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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 141–182 (Mi izv2429)  

This article is cited in 42 scientific papers (total in 42 papers)

The fundamental solution of a diffusion-wave equation of fractional order

A. V. Pskhu

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

Abstract: We construct a fundamental solution of a diffusion-wave equation with Dzhrbashyan–Nersesyan fractional differentiation operator with respect to the time variable. We prove reduction formulae and solve the problem of sign-determinacy for the fundamental solution. A general representation for solutions is constructed. We give a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions satisfying an analogue of Tychonoff's condition. It is shown that our fundamental solution yields the corresponding solutions for the diffusion and wave equations when the order of the fractional derivative is equal to 1 or tends to 2. The corresponding results for equations with Riemann–Liouville and Caputo derivatives are obtained as particular cases of our assertions.

Keywords: fundamental solution, diffusion equation of fractional order, wave equation of fractional order, diffusion-wave equation, Dzhrbashyan–Nersesyan fractional differentiation operator, Riemann–Liouville derivative, Caputo derivative, Tychonoff's condition, Wright's function, Cauchy problem.

DOI: https://doi.org/10.4213/im2429

Full text: PDF file (752 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:2, 351–392

Bibliographic databases:

UDC: 517.95
MSC: 26A33, 35A08, 35S10, 45K05
Received: 14.11.2006
Revised: 24.12.2007

Citation: A. V. Pskhu, “The fundamental solution of a diffusion-wave equation of fractional order”, Izv. RAN. Ser. Mat., 73:2 (2009), 141–182; Izv. Math., 73:2 (2009), 351–392

