On the uniqueness of the solution of the Cauchy problem for the equation of fractional diffusion with Bessel operator

F. G. Khushtova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 4, pp. 774-784.

Résumé

In this paper, we consider fractional diffusion equation involving the Bessel operator acting with respect to a spatial variable and the Riemann-Liouville fractional differentiation operator acting with respect to a time variable. When the order of the fractional derivative is unity, and the singularity of the Bessel operator is absent, this equation coincides with the classical heat equation. Earlier, a solution of the Cauchy problem has been considered for the considered equation and a uniqueness theorem has been proved for a class of functions satisfying the analog of the Tikhonov condition. In this paper, we have constructed an example to show that the exponent (power) at the condition of the uniqueness of the solution to the Cauchy problem cannot be raised under. Its increase leads to a non-uniqueness of the solution. Using the well-known properties of the Wright function, we have obtained estimates for constructed function, which satisfies the homogeneous equation and the zero Cauchy condition.

Détail

Mots-clés : fractional diffusion equation
Keywords: fractional differentiation operator, Bessel operator, Cauchy problem, solution uniqueness, Tikhonov condition, Wright function.

Bibliographie

[1] Nakhushev A. M., Drobnoe ischislenie i ego primenenie [Fractional calculus and its applications], Fizmatlit, Moscow, 2003, 271 pp. (In Russian) | Zbl

[2] Pskhu A. V., Uravneniia v chastnykh proizvodnykh drobnogo poriadka [Partial differential equations of fractional order], Nauka, Moscow, 2005, 199 pp. (In Russian) | Zbl

[3] Alexiades V., “Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems”, Arch. Rational Mech. Anal., 79:4 (1982), 325–350 | DOI | MR | Zbl

[4] Calton D., “Cauchy's problem for a singular parabolic partial differential equation”, J. Diff. Equations, 8:2 (1970), 250–257 | DOI | MR

[5] Kȩpiński S., “Über die Differentialgleichung $\frac{\partial^2 z }{\partial x^2}+\frac{m+1}{x}\frac{\partial z }{\partial x} -\frac{n}{x}\frac{\partial z }{\partial t}=0$”, Math. Ann., 61:3 (1905), 397–405 (In German) | MR | Zbl

[6] Tersenov S. A., Parabolicheskie uravneniia s meniaiushchimsia napravleniem vremeni [Parabolic Equations with a Changing Time Direction], Nauka, Moscow, 1985, 105 pp.

[7] Matiichuk M. I., Parabolichni singuliarni kraiovi zadachi [Parabolic Singular Boundary-Value Problems], Institute of Mathematics of the Ukrainian National Academy of Sciences, Kiev, 1999, 176 pp. (In Ukrainian)

[8] Kipriyanov I. A., Singuliarnye ellipticheskie kraevye zadachi [Singular Elliptic Boundary-Value Problems], Nauka, Moscow, 1997, 208 pp. (In Russian)

[9] Kipriyanov I. A., Katrakhov V. V., Lyapin V. M., “On boundary value problems in domains of general type for singular parabolic systems of equations”, Sov. Math., Dokl., 17 (1976), 1461–1464 | MR | Zbl

[10] Sitnik S. M., Application of the Bushman–Erdélyi transformation operators and their generalizations in the theory of differential equations with singularities in the coefficients, Dr. Phys. Math. Sci. Thesis, Voronezh, 2016, 307 pp. (In Russian)

[11] Katrakhov V. V., Sitnik S. M., “The transmutation method and boundary-value problems for singular elliptic equations”, Singular differential equations, CMFD, 64, no. 2, Peoples' Friendship University of Russia, Moscow, 2018, 211–426 (In Russian) | DOI

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

[13] Kochubei A. N., “The Cauchy problem for evolution equations of fractional order”, Differ. Equ., 25:8 (1989), 967–974 | MR

[14] Kochubeii A. N., Éidel'man S. D., “The Cauchy problem for evolution equations of fractional order”, Dokl. Akad. Nauk, 394:2 (2004), 159–161 (In Russian) | MR | Zbl

