Капуто-Фабрицио бөлшек туындылы фильтрация есебі үшін сандық әдістің қателігін бағалау
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.010Кілттік сөздер:
finite element method, fractional derivative of Caputo-Fabrizio, convergence, filtration problem, fractured porous mediumАннотация
Бұл мақалада жарықшалары көлемі бойынша біртекті таралуы болжамында кеуекті ортада сұйықтықтың қозғалыс үлгісі зерттеледі. Бұл үлгі Дарси заңының бөлшек-дифференциалды баламасын қолдануға, сонымен қатар тау жынысы мен сұйықтықтың қасиеттері қысымнан және оның бөлшек туындысынан тәуелділік болжамына негізделген. Алдыңғы зерттеулерге қарағанда, бұл мақалада сингулярлық емес ядросы бар Капуто-Фабрицио мағынасындағы бөлшек туынды қолданылады. Бұл мақалада осы бастапқы шекаралық есепті шешудің сандық әдісі ұсынылған және оның жинақтылық реті теориялық тұрғыдан зерттелген. Толық дискретті сұлбаның құрылуы уақыт бойынша бүтін және бөлшек туындыларына ақырлы айырымдық жуықтауды, ал кеңістіктік айнымалысы бойынша Галеркин әдісін қолдануға негізделген. Бүтін туынды және Капуто-Фабрицио мағынасындағы бөлшек туындыны жуықтау үшін екінші ретті формула қолданылды. Априорлық бағалау жартылай дискретті және толық дискретті сұлбалар үшін алынды, олардан уақыт және кеңістіктік айнымалылары бойынша екінші ретті жинақтылық шығады. Сұлбаның дәлдігін тексеру үшін үлгі есептің мысалында бірқатар есептеу тәжірибелері жүргізілді. Сандық тәжірибелердің нәтижелері теориялық талдау нәтижелерін толық растайды.
Библиографиялық сілтемелер
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[2] Chechkin A. V., Gorenflo R. and Sokolov I. M.,
"Fractional diffusion in inhomogeneous media", Journal of Physics A: Mathematical And General, 38 (2005): 679–984.
[3] Esen A., Ucar Y., Yagmurlu M. and Tasbozan O., "Solving fractional diffusion and fractional diffusionwave equations by Petrov Galerkin finite element method", Turkish World Mathematical Society Journal of Applied and Engineering Mathematics, 4, no. 2 (2014): 155–168.
[4] Zhu A., Wang Y. and Xu Q., "A weak Galerkin finite element approximation of two-dimensional subdiffusion equation with time fractional derivative", AIMS Mathematics, 5, no. 5 (2020): 4297–4310.
[5] Abiola O. D., Enamul H. M., Kaseem M. and Sidqi A. A., "A modified memory based mathematical model describing fluid flow in porous media", Computers and Mathematics with Applications, 73, no. 6 (2017): 1385–1402.
[6] Gazizov R. K. and Lukashchuk S. Yu., "Drobno-differentsialnyi podkhod k modelirovaniyu protsessov filtratsii v slozhnykh neodnorodnykh poristykh sredakh [Fractional differential approach to modeling filtration processes in complex inhomogeneous porous media]", Vestnik UGATU, 21, no. 4 (2017): 104–112 (in Russian).
[7] Caffarelli L. and Vazquez J. L., "Nonlinear porous medium flow with fractional potential pressure", Archive for Rational Mechanics and Analysis, 202, no. 2 (2011): 537–565.
[8] Gazizov R. K., Kasatkin A. A. and Lukashchuk S. Yu., "Symmetries and exact solutions of fractional filtration equations", AIP Conference Proceedings, 1907, no. 020010 (2017): 1–9.
[9] Meilanov R. R., Akhmedov E. N. and Beybalaev V. D.,
"To the theory of nonlocal nonisothermal filtration in porous medium", IOP Conf. Series: Journal of Physics: Conf. Series, 946, no. 012076 (2018): 1–8.
[10] El Amin M. F., Radwan A. G. and Sun S., "Analytical solution for fractional derivative gasflow equation in porous media", Results in Physics, 7 (2017): 2432–2438.
