Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

Nonlocal Boundary Value Problems in Differential and Difference Interpretations for the Generalized Loaded Moisture Transfer Equation

Author(s):

M. KH. Beshtokov

Department of Computational Methods,
Institute of Applied Mathematics and Automation,
Kabardino-Balkarskii Scientific center RAS

beshtokov_murat@yandex.ru

Abstract:

We study boundary value problems for a spatially one-dimensional loaded moisture transfer equation of fractional-order with nonlocal boundary conditions. Using the method of energy inequalities, and assuming the existence of a solution of the problem, we derive a priori estimates for the solutions of nonlocal boundary value problems in differential form. Difference schemes are constructed and analogues of a priori estimates in difference form are proved for them; error estimates are given under the assumption of sufficient smoothness of the solutions of the equation. From the obtained a priori estimates, the uniqueness and stability of the solution with respect to the initial data and the right-hand side follow, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem at the rate equal to the approximation order of the difference problem.

Keywords

References:

  1. Nakhushev, A. M. Uravneniya matematicheskoi biologii [Equations of Mathematical Biology]. Moscow, Vysshaya Shkola, 1995. 301 p. (in Russian)
  2. Dzhenaliyev, M. T. [On a quadratic functional in the Cauchy problem for a loaded first-order differential operator equation] Differents. uravneniya, 31:12 (1995), 2029-2037; II // 32:4 (1996), 518-522. (in Russian)
  3. Cannon, J. R., Yin, N. M. On a class of nonlinear nonclassical parabolic problems // J. Different. Equat., 79 (1989), 266-288
  4. Samarskiy, A. A. [On some problems of the theory of differential equations] Differents. uravneniya, 16:11 (1980), 1925-1935. (in Russian)
  5. Uchaikin, V. V. Metod drobnykh proizvodnykh [Method of Fractional Derivatives]. Ul’yanovsk: Artishok, 2008. 512 p. (in Russian)
  6. Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1982. 468 p
  7. Podlubny. I. Fractional Differential Equations, Academic Press, San Diego, 1999. 340 p
  8. Chukbar, K. V. [Stochastic transfer and fractional derivatives] ZHETF. 108:5(11) (1995), 1875-1884. (in Russian)
  9. Kochubey, A. N. [Diffusion of fractional order] Differents. uravneniya, 26:4 (1990), 660-670. (in Russian)
  10. Nigmatullin, R. R. Fractional integral and its physical interpretation // Theoret. and Math. Phys., 90:3 (1992), 242-251
  11. Barenblatt, G. I., Zheltov, Yu. P., and Kochina, I. N. [On basic concepts of the theory of filtration of homogeneous liquids in cracked rocks] Prikl. Mat. Mekh. , 25:5 (1960), 852-864 (in Russian)
  12. Dzektser, Ye. S. [The equations of motion of groundwater with a free surface in multilayer environments] DAN SSSR, 220:3 (1975), 540-543. (in Russian)
  13. Rubinshteyn, L. I. [To the question of the process of heat distribution in heterogeneous media] AN SSSR, ser. geogr. , 12:1 (1948), 27-45. (in Russian)
  14. Ting, T. W. A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl, 45:9 (1974)
  15. Hallaire, M. On a theory of moisture-transfer // Inst. Rech. Agronom. , 3 (1964), 60-72
  16. Chudnovskiy, A. F. Teplofizika pochv [Thermophysics of soils]. Moscow, Nauka, 1976. 352 p. (in Russian)
  17. Sveshnikov, A. A., Al’shin, A. B., Korpusov, M. O., and Pletner, Yu. D. Lineinye i nelineinye uravneniya sobolevskogo tipa [Linear and Nonlinear Sobolev-Type Equations], Moscow, Fizmatlit, 2007. 732 p. (in Russian)
  18. Bedanokova, S. YU. [The equation of motion of soil moisture and a mathematical model of the moisture content of the soil layer based on the Hillaire’s equation], Vestnik Adygeyskogo gosudarstvennogo universiteta. Seriya 4: Yestestvenno matematicheskiye i tekhnicheskiye nauki. 4 (2007), 68-71. (in Russian)
  19. Beshtokov, M. KH. On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation // Differential Equations, 52:10 (2016), 1341-1354
  20. Beshtokov, M. KH. Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients // Comput. Math. Math. Phys., 56:10 (2016), 1763-1777
  21. Beshtokov, M. KH. Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution // Comput. Math. Math. Phys., 57:12 (2017), 1973-1993
  22. Beshtokov, M. KH. Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a Riemann-Liouville fractional derivative // Differential Equations, 54:6 (2018), 758-774
  23. Beshtokov, M. KH. Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative // Differential Equations, 55:7 (2019), 1-10
  24. Taukenova, F. I. and Shkhanukov-Lafishev, M. Kh. Difference methods for solving boundary value problems for fractional differential equations, Comput. Math. Math. Phys., 46:10 (2006), 1785-1795
  25. Diethelm, K., Walz, G. Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms, 16 (1997), 231-253
  26. Alikhanov, A. A. A priori estimates for solutions of boundary value problems for fractional-order equations, Differ. Equations, 46:5 (2010), 660-666
  27. Alikhanov, A. A. A new difference scheme for the time fractional diffusion equation. // Journal of computational physics, 280 (2015), 424-438
  28. Beshtokov, M. KH. To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov-Caputo fractional derivative // Russian Mathematics, 62:10 (2018), 1-14
  29. Samarskii, A. A. Teoriya raznostnykh skhem [Theory of difference schemes]. Moscow, Nauka, 1983. 616 p. (in Russian)
  30. Samarskii, A. A. and Gulin, A. V. Stabil’nost’ raznostnykh skhem [Stability of difference schemes]. Moscow, Nauka, 1973. 416 p. (in Russian)

Full text (pdf)