Volume 19, Issue 2 (9-2024)
IJMSI 2024, 19(2): 127-153 |
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Soori Z, Aminataei A. Numerical Approach of Cattaneo Equation with Time Caputo-Fabrizio Fractional Derivative. IJMSI 2024; 19 (2) :127-153
URL: http://ijmsi.ir/article-1-1783-en.html
URL: http://ijmsi.ir/article-1-1783-en.html
Abstract:
In the paper, we consider a type of Cattaneo equation with time fractional derivative without singular kernel based on fourth-order compact finite difference (CFD) in the space directions. In case of two dimensional, two alternating direction implicit (ADI) methods are proposed to split the equation into two separate one dimensional equations. The time fractional derivation is described in the Caputo-Fabrizio’s sense with scheme of order O(τ2). The solvability, unconditional stability and H1 norm convergence of the scheme are proved. Numerical results confirm the theoretical results and the effectiveness of the proposed scheme.
Keywords: Caputo-Fabrizio fractional derivative, Compact finite difference, Cattaneo equation, Alternating direction implicit method.
Type of Study: Research paper |
Subject:
General
References
1. T. Akman, B. Yildiz, D. Baleanu, New Discretization of Caputo-Fabrizio Derivative, Comp. Appl. Math., 37, (2018), 3307-3333. [DOI:10.1007/s40314-017-0514-1]
2. A. Atangana, On the New Fractional Derivative and Application to Nonlinear Fishers Reaction-diffusion Equation, Appl. Math. Comput., 273, (2016), 948-956. [DOI:10.1016/j.amc.2015.10.021]
3. A. Atangana, B. S. T. Alkahtani, Extension of the Resistance, Inductance, Capacitance Electrical Circuit to Fractional Derivative without Singular Kernel, Adv. Mech. Eng., 7, DOI:10.1177/1687814055911937. [DOI:10.1177/1687814015591937]
4. A. Atangana, R. T. Alqahtani, Numerical Approximation of the Space-time CaputoFabrizio Fractional Derivative and Application to Groundwater Pollution Equation, Adv. Differ. Equ., 156, DOI:10.1186/s13662-016-0871-x. [DOI:10.1186/s13662-016-0871-x]
5. A. Atangana, J. J. Nieto, Numerical Solution for the Model of RLC Circuit via the Fractional Derivative without Singular Kernel, Adv. Mech. Eng., 7, (2015), 1-7. [DOI:10.1177/1687814015613758]
6. S. M. Aydogan, D. Baleanu, A. Mousalou, S. Rezapour, On Approximate Solutions for Two Higher-order Caputo-Fabrizio Fractional Integro-differential Equations, Adv. Differ. Equ., 221, (2017), 1-11. [DOI:10.1186/s13662-017-1258-3]
7. M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1, (2015), 73-85.
8. J. D. Djida, A. Atangana, More Generalized Groundwater Model with Space-time Caputo Fabrizio Fractional Differentiation, Numer. Methods Partial Differ. Equ., 33, (2017), 1616-1627. [DOI:10.1002/num.22156]
9. B. Fakhr Kazemi, H. Jafari, H, Error Estimate of the MQ-RBF Collocation Method for Fractional Differential Equations with Caputo-Fabrizio Derivative, Math. Sci., 11, (2017), 297-305. [DOI:10.1007/s40096-017-0232-2]
10. G. Femandz-Anaya, F. J. Valdes-Parada, J. Alvarea-Ramirez, On Generalized Fractional Cattaneo's Equation, Physica A: statical. Mech. Appl., 390, (2011), 4198-4202. [DOI:10.1016/j.physa.2011.07.001]
11. M. A. Firoozjaee, H. Jafari, A. Lia, D. Baleanu, Numerical Approach of Fokker-Planck Equation with Caputo-Fabrizio Fractional Derivative Using Ritz Approximation, J. Compu. Appl. Math., 339, (2018), 367-373. [DOI:10.1016/j.cam.2017.05.022]
12. H. P. Ghazizadeh, M. Maerefat, A. Azim, Explicit and Implicit Finite Difference Schemes for Fractional Cattaneo Equation, J. Comput, Phys, 229, (2010), 7042-7057. [DOI:10.1016/j.jcp.2010.05.039]
13. F. El-Ghenbazia, M. Tayeb Meftah, Solution of Nonlocal Schrodinger Equation via the Caputo-Fabrizio Definition for Some Quantum Systems, Reports. Math. Physics, 85, (2020), 57-67. [DOI:10.1016/S0034-4877(20)30010-0]
