Abstract: The work is devoted to nonlocal boundary value problems for one-dimensional space-loaded Hallaire equations with variable coefficients and two Caputo fractional differentiation operators with orders \(\alpha\) and \(\beta\). Similar problems arise in the practice of regulating the salt regime of soils, when the stratification of the upper layer is achieved by draining the water layer from the surface of a site flooded for some time. Difference schemes are constructed for the numerical solution of the problems posed on a uniform grid. Using the method of energy inequalities for various relations between the orders of the fractional Caputo derivative \(\alpha\) and \(\beta\), we obtain a priori estimates in differential and difference interpretations for solutions of nonlocal boundary value problems. The obtained a priori estimates imply uniqueness and stability of the solution with respect to the right-hand side and the initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding original differential problem (assuming the existence of a solution to the differential problem in the class of sufficiently smooth functions) at a rate of \(O(h^2+ \tau^2)\) for \(\alpha=\beta\) and \(O(h^2+\tau^{2-\max\{\alpha,\beta\}})\) for \(\alpha\neq\beta \). The paper also presents an algorithm for the numerical solution of a nonlocal boundary value problem for a loaded Hallaire equation with variable coefficients and a Bessel operator.
Keywords: nonlocal boundary value problems, a priori estimate, Hallaire equation, loaded equations, generalized moisture transfer equation, Caputo fractional derivative
For citation: Beshtokov, M. Kh. Numerical Methods for Solving Nonlocal Boundary Value Problems for Generalized Loaded Hallaire Equations, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 15-35 (in Russian).
DOI 10.46698/c8748-9711-0633-d
1. Nakhushev, A. M. Drobnoe ischislenie i ego primenenie [Fractional Calculus
and Its Application], Moscow, Fizmatlit, 2003, 272 p. (in Russian).
2. Uchaykin, V. V. Metod drobnykh proizvodnykh [Method of Fractional Derivatives],
Ul'yanovsk, Artishok, 2008, 512 p. (in Russian).
3. Samko, S. G., Kilbas, A. A. and Marichev, O. I. Integraly i proizvodnye drobnogo poryadka
i nekotorye ikh prilozheniya [Fractional Integrals and Derivatives and Some of Their Applications],
Minsk, Nauka i Tekhnika, 1987, 688 p. (in Russian).
4. Podlubny, I. Fractional Differential Equations, San Diego, Academic Press, 1999, 368 p.
5. Barenblatt, G. I., Entov, V. M. and Ryzhik, V. M. Dvizheniye zhidkostey i gazov v
prirodnykh plastakh [Movement of Liquids and Gases in Natural Reservoirs], Moscow, Nedra, 1984, 447 p. (in Russian).
6. Shkhanukov, M. Kh. Some Boundary Value Problems for a Third-Order Equation That Arise
in the Modeling of the Filtration if a Fluid in Porous Media, Differentsial'nye
Uravneniya [Differential Equations], 1982, vol. 18, no. 4, pp. 689-700 (in Russian).
7. Cuesta, C., van Duijn, C. J. and Hulshof, J. Infiltration in Porous Media with Dynamic
Capillary Pressure: Travelling Waves, European Journal of Applied Mathematics, 2000,
vol. 11, no. 4, pp. 381-397. DOI: 10.1017/s0956792599004210.
8. Chudnovskiy, A. F. Teplofizika pochv [Thermal Physics of Soils], Moscow, Nauka, 1976, 353 p. (in Russian).
9. Hallaire, M. Le Potentiel Efficace de L'eau Dans le Sol en Regime de Dessechement,
L'Eau et la Production Vegetale, Paris, Institut National de la Recherche Agronomique, 1964, vol. 9, pp. 27-62.
10. Colton, D. L. On the Analytic Theory of Pseudoparabolic Equations,
The Quarterly Journal of Mathematics, 1972, vol. 23, pp. 179-192.
11. Dzektser, E. S. Equations of Motion of Groundwater with a Free Surface in Multilayer Media,
Doklady Akademii Nauk SSSR, 1975, vol. 220, no. 3, pp. 540-543 (in Russian).
12. Chen, P. J. and Gurtin, M. E. On a Theory of Heat Conduction Involving
Two Temperatures, Journal of Applied Mathematics and Physics (ZAMP),
1968, vol. 19, no. 4, pp. 614-627. DOI: 10.1007/BF01594969.
13. Ting, T. W. Certain Non-Steady Flows of Second-Order Fluids,
Archive for Rational Mechanics and Analysis, 1963, vol. 14, pp. 1-26.
