Исследование начальной краевой задачи для двумерного уравнения конвекции-диффузии с дробной производной по времени в смысле Капуто-Фабрицио
DOI:
https://doi.org/10.26577/JMMCS.2021.v110.i2.10Аннотация
В настоящей работе исследуется начально-краевая задача для дифференциального уравнения с производной дробного порядка по времени в смысле Капуто-Фабрицио. Данное уравнение имеет большую прикладную значимость при моделировании процессов фильтрации и аномальной дисперсии. Доказаны единственность и непрерывная зависимость решения задачи от входных данных в дифференциальной форме. Предложена вычислительно эффек- тивная неявная разностная схема с весами. Получены априорные оценки для решения задачи в предположении существования решения в классе достаточно гладких функций. Из этих оценок следуют единственность решения и устойчивость разностной схемы по начальным данным и правой части уравнения. Доказана сходимость решения разностной задачи к решению дифференциальной задачи со вторым порядком по временной и пространственной переменным. Представлены результаты вычислительных экспериментов, подтверждающие достоверность теоретического анализа.
Ключевые слова: Дифференциальное уравнение дробного порядка, дробная производная в смысле Капуто-Фабрицио, метод конечных разностей, метод энергетических неравенств, устойчивость, сходимость, априорная оценка.
Библиографические ссылки
[1] Latawiec K.J., Stanislawski R., Lukaniszyn M., Czuczwara W., and Rydel M., "Fractional-order modeling of electric circuits: Modern empiricism vs. classical science" , Proceedings of Progress in Applied Electrical Engineering (2017).
[2] Oprzedkiewicz K., Mitkowski W., "A Memory-Efficient Noninteger-Order Discrete-Time State-Space Model of a Heat Transfer Process" , Int. J. Appl. Math. Comput. Sci., 28 (2018): 649–659.
[3] Wang K.L., Liu S.Y., "He’s fractional derivative and its application for fractional Fornberg-Whitham equation" , Therm. Sci., 21 (2017): 2049-2055.
[4] Altaf M., Atangana A., "Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives" , Entropy, 21 (2019): 303.
[5] Lichae B.H., Biazar J., Ayati Z., "The Fractional Differential Model of HIV-1 Infection of CD+T-Cells with Description of the Effect of Antiviral Drug Treatment" , Comput. Math. Methods Med., 2019:4059549 (2019).
[6] Caputo M., "Models of flux in porous media with memory" , Water Resources Research, 36 (2000): 693–705.
[7] Agarwal R., Yadav M. P., Baleanu D., Purohit S. D., "Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative" , AIMS Mathematics, 5:2 (2020): 1062–1073.
[8] Di Giuseppe E., Moroni M., Caputo M., "Flux in porous media with memory: models and experiments" , Transport in Porous Media, 83:3 (2010): 479–500.
[9] Gazizov R. K., Lukaschuk S. Yu., "Drobno-differencialny podhod k modelirovaniyu processov filtratsii v slozhnyh neodnorodnyh poristyh sredah [Fractional-differential approach to modeling filtration processes in complex inhomogeneous
porous media]" , Vestnik UGATU, 21:4 (2017): 104–112.
[10] Umarov S., Daum F., Nelson K., "Fractional nonlinear filtering problems and their associated fractional Zakai equations" , ArXiV, 1305.2658 (2013): 1-15.
[11] Zhong W., Li C., and Kou J.,. "Numerical Fractional-Calculus Model for Two-Phase Flow in Fractured Media" , Advances in Mathematical Physics, 2013:429835 (2013): 1-7.
[12] Lukaschuk V. O., Lukaschuk S. Yu, "Gruppovaya klassifikaciya, invariantnye reshenija i zakony sohraneniya nelineinogo dvumernogo ortotropnogo uravneniya filtratsii s drobnoi proizvodnoi Rimana–Liuvillya po vremeni [Group classification,
invariant solutions, and conservation laws of a nonlinear two-dimensional orthotropic filtration equation with a Riemann–Liouville fractional derivative in time]" , Vestnik SGTU. Seriya fiziko-matematicheskie nauki, 24:2 (2020): 226–248.
[13] Choudharya A., Kumarb D., Singh J., "A fractional model of fluid flow through porousmedia with mean capillary pressure" , Journal of the Association of Arab Universities for Basic and Applied Sciences, 21 (2016): 59–63.
[14] Uddin1 S., Mohamad M., "Caputo-Fabrizio Time Fractional Derivative Applied to Visco Elastic MHD Fluid Flow in the Porous Medium,"International Journal of Engineering & Technology, 7 (2018): 533-537.
