610–1623  crossref  mathscinet  zmath  isi  scopus
    21. A. V. Pskhu, “Uravnenie drobnoi diffuzii s operatorom diskretno raspredelennogo differentsirovaniya”, Sib. elektron. matem. izv., 13 (2016), 1078–1098  mathnet  crossref  mathscinet  zmath
    22. M. G. Mazhgikhova, “Zadacha Neimana dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka s zapazdyvayuschim argumentom”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 2016, no. 2, 15–20  mathnet
    23. Zh. A. Balkizov, “Dirichlet boundary value problem for a third order parabolic-hyperbolic equation with degenerating type and order in the hyperbolicity domain”, Ufa Math. J., 9:2 (2017), 25–39  mathnet  crossref  isi  elib
    24. F.T. Bogatyreva, “Zadacha Dirikhle dlya uravneniya drobnogo poryadka s postoyannymi koeffitsientami”, Chelyab. fiz.-matem. zhurn., 2:4 (2017), 401–411  mathnet  elib
    25. R. Kh. Makaova, “Kraevaya zadacha dlya giperbolicheskogo uravneniya tretego poryadka s vyrozhdeniem poryadka vnutri oblasti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:4 (2017), 651–664  mathnet  crossref  zmath  elib
    26. L. Kh. Gadzova, “Nelokalnaya kraevaya zadacha dlya obyknovennogo differentsialnogo uravneniya s operatorom diskretno raspredelennogo differentsirovaniya”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 2017, no. 1, 12–18  mathnet
    27. M. G. Mazhgikhova, “Zadacha Koshi dlya obyknovennogo differentsialnogo uravneniya s operatorom Rimana-Liuvillya s zapazdyvayuschim argumentom”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 2017, no. 1, 24–28  mathnet
    28. Gadzova L.Kh., “Boundary Value Problem For a Linear Ordinary Differential Equation With a Fractional Discretely Distributed Differentiation Operator”, Differ. Equ., 54:2 (2018), 178–184  crossref  mathscinet  zmath  isi  scopus
    29. Mazhgikhova M.G., “Dirichlet Problem For a Fractional-Order Ordinary Differential Equation With Retarded Argument”, Differ. Equ., 54:2 (2018), 185–192  crossref  mathscinet  zmath  isi  scopus
    30. M. G. Mazhgikhova, “Kraevye zadachi dlya lineinogo obyknovennogo differentsialnogo uravneniya drobnogo poryadka s zapazdyvayuschim argumentom”, Sib. elektron. matem. izv., 15 (2018), 685–695  mathnet  crossref
    31. M. G. Mazhgikhova, “Nachalnaya i kraevaya zadachi dlya obyknovennogo differentsialnogo uravneniya drobnogo poryadka s zapazdyvayuschim argumentom”, Chelyab. fiz.-matem. zhurn., 3:1 (2018), 27–37  mathnet  elib
    32. L. Kh. Gadzova, “Zadacha Koshi dlya obyknovennogo differentsialnogo uravneniya s operatorom drobnogo diskretno raspredelennogo differentsirovaniya”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2018, no. 3(23), 48–56  mathnet  crossref  elib
    33. R. Kh. Makaova, “Zadacha Trikomi dlya vyrozhdayuschegosya vnutri oblasti giperbolicheskogo uravneniya tretego poryadka”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2018, no. 3(23), 67–75  mathnet  crossref  elib
    34. Zh. A. Balkizov, “Nelokalnaya kraevaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa tretego poryadka s vyrozhdeniem tipa i poryadka v oblasti ego giperbolichnosti”, Materialy mezhdunarodnoi nauchnoi konferentsii Aktualnye problemy prikladnoi matematiki i fiziki Kabardino-Balkariya, Nalchik, 1721 maya 2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 149, VINITI RAN, M., 2018, 14–24  mathnet  mathscinet
    35. L. Kh. Gadzova, “Kraevaya zadacha so smescheniem dlya lineinogo obyknovennogo differentsialnogo uravneniya s operatorom diskretno raspredelennogo differentsirovaniya”, Materialy mezhdunarodnoi nauchnoi konferentsii Aktualnye problemy prikladnoi matematiki i fiziki Kabardino-Balkariya, Nalchik, 1721 maya 2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 149, VINITI RAN, M., 2018, 25–30  mathnet  mathscinet
    36. R. Kh. Makaova, “Kraevaya zadacha dlya vyrozhdayuschegosya vnutri oblasti giperbolicheskogo uravneniya tretego poryadka s operatorom Allera v glavnoi chasti”, Materialy mezhdunarodnoi nauchnoi konferentsii Aktualnye problemy prikladnoi matematiki i fiziki Kabardino-Balkariya, Nalchik, 1721 maya 2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 149, VINITI RAN, M., 2018, 64–71  mathnet  mathscinet
    37. L. Kh. Gadzova, “Nonlocal Boundary-Value Problem for a Linear Ordinary Differential Equation with Fractional Discretely Distributed Differentiation Operator”, Math. Notes, 106:6 (2019), 904–908  mathnet  crossref  crossref  isi  elib
    38. S. Kh. Gekkieva, M. A. Kerefov, “Dirichlet boundary value problem for Aller–Lykov moisture transfer equation with fractional derivative in time”, Ufa Math. J., 11:2 (2019), 71–81  mathnet  crossref  isi
    39. M. A. Kerefov, S. Kh. Gekkieva, “Vtoraya kraevaya zadacha dlya obobschennogo uravneniya vlagoperenosa Allera–Lykova”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 23:4 (2019), 607–621  mathnet  crossref
    40. M. G. Mazhgikhova, “Kraevaya zadacha so smescheniem dlya differentsialnogo uravneniya drobnogo poryadka s zapazdyvayuschim argumentom”, Vestnik KRAUNTs. Fiz.-mat. nauki, 28:3 (2019), 16–25  mathnet  crossref
    41. Kolokoltsov V.N., “The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations”, Fract. Calc. Appl. Anal., 22:3 (2019), 543–600  crossref  mathscinet  zmath  isi
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