Solutions of Fractional Diffusion Equations and Cattaneo-Hristov Diffusion Model
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Abstract
The analytical solutions of the fractional diffusion equations in one and two-dimensional spaces have been proposed. The analytical solution of the Cattaneo-Hristov diffusion model with the particular boundary conditions has been suggested. In general, the numerical methods have been used to solve the fractional diffusion equations and the Cattaneo-Hristov diffusion model. The Laplace and the Fourier sine transforms have been used to get the analytical solutions. The analytical solutions of the classical diffusion equations and the Cattaneo-Hristov diffusion model obtained when the order of the fractional derivative converges to 1 have been recalled. The graphical representations of the analytical solutions of the fractional diffusion equations and the Cattaneo-Hristov diffusion model have been provided.
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References
- B. S. T. Alkahtani and A. Atangana. A note on cattaneo-hristov model with non-singular fading memory. Therm. Sci., 21(1)(2017), 1-7.
- A. Atangana and D. Baleanu. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408, (2016).
- A. Atangana and JF. Gomez-Aguilar. Fractional derivatives with noindex law property: Application to chaos and statistics. Chaos Solitons Fractals, 114 (2018), 516-535.
- A. Atangana and I. Koca. Chaos in a simple nonlinear system with atanganabaleanu derivatives with fractional order. Chaos Solitons Fractals, 89 (2016), 447-454.
- L. Beghin. Fractional diffusion-type equations with exponential and logarithmic differential operators. Stoc. Proc. Appl., 128(7)(2018), 2427-2447.
- M. Caputo and M. Fabrizio. A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1(2)(2015), 1-13.
- A. C. Escamilla, JF. G. Aguilar, L. Torres, and RF. E. Jimnez. A numerical solution for a variable-order reactiondiffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A: Stat. Mech. Appl., 491(2018), 406-424.
- H. Delavari, D. Baleanu, and J. Sadati. Stability analysis of caputo fractional-order nonlinear systems revisited. Nonlinear Dyn., 67(4) (2012), 2433-2439.
- E. F. D. Goufo. Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: basic theory and applications. Chaos: An Inter. J. Nonlinear Sci., 26(8) (2016), 084-305.
- J. Hristov. On the atangana-baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation. Trends in theory and applications of fractional derivatives with Mittag-Leffler kernel, Springer. 2019.
- J. Hristov. Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Spect. Topics, 193(1)(2011), 229-243.
- J. Hristov. Transient heat diffusion with a non-singular fading memory: from the cattaneo constitutive equation with jeffrey's kernel to the caputo-fabrizio time-fractional derivative. Therm. Sci., 20(2) (2016), 757-762.
- J. Hristov. Derivation of the fractional dodson equation and beyond: Transient diffusion with a non-singular memory and exponentially fadingout diffusivity. Progr. Fract. Differ. Appl, 3(4) (2017), 1-16.
- J. Hristov. Multiple integral-balance method basic idea and an example with mullins model of thermal grooving. Therm. Sci., 21(2017), 1555-1560.
- J. Hristov. The non-linear dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization. Math. Nat. Sci., 1(1) (2017), 1-17.
- J. Hristov. Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. Math. Model. Nat. Phenom. 13(1)(2018), 6.
- J. Hristov. Integral-balance solution to nonlinear subdiffusion equation. Front. Fract. Calcu., 1(2018), 70.
- J. Hristov. The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method. Therm. Sci., 22(2) (2018), 777-788.
- J. Hristov. Integral balance approach to 1-d space-fractional diffusion models. Math. Meth. Eng., (2019), 111-131, Springer.
- J. Hristov. A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integralbalance approach. Math. Meth. Eng., (2019), 153-174, Springer.
- F. Jarad and T. Abdeljawad. A modified laplace transform for certain generalized fractional operators. Res. Nonlinear Anal., (2)(2018), 88-98.
- F. Jarad, E. Ugurlu, T. Abdeljawad, and Dumitru Baleanu. On a new class of fractional operators. Adv. Diff. Equa., (1)(2017), 247.
- H. Jordan. Steady-state heat conduction in a medium with spatial non-singular fading memory derivation of caputo-fabrizio spacefractional derivative from cattaneo concept with jeffrey's kernel and analytical solutions. Therm. Sci., 21(2) (2017), 827-839.
- U. N. Katugampola. A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl, 6(4)(2014), 115.
- A. A. Kilbas, M Rivero, L Rodriguez-Germa, and JJ Trujillo. Caputo linear fractional differential equations. IFAC Proc. 39(11) (2006), 52-57.
- I. Koca and Abdon Atangana. Solutions of cattaneo-hristov model of elastic heat diffusion with caputo-fabrizio and atangana-baleanu fractional derivatives. Therm. Sci., 21 (2017), 2299-2305.
- L. Li, J. G. Liu, and L. Wang. Cauchy problems for kellersegel type timespace fractional diffusion equation. J. Differ. Equ., 265(3)(2018), 1044-1096.
- Y. Li, Y. Q. Chen, and I. Podlubny. Mittagleffler stability of fractional order nonlinear dynamic systems. Auto., 45(8) (2009), 1965-1969.
- Y. Li, F. Liu, I. W. Turner, and T. Li. Time-fractional diffusion equation for signal smoothing. Appl. Math. Comp., 326 (2018), 108116.
- J. Losada and J. J. Nieto. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl, 1(2)(2015), 87-92.
- Y. Ma, F. Zhang, and C. Li. The asymptotics of the solutions to the anomalous diffusion equations. Comput. Math. Appl., 66(5)(2013), 682-692.
- T. Myers. Optimal exponent heat balance and refined integral methods applied to stefan problems. Int. J. Heat Mass Transfer, 53(5-6) (2010), 1119-1127.
- K. M. Owolabi and A. Atangana. Robustness of fractional difference schemes via the caputo subdiffusion-reaction equations. Chaos, Solitons Fractals, 111 (2018), 119-127.
- I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1998), 198. Acad. Press.
- I. Podlubny. Matrix approach to discrete fractional calculus ii: Partial fractional differential equations. (2009).
- S. Priyadharsini. Stability of fractional neutral and integrodifferential systems. J. Fract. Calc. Appl.,7(1) (2016), 87-102.
- Z. Ruan, W. Zhang, and Zewen Wang. Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation. Appl. Math. Comput., 328 (2018), 365-379.
- K. M. Saad, D. Baleanu, and A. Atangana. New fractional derivatives applied to the kortewegde vries and korteweg-de vries-burgers equations. Comput. Appl. Math., 37 (2018), 52035216.
- Y. Salehi, M. T. Darvishi, and W. E. Schiesser. Numerical solution of space fractional diffusion equation by the method of lines and splines. Appl. Math. Comput., 336 (2018), 465-480.
- N. Sene. Exponential form for lyapunov function and stability analysis of the fractional differential equations. J. Math. Comput. Sci. 18(4)(2018), 388-397.
- N. Sene. Lyapunov characterization of the fractional nonlinear systems with exogenous input. Fractal Fract., (2018), 2(2):17.
- N. Sene. Solutions for some conformable differential equations. Progr. Fract. Differ. Appl., 4(4)(2018), 493-501.
- N. Sene. Stokes first problem for heated flat plate with atangana-baleanu fractional derivative. Chaos Soli. Fract., 117 (2018), 68-75.
- S. Shen, F. Liu, and V. V. Anh. The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation. J. Comput. Appl. Math., 345 (2019), 515-534.
- J. Zhang, X. Zhang, and B. Yang. An approximation scheme for the time fractional convectiondiffusion equation. Appl. Math. Comput., 335(2018), 305-312.