[11] Ray S. S., "Exact solutions for time fractional diffusion wave equations by decomposition method", Phys. Scr., 75 (2007): 53–61.
[12] Zhang Y. N., Sun Z. Z. and Liao H. L., "Finite difference methods for the time fractional diffusion equation on nonuniform meshes", Journal of Computational Physics, 265 (2014): 195–210.
[13] Qiao H. L., Liu Z. G. and Cheng A. J., "Two unconditionally stable difference schemes for time distributed order differential equation based on Caputo–Fabrizio fractional derivative", Advances in Difference Equations, 2020 (2020): 1–17.
[14] Alikhanov A. A., "A new difference scheme for the time fractional diffusion equation", Journal of Computational Physics, 208 (2015): 424–438.
[15] Du R., Cao W. R. and Sun Z. Z., "A compact difference scheme for the fractional diffusion wave equation", Applied Mathematical Modelling, 34 (2010): 2998–3007.
[16] Xu T., Lu S., Chen W. and Chen H., "Finite difference scheme for multiterm variable order fractional diffusion equation", Advances in Difference Equations, 1 (2018): 1-13.
[17] Huang J., Tang Y., Wang W. and Yang J. "A compact difference scheme for time fractional diffusion equation with Neumann boundary conditions", Communications in Computer and Information Science, 323 (2012): 273–284.
[18] Zhang C., Liu H. and Zhou Z. J., "A priori error analysis for timestepping discontinuous Galerkin finite element approximation of time fractional optimal control problem", Journal of Scientific Computing, 80, no. 2 (2019): 993–1018
[19] Liu J. and Zhou Z., "Finite element approximation of time fractional optimal control problem with integral state constraint", AIMS Mathematics, 6, no. 1 (2020): 979–997.
[20] Liu K., Feckan M., O'Regan D. and Wang J. R., "Hyers–Ulam stability and existence of solutions for differential equations with Caputo–Fabrizio fractional derivative", Mathematics, 333, no. 7 (2019): 1-14.
[21] Coronel Escamilla A., Gomez Aguilar J.F., Torres L. and Escobar Jimenez R.F., "A numerical solution for a variable order reaction diffusion model by usingfractional derivatives with nonlocal and nonsingular kernel", Physica A: Statistical Mechanics and its Applications, 491 (2017): 406–424.
[22] Liu Y., Du Y. and Li H., "A twogrid mixed finite element method for a nonlinear fourthorder reaction diffusion problem with time fractional derivative", Computers and Mathematics with Application, 70 (2015): 2474–2492.
[23] Alimbekova N. B., Baigereyev D. R. and Madiyarov M. N., "Study of a numerical method for solving a boundary value problem for a differential equation with a fractional time derivative", Izvestiya of Altai State University, 114, no. 4 (2020): 64–69.
[24] Alimbekova N. B., Berdyshev A. S. and Baigereyev D. R., "Study of initial boundary value problem for two dimensional differential equation with fractional time derivative in the sense of Caputo", Third International Conference on Material Science, Smart Structures and Applications: (ICMSS 2020) (2021): 1–6.
[25] Baigereyev D., Alimbekova N., Berdyshev A. and Madiyarov M., "Convergence analysis of a numerical method for a fractional model of fluid flow in fractured porous media", Mathematics, 9, no. 2179 (2021): 1-24.
[26] Caputo M., "Models of flux in porous media with memory", Water Resources Research, 36, no. 3 (2000): 693– 705.
[27] Agarwal R., Yadav M. P., Baleanu D. and Purohit S. D., "Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative", AIMS Mathematics, 5, no. 2 (2020): 1062–1073.
[28] Akman T., Yildiz B. and Baleanu D., "New discretization of Caputo-Fabrizio derivative", Computational and Applied Mathematics, 37, no. 3 (2017): 3307-3333.
Жүктелулер
Как цитировать
Baigereyev, D., Alimbekova, N., & Oskorbin, N. (2022). Капуто-Фабрицио бөлшек туындылы фильтрация есебі үшін сандық әдістің қателігін бағалау. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 114(2). https://doi.org/10.26577/JMMCS.2022.v114.i2.010
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