Geodesic
The first boundary-value problem for a~fractional diffusion-wave equation in a~non-cylindrical domain
Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1212-1233.

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We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan–Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient condition for unique solubility is the condition of Hölder smoothness for the lateral boundary of the domain. The corresponding results for equations with Riemann–Liouville and Caputo derivatives are particular cases of results obtained here.
Keywords: first boundary-value problem, fractional derivative, Dzhrbashyan–Nersesyan operator
Mots-clés : diffusion-wave equation, non-cylindrical domain. 
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A. V. Pskhu. The first boundary-value problem for a~fractional diffusion-wave equation in a~non-cylindrical domain. Izvestiya. Mathematics , Tome 81 (2017) no. 6, pp. 1212-1233. http://geodesic.mathdoc.fr/item/IM2_2017_81_6_a7/

[1] M. M. Dzhrbashyan, A. B. Nersesyan, “Drobnye proizvodnye i zadachi Koshi dlya differentsialnykh uravnenii drobnogo poryadka”, Izv. AN Arm. SSR, 3:1 (1968), 3–29 | MR | Zbl

[2] A. M. Nakhushev, Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp. | Zbl

[3] V. V. Uchaikin, Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008, 512 pp.

[4] F. Mainardi, Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models, Imperial College Press, London, 2010, xx+347 pp. | DOI | MR | Zbl

[5] V. E. Tarasov, Modeli teoreticheskoi fiziki s integro-differentsirovaniem drobnogo poryadka, In-t kompyuternykh issledovanii, M.–Izhevsk, 2011, 568 pp.

[6] T. M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional calculus with applications in mechanics. Vibrations and diffusion processes, Mech. Eng. Solid Mech. Ser., ISTE, London; John Wiley Sons, Inc., Hoboken, NJ, 2014, xiii+315 pp. ; Fractional calculus with applications in mechanics. Wave propagation, impact and variational principles, xvi+406 pp. | DOI | Zbl | DOI | MR | Zbl

[7] W. Wyss, “The fractional diffusion equation”, J. Math. Phys., 27:11 (1986), 2782–2785 | DOI | MR | Zbl

[8] W. R. Schneider, W. Wyss, “Fractional diffusion and wave equations”, J. Math. Phys., 30:1 (1989), 134–144 | DOI | MR | Zbl

[9] A. N. Kochubei, “Diffusion of fractional order”, Differential Equations, 26:4 (1990), 485–492 | MR | Zbl

[10] Ya. Fujita, “Integrodifferential equation which interpolates the heat equation and the wave equation”, Osaka J. Math., 27:2 (1990), 309–321 ; II, 27:4, 797–804 | MR | Zbl | MR | Zbl

[11] S. Kh. Gekkieva, “Kraevaya zadacha dlya obobschennogo uravneniya perenosa s drobnoi proizvodnoi po vremeni”, Dokl. Adygskoi (Cherkesskoi) mezhdunar. AN, 1:1 (1994), 17–18

[12] F. Mainardi, “The fundamental solutions for the fractional diffusion-wave equation”, Appl. Math. Lett., 9:6 (1996), 23–28 | DOI | MR | Zbl

[13] V. Kh. Shogenov, S. K. Kumykova, M. Kh. Shkhanukov-Lafishev, “Obobschennoe uravnenie perenosa i drobnye proizvodnye”, Dokl. NAN Ukrainy, 1997, no. 12, 47–54 | MR | Zbl

[14] E. Buckwar, Yu. Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations”, J. Math. Anal. Appl., 227:1 (1998), 81–97 | DOI | MR | Zbl

[15] R. Gorenflo, Yu. Luchko, F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation”, J. Comput. Appl. Math., 118:1-2 (2000), 175–191 | DOI | MR | Zbl

[16] R. Gorenflo, A. Iskenderov, Yu. Luchko, “Mapping between solutions of fractional diffusion-wave equations”, Fract. Calc. Appl. Anal., 3:1 (2000), 75–86 | MR | Zbl

[17] O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynam., 29:1-4 (2002), 145–155 | DOI | MR | Zbl

[18] A. V. Pskhu, “Solution of the first boundary value problem for a fractional-order diffusion equation”, Differ. Equ., 39:9 (2003), 1359–1363 | DOI | MR | Zbl

[19] A. V. Pskhu, “Solution of boundary value problems for the fractional diffusion equation by the Green function method”, Differ. Equ., 39:10 (2003), 1509–1513 | DOI | MR | Zbl

[20] S. D. Eidelman, A. N. Kochubei, “Cauchy problem for fractional diffusion equations”, J. Differential Equations, 199:2 (2004), 211–255 | DOI | MR | Zbl

[21] A. A. Voroshilov, A. A. Kilbas, “The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative”, Differ. Equ., 42:5 (2006), 638–649 | DOI | MR | Zbl

[22] A. V. Pskhu, “Pervaya kraevaya zadacha dlya diffuzionno-volnovogo uravneniya drobnogo poryadka”, Issledovaniya po differentsialnym uravneniyam i matematicheskomu modelirovaniyu, VNTs RAN, Vladikavkaz, 2009, 235–242

[23] A. V. Pskhu, “The fundamental solution of a diffusion-wave equation of fractional order”, Izv. Math., 73:2 (2009), 351–392 | DOI | DOI | MR | Zbl

[24] Yu. Luchko, “Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation”, Fract. Calc. Appl. Anal., 15:1 (2012), 141–160 | DOI | MR | Zbl

[25] M. O. Mamchuev, “Neobkhodimye nelokalnye usloviya dlya diffuzionno-volnovogo uravneniya”, Vestn. SamGU. Estestvennonauchn. ser., 2014, no. 7(118), 45–59 | Zbl

[26] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., 204, Elsevier Science B.V., Amsterdam, 2006, xvi+523 pp. | MR | Zbl

[27] A. V. Pskhu, Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005, 200 pp. | MR | Zbl

[28] M. Gevrey, “Sur les équations aux dérivées partielles du type parabolique”, J. Math. Pures Appl. (6), 9 (1913), 305–476 | Zbl

[29] I. Petrowsky, “Zur ersten Randwertaufgabe der Wärmeleitungsgleichung”, Compositio Math., 1 (1935), 383–419 | MR | Zbl

[30] E. M. Wright, “On the coefficients of power series having exponential singularities”, J. London Math. Soc., 8:1 (1933), 71–79 | DOI | MR | Zbl

[31] E. M. Wright, “The generalized Bessel function of order greater than one”, Quart. J. Math., Oxford Ser., 11 (1940), 36–48 | DOI | MR | Zbl

[32] V. A. Ilin, V. A. Sadovnichii, Bl. X. Sendov, Matematicheskii analiz, v. 2, Prodolzhenie kursa, 2-e izd., Izd-vo MGU, M., 1987, 358 pp. | MR | Zbl

[33] A. V. Pskhu, “Solution of a boundary value problem for a fractional partial differential equation”, Differ. Equ., 39:8 (2003), 1150–1158 | DOI | MR | Zbl

[34] A. V. Pskhu, “On a boundary value problem for a fractional partial differential equation in a domain with curvilinear boundary”, Differ. Equ., 51:8 (2015), 1072–1078 | DOI | DOI | MR | Zbl