[15] Caputo M., Fabrizio M., "A New Definition of Fractional Derivative without Singular Kernel" , Progress in Fractional Differentiation and Applications, 2 (2015): 73–85.
[16] Losada J., Nieto J. J., "Properties of a New Fractional Derivative without Singular Kernel,"Progress in Fractional Differentiation and Applications, 1:2 (2015): 87–92.
[17] AlSalti N., Karimov E., Kerbal S., "Boundary value problems for fractional heat equation involving Caputo-Fabrizio derivative" , New Trends in Mathematical Sciences, 4:4 (2016): 79–89.
[18] Feulefack P. A., Djida J. D., Atangana A., "A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative" , American Institute of Mathematical Sciences, 24:7 (2019): 3227–3247.
[19] Atangana A., Baleanu D., "Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer" , Journal of Engineering Mechanics, 143:5 (2016): D4016005.
[20] Yu F., Chen M., "Finite difference/spectral approximations for the twodimensional time Caputo-Fabrizio fractional diffusion equation" , arXiv, 1906.00328 (2019): 1-18.
[21] Aydogan M., Baleanu D., Mousalou A., Rezapour S., "On high order fractional integro differential equations including the Caputo–Fabrizio derivative" , Boundary Value Problems, 2018:90 (2018): 1–15.
[22] Gong C., Bao W., Tang G. et al., "A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method" , The Scientific World Journal, 219580 (2014): 1–8.
[23] Atangana A., "On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation" , Applied Mathematics and Computation, 273 (2016): 948–956.
[24] Yang X. J., Srivastava H. M., "Machado Tenreiro J. A. A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow" , Thermal Science, 20:2 (2016): 753–756.
[25] Atangana A., Alkahtani B., "New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative" , Arabian Journal of Geosciences volume, 9:8 (2016): 1–6.
[26] Alkahtani B., Atangana A., "Chaos on the Vallis Model for El Nino with Fractional Operators" , Entropy, 18:100 (2016): 1–17.
[27] Alqahtani R. T., "Fixed point theorem for Caputo–Fabrizio fractional Nagumo equation with nonlinear diffusion and convection" , Journal of Nonlinear Sciences and Application, 9:5 (2016): 1991–1999.
[28] Atangana A., Fractional Operators with Constant and Variable Order with Application to GeoHydrology (Elsevier, 2018).
[29] Alikhanov A. A., "A priori estimates for solutions of boundary value problems for fractional order equations" , Differential Equations, 46 (2010): 660–666.
[30] Alikhanov A. A., "A new difference scheme for the time fractional diffusion equation" , Journal of Computational Physics, 280 (2015): 424–438.
[31] Zhumagulov B. T., Temirbekov N. M., Baigereyev D. R. "Efficient difference schemes for the three-phase non-isothermal flow problem" , AIP Conference Proceedings, 1880:060001 (2017): 1-10.
[2] Oprzedkiewicz K., Mitkowski W., "A Memory-Efficient Noninteger-Order Discrete-Time State-Space Model of a Heat Transfer Process" , Int. J. Appl. Math. Comput. Sci., 28 (2018): 649–659.
[3] Wang K.L., Liu S.Y., "He’s fractional derivative and its application for fractional Fornberg-Whitham equation" , Therm. Sci., 21 (2017): 2049-2055.
[4] Altaf M., Atangana A., "Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives" , Entropy, 21 (2019): 303.
[5] Lichae B.H., Biazar J., Ayati Z., "The Fractional Differential Model of HIV-1 Infection of CD+T-Cells with Description of the Effect of Antiviral Drug Treatment" , Comput. Math. Methods Med., 2019:4059549 (2019).
[6] Caputo M., "Models of flux in porous media with memory" , Water Resources Research, 36 (2000): 693–705.
[7] Agarwal R., Yadav M. P., Baleanu D., Purohit S. D., "Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative" , AIMS Mathematics, 5:2 (2020): 1062–1073.
[8] Di Giuseppe E., Moroni M., Caputo M., "Flux in porous media with memory: models and experiments" , Transport in Porous Media, 83:3 (2010): 479–500.
[9] Gazizov R. K., Lukaschuk S. Yu., "Drobno-differencialny podhod k modelirovaniyu processov filtratsii v slozhnyh neodnorodnyh poristyh sredah [Fractional-differential approach to modeling filtration processes in complex inhomogeneous
porous media]" , Vestnik UGATU, 21:4 (2017): 104–112.
[10] Umarov S., Daum F., Nelson K., "Fractional nonlinear filtering problems and their associated fractional Zakai equations" , ArXiV, 1305.2658 (2013): 1-15.
