Study of the initial boundary value problem for a two-dimensional convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio
DOI:
https://doi.org/10.26577/JMMCS.2021.v110.i2.10Abstract
In this paper, we study an initial boundary value problem for a differential equation with a fractional order derivative in time in the Caputo-Fabrizio sense. This equation is of great practical importance in modeling the processes of fluid motion in porous media and anomalous dispersion. The uniqueness and continuous dependence of the solution of the problem on the input data in differential form is proved. A computationally efficient implicit difference scheme with weights is
proposed. A priori estimates are obtained for the solution of the problem under the assumption that the solution exists in the class of sufficiently smooth functions. The uniqueness of the solution and the stability of the difference scheme with respect to the initial data and the right-hand side of the equation follows from the obtained estimates. The convergence of the difference problem solution to the differential problem solution with the second order in time and space variables
is proved. The results of computational experiments confirming the reliability of the theoretical analysis are presented.
Key words: Fractional differential equation, fractional derivative in the sense of Caputo-Fabrizio, finite difference method, energy inequality method, stability, convergence, a priori estimate.
References
[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.
Downloads
Published
2021-09-27
Issue
Section
Applied Mathematics
