Far Eastern Mathematical Journal

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Difference methods for solving nonlocal boundary value problems for fractional-order differential convection-diffusion equations with memory effect


Beshtokov M.KH., KHudalov M.Z.

2021, issue 1, Ñ. 3-25
DOI: https://doi.org/10.47910/FEMJ202101


Abstract
In the present paper, in a rectangular domain, we study nonlocal boundary value problems for one-dimensional in space differential equations of convection-diffusion of fractional order with a memory effect, in which the unknown function appears in the differential expression and at the same time appears under the integral sign. The emergence of the integral term in the equation is associated with the need to take into account the dependence of the instantaneous values of the characteristics of the described object on their respective previous values, i.e. the effect of its prehistory on the current state of the system.

Keywords:
Boundary value problems, a priori estimate, differential equation of fractional order, fractional Caputo derivative, convection-diffusion equation, memory effect

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References

[1] K. B. Oldham, J. Spanier, “The fractional calculus. Theory and applications of differentia- tion and integration to arbitrary order”, New York: Academic Press, 1974.
[2] K. S. Miller, B. Ross, “An introduction to the fractional calculus and fractional differential equations”, New York: Wiley, Wiley and Sons, 1993.
[3] I. Podlubny, “Fractional differential equations”, San. Diego: Academic Press, 1999.
[4] V. E. Tarasov, Modeli teoreticheskoi fiziki s integro-differentsirovaniem drobnogo poriadka, Izhevsk: Izhevskii institut komp'iuternykh issledovanii, M., 2011.
[5] A. M. Nakhushev, Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003.
[6] V. V. Uchaikin, Metod drobnykh proizvodnykh, Artishok, Ul'ianovsk., 2008.
[7] B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, New York., 1982.
[8] S. V. Nerpin, A. F. Chudnovskii, Energo i massoobmen v sisteme pochva rastenie-vozdukh, Gidrometeoizdat, L., 1975.
[9] R. R. Nigmatullin, “Osobennosti relaksatsii sistemy s “ostatochnoi” pamiat'iu”, Fiz. tverdogo tela, 27:5 (1985), 1583–1585.
[10] A. A. Alikhanov, “Boundary value problems for the diffusion equation of the variable order in differential and difference settings”, Appl. Math., 219 (2012), 3938–394.
[11] A. A. Alikhanov, “A new difference scheme for the time fractional diffusion equation”, J. Comput. Phys, 280 (2015), 424–438.
[12] M. KH. Beshtokov, “To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov-Caputo fractional derivative”, Russian Mathematic, 62:10 (2018), 1–14.
[13] C. Ji, Z. Z. Sun, “A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion Equation”, Journal of Scientific ComputingRussian Mathematic, 64:3 (2014), 959–985.
[14] Y. Yan, Z. Z. Sun, J. Zhang, “Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme the Fractional Sub- diffusion Equation”, Commun. Comput. Phys., 22:4 (2017), 1028–1048.
[15] G. Gao, A. A. Alikhanov, Z. Z. Sun, “The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-Order Fractional Sub-diffusion Equations”, Journal of Scientific Computing, 73:1 (2017), 93–121.
[16] X. M. Gu, T. Zh. Huang, C. C. Ji, B. Carpentieri, A. A. Alikhanov, “Fast iterative method with a second-order implicit difference scheme for time-space fractional convection-diffusion equation”, J. Sci. Comp., 72 (2017), 957–985.
[17] H.Y. Jian, T.Z. Huang, X.L. Zhao, X.L. Zhao, “A fast second-order accurate difference schemes for time distributed-order and Riesz space fractional diffusion equations”, J. Appl. Anal. and Comp., 9:4 (2019), 1028–1048.
[18] M. KH. Beshtokov, “Local and nonlocal boundary value problems for degenerating and non- degenerating pseudoparabolic equations with a Riemann-Liouville fractional derivative”, Differential Equations, 54:6 (2018), 758–774.
[19] M.Kh. Beshtokov, “ Kraevye zadachi dlia psevdoparabolicheskogo uravneniia s drobnoi proizvodnoi Kaputo” , Differents. uravneniia, 55:7 (2019), 919–928.
[20] M. KH. Beshtokov, V. A. Vodakhova, “Nonlocal boundary value problems for a fractional order convection–diffusion equation”, Vestnik Udmurtskogo Universiteta. Matem- atika. Mekhanika. Komp’yuternye Nauki, 29:4 (2019), 459–482.
[21] M. Kh. Beshtokov, F. A. Erzhibova, “ K kraevym zadacham dlia integro- differentsial'nykh uravnenii drobnogo poriadka” , Matematicheskie trudy, 23:1 (2020), 16–36.
[22] M. KH. Beshtokov, M. Z. Khudalov, “Difference methods of the solution of local and non- local boundary value problems for loaded equation of thermal conductivity of fractional order”, Stability, Control and Differential Games, Springer Nature, 2020.
[23] A. A. Samarskii, Teoriia raznostnykh skhem, Nauka, M., 1983.
[24] A. A. Samarskii, A. V. Gulin, Ustoichivost' raznostnykh skhem, Nauka, M., 1973.

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