14. Sveshnikov, A. G., Alshin, A. B., Korpusov, M. O. and Pletner, Yu. D.
Lineynyye i nelineynyye uravneniya sobolevskogo tipa [Linear and Nonlinear quations
of the Sobolev Type], Moscow, Fizmatlit, 2007, 736 p. (in Russian).
15. Bedanokova, S. Yu. The Equation of Soil Moisture Movement and the Mathematical Model
of Moisture Content of the Soil Layer, Based on the Hallaire's Equation, Vestnik Adygeyskogo
Gosudarstvennogo Universiteta. Seriya 4: Yestestvenno-Matematicheskiye i Tekhnicheskiye Nauki,
2007, vol. 4, pp. 68-71 (in Russian).
16. Goloviznin, V. M., Kiselev, V. P. and Korotkiy, I. A. Chislennye metody resheniya
uravneniya drobnoy diffuzii s drobnoy proizvodnoy po vremeni v odnomernom sluchae
[Computational Methods for One-Dimensional Fractional Diffusion Equations], Moscow,
Nuclear Safety Institute of the Russian Academy of Sciences, 2003, 35 p. (in Russian).
17. Lafisheva, M. M., Shhanukov-Lafishev, M. H. Locally One-Dimensional Difference
Schemes for the Fractional Order Diffusion Equation, Comput. Math. Math. Phys., 2008,
vol. 48, no.10, pp. 1875-1884.
18. Diethelm, K. and Walz, G. Numerical Solution of Fractional Order Differential
Equations by Extrapolation, Numerical Algorithms, 1997, vol. 16, pp. 231-253.
DOI: 10.1023/a:1019147432240.
19. Alikhanov, A. A. A Priori Estimates for Solutions of Boundary Value Problems
for Fractional-Order Equations, Differential Equations, 2010, vol. 46, no. 5, pp. 660-666.
DOI: 10.1134/S0012266110050058.
20. Alikhanov, A. A. A New Difference Scheme for the Time Fractional Diffusion
Equation, Journal of Computational Physics, 2015, vol. 280, pp. 424-438.
DOI: 10.1016/j.jcp.2014.09.031.
21. Nerpin, S. V. and Chudnovskiy, A. F. Energo- i massoobmen v sisteme
rasteniye-pochva-vozdukh [Energy and Mass Transfer in the Plant-Soil-Air System],
Leningrad, Gidrometeoizdat, 1975, 358 p. (in Russian).
22. Nigmatulin, R. R. Relaxation Features of a System with Residual Memory,
Fizika Tverdogo Tela [Physics of the Solid State], 1985, vol. 27, no. 5, pp. 1583-1585 (in Russian).
23. Beshtokov, M. Kh. Boundary Value Problems for Degenerating and Non-Degenerating Sobolev-Type
Equations with a Nonlocal Source in Differential and Difference Forms, Differential Equations,
2018, vol. 54, no. 2, pp. 250-267. DOI: 10.1134/S0012266118020118.
24. Beshtokov, M. Kh. The Third Boundary Value Problem for Loaded Differential
Sobolev Type Equation and Grid Methods of Their Numerical Implementation,
IOP Conference Series: Materials Science and Engineering, 2016, vol. 58, no. 1, pp. 1-6.
DOI: 10.1088/1757-899X/158/1/012019.
25. Beshtokov, M. Kh. To Boundary Value Problems for Degenerating Pseudoparabolic Equations with
Gerasimov-Caputo Fractional Derivative, Russian Mathematics, 2018, vol. 62, pp. 1-14.
DOI: 10.3103/S1066369X18100018.
26. Beshtokov, M. Kh. Boundary Value Problems for the Generalized Modified Moisture Transfer
Equation and Difference Methods for Their Numerical Realization, Prikladnaya Matematika
i Fizika [Applied Mathematics and Physics], 2020, vol. 52, no. 2, pp. 128-138 (in Russian).
DOI: 10.18413/2687-0959-2020-52-2-128-138.
27. Beshtokov, M. Kh. Boundary Value Problems for a Loaded Modified Time-Fractional
Moisture Transfer Equation with a Bessel Operator and Difference Methods for Their Solution,
Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternyye Nauki,
2020, vol. 30, no. 2, pp. 158-175 (in Russian).
28. Samarskii, A. A. Teoriya raznostnykh skhem [Theory of Difference Schemes],
Moscow, Nauka, 1983, 616 p. (in Russian).
29. Voyevodin, A. F. and Shugrin, S. M. Chislennyye metody rascheta odnomernykh sistem
[Numerical Methods for Calculating One-Dimensional Systems], Novosibirsk, Nauka,
Siberian Branch, 1981, 208 p. (in Russian).