[11] Zhong W., Li C., and Kou J.,. "Numerical Fractional-Calculus Model for Two-Phase Flow in Fractured Media" , Advances in Mathematical Physics, 2013:429835 (2013): 1-7.
[12] Lukaschuk V. O., Lukaschuk S. Yu, "Gruppovaya klassifikaciya, invariantnye reshenija i zakony sohraneniya nelineinogo dvumernogo ortotropnogo uravneniya filtratsii s drobnoi proizvodnoi Rimana–Liuvillya po vremeni [Group classification,
invariant solutions, and conservation laws of a nonlinear two-dimensional orthotropic filtration equation with a Riemann–Liouville fractional derivative in time]" , Vestnik SGTU. Seriya fiziko-matematicheskie nauki, 24:2 (2020): 226–248.
[13] Choudharya A., Kumarb D., Singh J., "A fractional model of fluid flow through porousmedia with mean capillary pressure" , Journal of the Association of Arab Universities for Basic and Applied Sciences, 21 (2016): 59–63.
[14] Uddin1 S., Mohamad M., "Caputo-Fabrizio Time Fractional Derivative Applied to Visco Elastic MHD Fluid Flow in the Porous Medium,"International Journal of Engineering & Technology, 7 (2018): 533-537.
[15] Caputo M., Fabrizio M., "A New Definition of Fractional Derivative without Singular Kernel" , Progress in Fractional Differentiation and Applications, 2 (2015): 73–85.
[16] Losada J., Nieto J. J., "Properties of a New Fractional Derivative without Singular Kernel,"Progress in Fractional Differentiation and Applications, 1:2 (2015): 87–92.
[17] AlSalti N., Karimov E., Kerbal S., "Boundary value problems for fractional heat equation involving Caputo-Fabrizio derivative" , New Trends in Mathematical Sciences, 4:4 (2016): 79–89.
[18] Feulefack P. A., Djida J. D., Atangana A., "A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative" , American Institute of Mathematical Sciences, 24:7 (2019): 3227–3247.
[19] Atangana A., Baleanu D., "Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer" , Journal of Engineering Mechanics, 143:5 (2016): D4016005.
[20] Yu F., Chen M., "Finite difference/spectral approximations for the twodimensional time Caputo-Fabrizio fractional diffusion equation" , arXiv, 1906.00328 (2019): 1-18.
[21] Aydogan M., Baleanu D., Mousalou A., Rezapour S., "On high order fractional integro differential equations including the Caputo–Fabrizio derivative" , Boundary Value Problems, 2018:90 (2018): 1–15.
[22] Gong C., Bao W., Tang G. et al., "A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method" , The Scientific World Journal, 219580 (2014): 1–8.
[23] Atangana A., "On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation" , Applied Mathematics and Computation, 273 (2016): 948–956.
[24] Yang X. J., Srivastava H. M., "Machado Tenreiro J. A. A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow" , Thermal Science, 20:2 (2016): 753–756.
[25] Atangana A., Alkahtani B., "New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative" , Arabian Journal of Geosciences volume, 9:8 (2016): 1–6.
[26] Alkahtani B., Atangana A., "Chaos on the Vallis Model for El Nino with Fractional Operators" , Entropy, 18:100 (2016): 1–17.
[27] Alqahtani R. T., "Fixed point theorem for Caputo–Fabrizio fractional Nagumo equation with nonlinear diffusion and convection" , Journal of Nonlinear Sciences and Application, 9:5 (2016): 1991–1999.
[28] Atangana A., Fractional Operators with Constant and Variable Order with Application to GeoHydrology (Elsevier, 2018).
[29] Alikhanov A. A., "A priori estimates for solutions of boundary value problems for fractional order equations" , Differential Equations, 46 (2010): 660–666.
[30] Alikhanov A. A., "A new difference scheme for the time fractional diffusion equation" , Journal of Computational Physics, 280 (2015): 424–438.
[31] Zhumagulov B. T., Temirbekov N. M., Baigereyev D. R. "Efficient difference schemes for the three-phase non-isothermal flow problem" , AIP Conference Proceedings, 1880:060001 (2017): 1-10.
Загрузки
Как цитировать
Alimbekova, N. B., & Oskorbin, N. M. (2021). Исследование начальной краевой задачи для двумерного уравнения конвекции-диффузии с дробной производной по времени в смысле Капуто-Фабрицио. Вестник КазНУ. Серия математика, механика, информатика, 110(2), 97–112. https://doi.org/10.26577/JMMCS.2021.v110.i2.10
Выпуск
Раздел
Прикладная математика