14. B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, China, 2011.
15. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [DOI:10.1142/9789812817747]
16. Z. Liu, A. Cheng, X. Li, A Second Order Crank-Nicolson Scheme for Fractional Cattaneo Equation Based on New Fractional Derivative, Appl. Math. Comput., 311, (2017), 361-374. [DOI:10.1016/j.amc.2017.05.032]
17. J. R. Loh, Y. T. Toh, On the New Properties of Caputo-Fabrizio Operator and its Application in Deriving Shifted Legendre Operational Matrix, Appl. Numer. Math., 132, (2018), 138-153. [DOI:10.1016/j.apnum.2018.05.016]
18. J. Losada, J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1, (2015), 87-92.
19. K. M. Owolabi, A. Atangana, Analysis and Application of New Fractional AdamsBashforth Scheme with Caputo-Fabrizio Derivative, Chaos. Solitons. Fract, 105, (2017), 111-119. [DOI:10.1016/j.chaos.2017.10.020]
20. D. W. Peaceman, H. H. Rachford, The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Ind. Appl. Math., 3, (1955), 23-41. [DOI:10.1137/0103003]
21. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
22. H. T. Qi, X. W. Guo, The Cattaneo-type Time Fractional Heat Conduction Equation for Laser Heating, Fract. Difer. Appl., 66, (2013), 834-831. [DOI:10.1016/j.camwa.2012.11.021]
23. H. Qi, X. Jiang, Solution of the Space-time Fractional Cattaneo Diffusion Equation, Physica A: statical. Mech. Appl., 390, (2011), 1876-1883. [DOI:10.1016/j.physa.2011.02.010]
24. J. Ren, G. H. Gao, Efficient and Stable Numerical Methods for the Two-dimensional Fractional Cattaneo Equation, Numer. Algor., 69, (2015), 795-818. [DOI:10.1007/s11075-014-9926-9]
25. J. Ren, Z. Sun, Efficient Numerical Solution of the Multi-term Time Fractional Diffusionwave Equation, East Asian J. Appl. Math., 5, (2015), 1-28. [DOI:10.4208/eajam.080714.031114a]
26. A. A. Samarskii, V. B. Andreev, Difference Methods for Elliptic Equation, Nauka, Moscow, 1976.
27. J. She, M. Chen, A Second-order Accurate Scheme for Two-dimensional Space Fractional Diffusion Equations with Time Caputo-Fabrizio Fractional Derivative, Appl. Numer. Math., 151, (2020), 246-262. [DOI:10.1016/j.apnum.2020.01.007]
28. Z. Soori, A. Aminataei, A New Approximation to Caputo-type Fractional Diffusion and Advection Equations on Non-uniform Meshes, Appl. Numer. Math., 144, (2019), 21-41. [DOI:10.1016/j.apnum.2019.05.014]
29. Z. Soori, A. Aminataei, Effect of the Nodes Near Boundary Points on the Stability Analysis of Sixth-order Compact Finite Difference ADI Scheme for the Two-dimensional Time Fractional Diffusion-wave Equation, Transactions of A. Razmadze Mathematical Institute, 172, (2018), 582-605. [DOI:10.1016/j.trmi.2018.03.003]
30. Z. Soori, A. Aminataei, Numerical Solution of Space Fractional Diffusion Equation by Spline Method Combined with Richardson Extrapolation, Comput. Appl. Mathe., 139, (2020), 1-18. [DOI:10.1007/s40314-020-01160-4]
31. Z. Soori, A. Aminataei, Sixth-order Non-uniform Combined Compact Difference Scheme for Multi-term Time Fractional Diffusion-wave Equation, Appl. Numer. Math., 131, (2018), 72-94. [DOI:10.1016/j.apnum.2018.04.006]
32. J. V. C. Sousa, E. C. Oliveira, Two New Fractional Derivatives of Variable Order with Non-singular Kernel and Fractional Differential Equation, Comp. Appl. Math., 37, (2018), 5375-5394. [DOI:10.1007/s40314-018-0639-x]
33. Z. Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.
34. X-J. Yang, H. M. Machado, A New Fractional Operator of Variable Order: Application in the Description of Anomalous Diffusion, Phys A. Stat. Mech. Appl., 481, (2017), 276-283. [DOI:10.1016/j.physa.2017.04.054]
35. X-J. Yang, H. M. Srivastava, T. J. A. Machado, A New Fractional Derivative without Singular Kernel, Therm. Sci., 20, (2016), 753-756. [DOI:10.2298/TSCI151224222Y]
36. Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact Alternating Direction Implicit Scheme for the Two-dimensional Fractional Diffusion-wave Equation, Siam J. Numer. Anal., 50, (2012), 1535-1555. [DOI:10.1137/110840959]
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