Abstract
For the biparabolic partial differential evolution equation and its fractional differential generalization, statements are made and closed-form solutions of some boundary-value problems with nonlocal boundary conditions are obtained. Variants of direct and inverse problem statements are considered. The mathematical formulation of the inverse problem involves the search, together with the solution of the original integro-differential equation of fractional order, of its unknown right-hand side as well, which functionally depends only on the geometric variable.
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References
G. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications), Oxford University Press (1986).
E. I. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids [in Russian], Vysshaya Shkola, Moscow (1979).
A. V. Lykov, Heat and Mass Exchange [in Russian], Energiya, Moscow (1978).
C. Cattaneo, “Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Vol. 247, No. 4, 431–433 (1958).
V. I. Fushchich, A. S. Galitsyn, and A. S. Polubinskii, “A new mathematical model of heat conduction processes,” Ukr. Math. J., Vol. 42, No. 2, 210–216 (1990).
V. I. Fushchich, “Symmetry and partial solutions to some multidimensional equations of mathematical physics,” Theoretical-Algebraic Methods in Problems of Mathematical Physics [in Russian], Inst. Math. AS UkrSSR (1983), pp. 4–22.
V. M. Bulavatsky, “A biparabolic mathematical model of the filtration consolidation process,” Dopov. Nac. Akad. Nauk. Ukr., No. 8, 13–17 (1997).
V. M. Bulavatsky, “Mathematical modeling of filtrational consolidation of soil under motion of saline solutions on the basis of biparabolic model,” J. Autom. Inform. Sci., Vol. 35, No. 8, 13–22 (2003).
V. M. Bulavatsky and V. V. Skopetsky, “Generalized mathematical model of the dynamics of consolidation processes with relaxation,” Cybern. Syst. Analysis, Vol. 44, No. 5, 646–654 (2008).
V. V. Uchaikin, The Method of Fractional Derivatives [in Russian], Artishok, Ulyanovsk (2008).
M. M. Djrbashian, Harmonic Analysis and Boundary-Value Problems in the Complex Domain, Springer Basel AG, Basel (1993).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Sci. Publ., Philadelphia (1993).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).
M. Caputo, “Models of flux in porous media with memory,” Water Resources Research, Vol. 36, 693–705 (2000).
A. M. Nakhushev, Fractional Calculus and its Application [in Russian], Fizmatlit, Moscow (2003).
R. P. Meilanov, V. D. Beibalaev, and M. R. Shibanova, Applied Aspects of Fractional Calculus, Palmarium Acad. Publ., Saarbrucken (2012).
T. Sandev, R. Metzler, and Z. Tomovski, “Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative,” J. of Physics A, Vol. 44, 5–52 (2011).
Z. Tomovski, T. Sandev, R. Metzler, and J. Dubbeldam, “Generalized space–time fractional diffusion equation with composite fractional time derivative,” Physica A, Vol. 391, 2527–2542 (2012).
K. M. Furati, O. S. Iyiola, and M. Kirane, “An inverse problem for a generalized fractional diffusion,” Applied Mathematics and Computation, Vol. 249, 24–31 (2014).
V. M. Bulavatsky and V. A. Bogaenko, “Mathematical modelling of the fractional differential dynamics of the relaxation process of convective diffusion under conditions of planed filtration,” Cybern. Syst. Analysis, Vol. 51, No. 6, 886–895 (2015).
V. M. Bulavatsky, “Fractional differential analog of biparabolic evolution equation and some its applications,” Cybern. Syst. Analysis, Vol. 52, No. 5, 737–747 (2016).
N. I. Ionkin, “Solution of one boundary-value problem of the theory of thermal conductivity with the nonclassical boundary condition,” Diff. Uravneniya, Vol. 13, No. 2, 294–304 (1977).
E. I. Moiseyev, “Solving one nonlocal boundary-value problem by the spectral method,” Diff. Uravneniya, Vol. 35, No. 8, 1094–1100 (1999).
V. M. Bulavatsky, Iu. G. Kryvonos, and V. V. Skopetsky, Nonclassical Mathematical Models of Heat and Mass Transfer Processes [in Ukrainian], Naukova Dumka, Kyiv (2005).
I. A. Kaliev and M. M. Sabitova, “Problems of determining the temperature and density of heat sources from the initial and terminal temperatures,” Sibirskii Zhurnal Industr. Matem., Vol. 12, No. 1 (37), 89–97 (2009).
R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin (2014).
A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Integral Transforms and Special Functions, Vol. 15, No. 1, 31–49 (2004).
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Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2019, pp. 106–114.
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Bulavatsky, V.M. Some Nonlocal Boundary-Value Problems for the Biparabolic Evolution Equation and Its Fractional-Differential Analog. Cybern Syst Anal 55, 796–804 (2019). https://doi.org/10.1007/s10559-019-00190-z
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DOI: https://doi.org/10.1007/s10559-019-00190-z