[15] Kochubei A. N., “Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation”, Fract. Calc. Appl. Anal., 17:3 (2014), 881–896 | DOI | MR | Zbl

[16] Kochubei A. N., “Cauchy problem for fractional diffusion-wave equations with variable coefficients”, Applicable Analysis, 93:10 (2014), 2211–2242 | DOI | MR | Zbl

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

[18] Pskhu A. V., “Fractional diffusion equation with discretely distributed differentiation operator”, Sib. Èlektron. Mat. Izv., 13 (2016), 1078–1098 (In Russian) | DOI | MR | Zbl

[19] Pskhu A. V., “Multi-time fractional diffusion equation”, Eur. Phys. J. Spec. Top., 222:8 (2013), 1939–1950 | DOI

[20] Pskhu A. V., “The first boundary-value problem for a fractional diffusion-wave equation in a non-cylindrical domain”, Izv. Math., 81:6 (2017), 1212–1233 | DOI | DOI | MR | Zbl

[21] Voroshilov A. A., Kilbas A. A., “A Cauchy-type problem for a diffusion-wave equation with a Riemann-Liouville partial derivative”, Dokl. Akad. Nauk, 406:1 (2006), 12–16 (In Russian) | MR | Zbl

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

[23] Gekkieva S. Kh., “The Cauchy problem for the generalized transport equation with time-fractional derivative”, Dokl. Adyg. (Cherkess.) Mezdunar. Akad. Nauk, 5:1 (2000), 16–19 (In Russian)

[24] Gekkieva S. Kh., “A boundary value problem for the generalized transfer equation with a fractional derivative in a semi-infinite domain”, Izv. KBNTs RAN, 2002, no. 1(8), 6–8 (In Russian)

[25] Mainardi F., “The fundamental solutions for the fractional diffusion-wave equation”, Appl. Math. Letters, 6:1 (1996), 23–28 | DOI | MR | Zbl

[26] Mainardi F., “The time fractional diffusion-wave equation”, Radiophys. Quantum Electron., 38:1-2 (1995), 13–24 | DOI | MR

[27] Mainardi F., Luchko Yu., Pagnini G., “The fundamental solution of the space-time fractional diffusion equation”, Fract. Calc. Appl. Anal., 4:2 (2001), 153–192, arXiv: [cond-mat.stat-mech] cond-mat/0702419 | MR | Zbl

[28] Pagnini G., “The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes”, Fract. Calc. Appl. Anal., 16:2 (2013), 436–453 | DOI | MR | Zbl

[29] Pagnini G., Paradisi P., “A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation”, Fract. Calc. Appl. Anal., 19:2 (2016), 408–440 | DOI | MR | Zbl

[30] Podlubny I., Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999, xxiv+340 pp. | MR | Zbl

[31] Mathai A. M., Saxena R. K., Haubold H. J., The $H$-function. Theory and Applications, Springer, Dordrecht, 2010, xiv+268 pp. | DOI | MR | Zbl

[32] Khushtova F. G., “Cauchy problem for a parabolic equation with Bessel operator and Riemann–Liouville partial derivative”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 74–84 | DOI | Zbl

[33] Gorenflo R., Luchko Y., Mainardi F., “Analytical properties and applications of the Wright function”, Fract. Calc. Appl. Anal., 2:4 (1999), 383–414, arXiv: math-ph/0701069 | MR | Zbl

[34] Tychonoff A., “Théorèmes d'unicité pour l'équation de la chaleur”, Mat. Sb., 42:2 (1935), 199–216 (In French) | Zbl

[35] Tikhonov A. N., “Uniqueness theorems for the heat equation”, Collection of scientific works. In ten volumes, Vol. 2: Mathematics. Part 2: Numerical mathematics 1956–1979. Mathematical physics 1933–1948. Edited by T. A. Sushkevich and A. V. Gulin, Nauka, Moscow, 2009, 371–382 (In Russian and French) | Zbl

[36] Wright E. M., “The asymptotic expansion of the generalized hypergeometric function”, Proc. Lond. Math. Soc., II. Ser., 46:1 (1940), 389–408 | DOI | MR | Zbl