A new approach for the solutions of the fractional generalized Casson fluid model described by Caputo fractional operator

Year 2020, Volume: 4 Issue: 4, 373 - 384, 30.12.2020

Ndolane Sene

https://doi.org/10.31197/atnaa.752330

Cited By: 1

Abstract

The fractional Casson fluid model has been considered in this paper in the context of the Goodman boundary conditions. A new approach for getting the solutions of the Casson fluid models have been proposed. There is the Double integral method and the Heat balance integral method. These two methods constitute the integral balance method. In these methods, the exponent of the approximate solutions is an open main problem, but this issue is intuitively solved by using the so-called matching method. The graphical representations of the solutions of the fractional Casson fluid model support the main results that have been presented. In our investigations, the Caputo derivative has been used.

Keywords

Fractional Casson fluid models, Fractional Heat equation, Integral balance methods

References

• [1] S. Ghosh and S. Mukhopadhyay, MHD slip flow and heat transfer of Casson nanofluid over an exponentially stretching permeable sheet, International Journal of Automotive and Mechanical Engineering (2017), 14(4), 4785- 4804.
• [2] M. Hamid, T. Zubair, M. Usman, and R. U. Haq, Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel, AIMS Mathematics, (2019), 4(5), 1416-1429.
• [3] N. A. Sheikh, D. L. C. Ching and I. Khan, A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis, Symmetry (2020), 12, 725.
• [4] N. A. Sheikh, D. L. C. Ching, I. Khan, D. Kumar, K. S. Nisar, A new model of fractional Casson fluid based on generalized Fick’s and Fourier’s laws together with heat and mass transfer, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.12.023
• [5] M. Saqib, A. R. M. Kasim, N. F. Mohammad, Dennis Ling Chuan Ching 4 and Sharidan Shafie Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate, Symmetry (2020), 12, 768.
• [6] A. Atangana, S. I. Araz, Extension of Atangana-Seda numerical method to partial differential equations with integer and non-integer order, Alexandria Eng. J. (2020), https://doi.org/10.1016/j.aej.2020.02.031.
• [7] Atangana, A. and T. Mekkaoui, Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus, Chaos, Solitons and Fractals, 128, 366-381, (2019).
• [8] A. Atangana, and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Sci. (2016), 20(2), 763-769.
• [9] Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1(2), 1-15, (2015).
• [10] D. Avci, M. Yavuz, N. Ozdemir, Fundamental Solutions to the Cauchy and Dirichlet Problems for a Heat Conduction Equation Equipped with the Caputo-Fabrizio Differentiation, Nova Science Publishers (2019), 95-107.
• [11] S. Aman, I. Khan, Z. Ismail and M. Z. Salleh, Application of fractional derivatives to nanofluids: exact and numerical solutions, Math. Model. Nat. Phenom. (2018), 13, 2.
• [12] Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1(2), 1-15, (2015).
• [13] J. Fahd, T. Abdeljawad, and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10, 2607-2619, (2017).
• [14] J. Fahd, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems-S, 1775-1786, (2019).
• [15] Roshdi Khalil, M Al Horani, Abdelrahman Yousef, and M Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
• [16] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands (2006), 204.
• [17] K. M. Owolabi, A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos (2019);, 29, 023111.
• [18] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA(1999), 198.
• [19] N. Sene, Stokes’ first problem for heated flat plate with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 117, 68-75, (2018).
• [20] N. Sene, Integral Balance Methods for Stokes’ First, Equation Described by the Left Generalized Fractional Derivative, Physics, 1, 154-166, (2019).
• [21] N. Sene, Analytical solutions and numerical schemes of certain generalized fractional diffusion models, Eur. Phys. J. Plus, 134, 199, (2019).
• [22] N. Sene, Integral-Balance Methods for the Fractional Diffusion Equation Described by the Caputo-Generalized Fractional Derivative, Methods of Mathematical Modelling: Fractional Differential Equations, (2019), 87.
• [23] N. Sene, Second-grade fluid model with Caputo-Liouville generalized fractional derivative, Chaos, Solitons & Fractals, 133, 109631, (2020).
• [24] M Yavuz, Characterization of two different fractional operators without singular kernel, Math. Model. Nat. Phen., 14(3), 302, (2019).
• [25] D. Kumar, J. Singh, D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences (2019), 43(1), 443-457.
• [26] D. Kumar, J. Singh, K. Tanwar, D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws, International Journal of Heat and Mass Transfer (2019), 138, 1222-1227.
• [27] A. Goswami, J. Singh, D. Kumar, Sushila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A: Statistical Mechanics and its Applications (2019), 524, 563-575.
• [28] Kolade M. Owolabi, Zakia Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 1, 013145, (2019).
• [29] C. Fetecau and J. Zierep, The Rayleigh-Stokes-problem for a Maxwell fluid, Zeitschrift für angewandte Mathematik und Physik ZAMP(2003), 54, 1086-1093.