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. O. Mamchuev, “Fundamentalnoe reshenie uravneniya drobnoi diffuzii s peremennymi koeffitsientami”, Trudy sedmoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem (3–6 iyunya 2010 g.). Chast 3, Differentsialnye uravneniya i kraevye zadachi, Matem. modelirovanie i kraev. zadachi, Samarskii gosudarstvennyi tekhnicheskii universitet, Samara, 2010, 170–173  mathnet
    2. Mamchuev M.O., “Fundamentalnoe reshenie uravneniya drobnoi diffuzii s peremennymi koeffitsientami”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 13:2 (2011), 33–37  elib
    3. Nakhusheva V.A., “O nelokalnykh differentsialnykh uravneniyakh matematicheskikh modelei nekotorykh stokhasticheskikh protsessov”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 13:1 (2011), 90–97  mathscinet  elib
    4. Kochubei A.N., “Fractional-parabolic systems”, Potential Anal., 37:1 (2012), 1–30  crossref  mathscinet  zmath  isi  elib
    5. Alikhanov A.A., “Boundary value problems for the diffusion equation of the variable order in differential and difference settings”, Appl. Math. Comput., 219:8 (2012), 3938–3946  crossref  mathscinet  zmath  isi  elib
    6. Mamchuev M.O., “Vidoizmenennaya zadacha Koshi dlya nagruzhennogo parabolicheskogo uravneniya vtorogo poryadka s postoyannymi koeffitsientami”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 14:2 (2012), 22–28  elib
    7. 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
    8. A. V. Pskhu, “Multi-time fractional diffusion equation”, Eur. Phys. J. Spec. Top., 222:8 (2013), 1939–1950  crossref  isi
    9. A. N. Kochubei, “Fractional-hyperbolic systems”, Fract. Calc. Appl. Anal., 16:4 (2013), 860–873  crossref  mathscinet  zmath  isi
    10. M. Krasnoschok, N. Vasylyeva, “Existence and Uniqueness of the Solutions for Some Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition”, International Journal of Partial Differential Equations, 2013 (2013), 796430, 20 pp.  crossref  zmath
    11. Masaeva O.Kh., “Dirichlet Problem for a Nonlocal Wave Equation”, Differ. Equ., 49:12 (2013), 1518–1523  crossref  mathscinet  zmath  isi
    12. Nakhusheva V.A., “On Some Fractal Differential Equations of Mathematical Models of Catastrophic Situations”, Differ. Equ., 49:4 (2013), 487–493  crossref  mathscinet  zmath  isi  elib
    13. A. N. Kochubei, “Cauchy problem for fractional diffusion-wave equations with variable coefficients”, Appl. Anal., 93:10 (2014), 2211–2242  crossref  mathscinet  zmath
    14. N. V. Vasil'eva, N. V. Krasnoshchek, “On the local solvability of the two-dimensional Hele–Shaw problem with fractional derivative with respect to time”, Siberian Adv. Math., 25:4 (2015), 276–296  mathnet  crossref  mathscinet
    15. M. O. Mamchuev, “Neobkhodimye nelokalnye usloviya dlya diffuzionno-volnovogo uravneniya”, Vestn. SamGU. Estestvennonauchn. ser., 2014, no. 7(118), 45–59  mathnet
    16. M. O. Mamchuev, “Obschee predstavlenie reshenii drobnogo telegrafnogo uravneniya”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 16:2 (2014), 47–51  elib
    17. Krasnoschok M., Vasylyeva N., “On a Nonclassical Fractional Boundary-Value Problem For the Laplace Operator”, J. Differ. Equ., 257:6 (2014), 1814–1839  crossref  mathscinet  zmath  isi
    18. Kochubei A.N., “Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation”, Fract. Calc. Appl. Anal., 17:3 (2014), 881–896  crossref  mathscinet  zmath  isi
    19. O. Kh. Masaeva, “Necessary and sufficient conditions for the uniqueness of the Dirichlet problem for nonlocal wave equation”, Bulletin KRASEC. Phys. & Math. Sci., 11:2 (2015), 19–23  mathnet  crossref  crossref  elib
    20. F. G. Khushtova, “Fundamentalnoe reshenie modelnogo uravneniya anomalnoi diffuzii drobnogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:4 (2015), 722–735  mathnet  crossref  zmath  elib
    21. F. G. Khushtova, “Zadacha Koshi dlya uravneniya parabolicheskogo tipa s operatorom Besselya i chastnoi proizvodnoi Rimana–Liuvillya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:1 (2016), 74–84  mathnet  crossref  zmath  elib
    22. 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
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    24. L. L. Karasheva, “Otsenka fundamentalnogo resheniya uravneniya parabolicheskogo tipa vysokogo poryadka s proizvodnoi Rimana-Liuvillya po vremennoi peremennoi”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2016, no. 4-1(16), 32–37  mathnet  crossref  mathscinet  elib
    25. F. G. Khushtova, “Second boundary-value problem in a half-strip for equation of parabolic type with the Bessel operator and Riemann–Liouville derivative”, Russian Math. (Iz. VUZ), 61:7 (2017), 73–82  mathnet  crossref  isi
    26. Roscani S.D., “Moving-Boundary Problems For the Time-Fractional Diffusion Equation”, Electron. J. Differ. Equ., 2017, 44  mathscinet  zmath  isi
    27. Chen L., Hu G., Hu Ya., Huang J., “Space-Time Fractional Diffusions in Gaussian Noisy Environment”, Stochastics, 89:1 (2017), 171–206  crossref  mathscinet  zmath  isi  elib  scopus
    28. A. V. Pskhu, “The first boundary-value problem for a fractional diffusion-wave equation in a non-cylindrical domain”, Izv. Math., 81:6 (2017), 1212–1233  mathnet  crossref  crossref  adsnasa  isi  elib
    29. F. G. Khushtova, “Dirichlet boundary value problem in half-strip for fractional differential equation with Bessel operator and Riemann–Liouville partial derivative”, Ufa Math. J., 9:4 (2017), 114–126  mathnet  crossref  isi  elib
    30. Mathai A.M., Haubold H.J., “Essentials of Fractional Calculus”: Mathai, AM Haubold, HJ, Fractional and Multivariable Calculus: Model Building and Optimization Problems, Springer Optimization and Its Applications, 122, Springer International Publishing Ag, 2017, 1–37  crossref  mathscinet  isi
    31. Yaseen M., Abbas M., Nazir T., Baleanu D., “A Finite Difference Scheme Based on Cubic Trigonometric B-Splines For a Time Fractional Diffusion-Wave Equation”, Adv. Differ. Equ., 2017, 274  crossref  mathscinet  isi
    32. Kim I., Kim K.-H., Lim S., “An l-Q(l-P)-Theory For the Time Fractional Evolution Equations With Variable Coefficients”, Adv. Math., 306 (2017), 123–176  crossref  mathscinet  zmath  isi
    33. Rekhviashvili S.Sh., Alikhanov A.A., “Simulation of Drift-Diffusion Transport of Charge Carriers in Semiconductor Layers With a Fractal Structure in An Alternating Electric Field”, Semiconductors, 51:6 (2017), 755–759  crossref  isi
    34. Chen H., Lu Sh., Chen W., “A Unified Numerical Scheme For the Multi-Term Time Fractional Diffusion and Diffusion-Wave Equations With Variable Coefficients”, J. Comput. Appl. Math., 330 (2018), 380–397  crossref  mathscinet  zmath  isi
    35. Roscani S.D., Tarzia D.A., “Explicit Solution For a Two-Phase Fractional Stefan Problem With a Heat Flux Condition At the Fixed Face”, Comput. Appl. Math., 37:4 (2018), 4757–4771  crossref  mathscinet  isi  scopus
    36. L. L. Karasheva, “Zadacha Koshi dlya parabolicheskogo uravneniya vysokogo chetnogo poryadka s drobnoi proizvodnoi po vremennoi peremennoi”, Sib. elektron. matem. izv., 15 (2018), 696–706  mathnet  crossref
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    38. Pskhu A.V., “On Solution Uniqueness of the Cauchy Problem For a Third-Order Partial Differential Equation With Time-Fractional Derivative”, AIP Conference Proceedings, 1997, eds. Ashyralyev A., Lukashov A., Sadybekov M., Amer Inst Physics, 2018, UNSP 020059-1  crossref  isi  scopus
    39. F. G. Khushtova, “K probleme edinstvennosti resheniya zadachi Koshi dlya uravneniya drobnoi diffuzii s operatorom Besselya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:4 (2018), 774–784  mathnet  crossref  elib
    40. R. M. Dzhafarov, N. V. Krasnoshchek, “The Cauchy problem for the fractional diffusion equation in a weighted Hölder space”, Siberian Math. J., 59:6 (2018), 1034–1050  mathnet  crossref  crossref  isi
    41. Jin B., Lazarov R., Zhou Zh., “Numerical Methods For Time-Fractional Evolution Equations With Nonsmooth Data: a Concise Overview”, Comput. Meth. Appl. Mech. Eng., 346 (2019), 332–358  crossref  mathscinet  isi
    42. Pskhu A., “Fundamental Solutions and Cauchy Problems For An Odd-Order Partial Differential Equation With Fractional Derivative”, Electron. J. Differ. Equ., 2019, 21  zmath  isi
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