Abstract: Interest in fractional order equations, both ordinary and partial, has been steadily growing in recent decades. This is due to the need to model processes in which the current state significantly depends on the previous states of the process, i.e. the so-called systems with "residual'' memory. The paper considers the Cauchy problem for a one-dimensional, homogeneous Euler-Poisson-Darboux equation with a differential operator of fractional order in time, which is a left-sided Bessel operator of fractional order. At the same time, the usual second-order differential operator is used for the spatial variable. The connection between the Meyer and Laplace transformation obtained using the Poisson transformation, which is a special case of the relation with the Obreshkov transformation, is shown. A theorem is proved that determines the conditions for the existence of a solution to the problem under consideration. When proving the theorem of the existence of a solution, the Meyer transform was used. In this case, the solution of the problem is presented explicitly through the generalized Green's function. The Green function constructed to solve the problem under consideration is defined by means of the generalized hypergeometric Fox \(H\)-function.
Keywords: fractional powers of Bessel operator, fractional Euler-Poisson-Darboux equation, Meijer integral transform, \(H\)-function.
For citation: Dzarakhohov, A. V. and Shishkina, E. L. Solution to the Fractional Order Euler-Poisson-Darboux Equation, Vladikavkaz Math. J., 2022, vol. 24, no. 2, pp. 85-100 (in Russian). DOI 10.46698/t3110-3630-4771-f
1. Kipriyanov, I. A. Singulyarnyye ellipticheskiye krayevyye zadachi
[Singular Elliptic Boundary Value Problems], Moscow, Nauka, 1997, 204 p. (in Russian).
2. Goldstein, S. On Diffusion by Discontinuous Movements and Thetelegraph Equation,
The Quarterly Journal of Mechanics and Applied Mathematics, 1951,
vol. 4, pp. 129-156. DOI: 10.1093/qjmam/4.2.129.
3. Katz, M. A Stochastic Model Related to the Telegrapher's Equation,
Rocky Mountain Journal of Mathematics, 1974, vol. 4, pp. 497-509.
DOI: 10.1216/RMJ-1974-4-3-497.
4. Orsingher, E. Hyperbolic Equations Arising in Random Models,
Stochastic Processes and their Applications, 1985, no. 21, pp. 93-106.
DOI: 10.1016/0304-4149(85)90379-5.
5. Orsingher, E. E. A Planar Random Motion Governed by the Two-Dimensional Telegraph Equation,
Stochastic Processes and their Applications, 1986, no. 23, pp. 385-397.
DOI: 10.2307/3214181.
6. Orsingher, E. Probability Law, Flow Function, Maximum Distribution
of Wave-Governed Random Motions, and their Connections with Kirchhoff's Laws,
Stochastic Processes and their Applications, 1990, no. 34, pp. 49-66. DOI: 10.1016/0304-4149(90)90056-X.
7. De Gregorio, A. and Orsingher, E. Random Flights Connecting Porous Medium and Euler-Poisson-Darboux Equations,
Journal of Mathematical Physics, 2020, vol. 61, no. 4, 041505, 18 pp. DOI: 10.1063/1.5121502.
8. Garra, R. and Orsingher, E. Random Flights Related to the Euler-Poisson-Darboux Equation,
Markov Processes and Related Fields, 2016, no. 22, pp. 87-110.
9. Iacus, S. Statistical Analysis of the Inhomogeneous Telegrapher's Process,
Statistics & Probability Letters, 2001, no. 55, pp. 83-88.
DOI: 10.1016/S0167-7152(01)00133-X.
10. Metzler, R. and Klafter, J. The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach,
Physics Reports, 2000, no. 339, pp. 1-77. DOI: 10.1016/S0370-1573(00)00070-3.
11. Gorenflo, R. R., Vivoli, A. and Mainardi, F. Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion, Nonlinear Dynamics, 2004, vol. 38, pp. 101-116. DOI: 10.1007/s10958-006-0006-0.
12. De Gregorio, A. and Orsingher, E. Flying Randomly in \(R^d\) with Dirichlet Displacements,
Stochastic Processes and their Applications, 2012, vol. 122, no. 2, pp. 676-713.
DOI: 10.1016/j.spa.2011.10.009.
13. Watson, G. N.Teoriya besselevykh funktsiy [A Treatise on the Theory of Bessel Functions],
Moscow, IL, 1949, 728 p. (in Russian).
14. Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions with Formulas,
Graphs and Mathematical Tables, New York, Dover Publ. Inc., 1972, 1060 p.
15. Kiryakova, V.Generalized Fractional Calculus and Applications,
New York, Pitman Res. Notes Math., Longman Scientific & Technical, Harlow, Co-publ. John Wiley,
301, 1994, 388 p.
16. Gorenflo, R., Kilbas, A. A., Mainardi, F. and Rogosin, S. V. Mittag-Leffler Functions, Related Topics and Applications,
Berlin/Heidelberg, Germany, Springer, 2016, 443 p.
17. Luchko, Yu. Algorithms for Evaluation of the Wright Function for the Real Arguments’ Values,
Fractional Calculus and Applied Analysis, 2008, vol. 11, pp. 57-75.
18. Kilbas, A. A. and Saigo, M.H-Transforms. Theory and Applications, Boca Raton, Florida, Chapman and Hall,
2004, 408 p. DOI: 10.1201/9780203487372.
19. Stankovic, B. On the Function of E. M. Wright,
Publications de l'Institut Mathematique, Nouvelle Ser.,
1970, vol. 10, no. 24, pp. 113-124.
20. Glaeske, H. J., Prudnikov, A. P. and Skornik, K. A.Operational Calculus and Related Topics, New York,
Chapman and Hall/CRC, 2006, 424 p. DOI: 10.1201/9781420011494.
21. Samko, S. G., Kilbas, A. A. and Marichev, O. I. Integraly i proizvodnyye drobnogo poryadka i nekotoryye ikh prilozheniya
[Integrals and Derivatives of Fractional Order and Some of their Applications],
Minsk, Nauka i tekhnika, 1987, 688 ñ.
22. Sprinkhuizen-Kuyper, I. G. A Fractional Integral Operator Corresponding to Negative Powers of a Certain Second-Order Differential Operator, Journal of Mathematical Analysis and Applications,
1979, vol. 72, pp. 674-702. DOI: 10.1016/0022-247x(79)90257-9.
23. McBride, A. C.Fractional calculus and Integral Transforms of Generalized Functions,
London, Pitman, 1979, 179 p.
24. Shishkina, E. L. and Sitnik, S. M. On Fractional Powers of Bessel Operators,
Journal of Inequalities and Special Functions,
Special Issue to Honor Prof. Ivan Dimovski's Contributions, 2017, vol. 8, no. 1,
pp. 49-67.
25. Shishkina, E. L. and Sitnik, S. M. A Fractional Equation with
Left-Sided Fractional Bessel Derivatives of Gerasimov-Caputo Type,
Mathematics, 2019, vol. 7, no. 12, pp. 1-21. DOI: 10.3390/math7121216.
26. Gerasimov, A. N. A Generalization of Linear Laws of Deformation
and its Applicationto Problems of Internal Friction, Akad. Nauk SSSR, Prikl. Mat. Mekh. T. 12.
[Academy of Sciences of the USSR. Applied Mathematics and Mechanics, vol. 12], 1948, pp. 529-539
(in Russian).
27. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations,
Amsterdam, Elsevier, 2006, 523 p.
