A Fractional Model for Diffusion Equation Using Generalized Fick??s Law: Exact Solution with Laplace Transform

Nadeem Ahmad Sheikh, Dennis Ling Chuan Ching, Ilyas Khan, Afnan Ahmad, Syed Ammad


Fractional calculus is the generalization of classical calculus. Many researchers have used different definitions in their studies. The most common definition is Caputo fractional derivatives operator. In this article the concentration equation is converted to fractional form using the generalized Fick??s law. The fractional partial differential is then transformed with an appropriate transformation. The Laplace and Fourier sine transformations are jointly used to solve the equation. The impact of fractional parameter and Schmidt number is checked on the concentration profile and presented in graphs and tabular form. The results show that diffusion is decreasing with increasing values of Schmidt number.

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G. W. Leibnitz, "Letter from hanover, germany, september 30, 1695 to ga l??hospital. Leibnizen Mathematische Schriften," ed: Olms Verlag, Hildesheim, Germany, 1962.

M. Axtell and M. E. Bise, "Fractional calculus application in control systems," in IEEE Conference on Aerospace and Electronics, 1990, pp. 563-566: IEEE.

K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon Yverdon-les-Bains, Switzerland, 1993.

S. Das, Functional fractional calculus. Springer Science & Business Media, 2011.

R. L. Magin, Fractional calculus in bioengineering. Begell House Redding, 2006.

Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Applied Mechanics Reviews, vol. 50, no. 1, pp. 15-67, 1997.

A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics. Springer, 2014.

J. T. Machado, V. Kiryakova, F. J. C. i. n. s. Mainardi, and n. simulation, "Recent history of fractional calculus," vol. 16, no. 3, pp. 1140-1153, 2011.

B. Mandelbrot, "The fractal geometry of nature," Earth Surface Processes and Landforms, vol. 44, no. 12, pp. 406-406, 1982.

I. Petr????, Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media, 2011.

R. L. Bagley and P. Torvik, "A theoretical basis for the application of fractional calculus to viscoelasticity," Journal of Rheology, vol. 27, no. 3, pp. 201-210, 1983.

S. Momani and N. Shawagfeh, "Decomposition method for solving fractional Riccati differential equations," Applied Mathematics Computation, vol. 182, no. 2, pp. 1083-1092, 2006.

S. Momani and M. A. Noor, "Numerical methods for fourth-order fractional integro-differential equations," Applied Mathematics Computation, vol. 182, no. 1, pp. 754-760, 2006.

V. Daftardar-Gejji and H. Jafari, "Solving a multi-order fractional differential equation using Adomian decomposition," Applied Mathematics Computation, vol. 189, no. 1, pp. 541-548, 2007.

S. S. Ray, K. Chaudhuri, and R. Bera, "Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method," Applied mathematics computation, vol. 182, no. 1, pp. 544-552, 2006.

Q. Wang, "Numerical solutions for fractional KdV??Burgers equation by Adomian decomposition method," Applied Mathematics Computation, vol. 182, no. 2, pp. 1048-1055, 2006.

M. Inc, "The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method," Journal of Mathematical Analysis Applications, vol. 345, no. 1, pp. 476-484, 2008.

S. Momani and Z. Odibat, "Analytical approach to linear fractional partial differential equations arising in fluid mechanics," Physics Letters A, vol. 355, no. 4-5, pp. 271-279, 2006.

?. Odibat and S. Momani, "Application of variational iteration method to nonlinear differential equations of fractional order," International Journal of Nonlinear Sciences Numerical Simulation, vol. 7, no. 1, pp. 27-34, 2006.

S. Momani and Z. Odibat, "Homotopy perturbation method for nonlinear partial differential equations of fractional order," Physics Letters A, vol. 365, no. 5-6, pp. 345-350, 2007.

N. Sweilam, M. Khader, and R. Al-Bar, "Numerical studies for a multi-order fractional differential equation," Physics Letters A, vol. 371, no. 1-2, pp. 26-33, 2007.

Z. Odibat and S. Momani, "Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order," Chaos Solitons and Fractals, vol. 36, no. 1, pp. 167-174, 2008.

H. Qi and H. Jin, "Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders," Acta Mechanica Sinica, vol. 22, no. 4, pp. 301-305, 2006.

T. Wenchang, P. Wenxiao, and X. Mingyu, "A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates," International Journal of Non-Linear Mechanics, vol. 38, no. 5, pp. 645-650, 2003.

N. A. Shah, D. Vieru, and C. Fetecau, "Effects of the fractional order and magnetic field on the blood flow in cylindrical domains," Journal of Magnetism Magnetic Materials, vol. 409, pp. 10-19, 2016.

F. Liu, V. Anh, and I. Turner, "Numerical solution of the space fractional Fokker??Planck equation," Journal of Computational Applied Mathematics, vol. 166, no. 1, pp. 209-219, 2004.

S. B. Yuste, "Weighted average finite difference methods for fractional diffusion equations," Journal of Computational Physics, vol. 216, no. 1, pp. 264-274, 2006.

S.-Y. Lee, H. Ke, and Y. Kuo, "Analysis of non-uniform beam vibration," Journal of Sound Vibration, vol. 142, no. 1, pp. 15-29, 1990.

E. M. Wright, "On the coefficients of power series having exponential singularities," Journal of the London Mathematical Society, vol. 1, no. 1, pp. 71-79, 1933.

E. Wright, "The generalized Bessel function of order greater than one," The Quarterly Journal of Mathematics, vol. 11, no. 1, pp. 36-48, 1940.

S. Aman, I. Khan, Z. Ismail, M. Z. Salleh, and I. Tlili, "A new Caputo time fractional model for heat transfer enhancement of water based graphene nanofluid: An application to solar energy," Results in Physics, vol. 9, pp. 1352-1362, 2018.

F. Ali, N. A. Sheikh, I. Khan, and M. Saqib, "Solutions with Wright function for time fractional free convection flow of Casson fluid," Arabian Journal for Science Engineering, vol. 42, no. 6, pp. 2565-2572, 2017.

M. Saqib, F. Ali, I. Khan, N. A. Sheikh, and A. Khan, "Entropy Generation in Different Types of Fractionalized Nanofluids," Arabian Journal for Science and Engineering, vol. 44, no. 1, pp. 531-540, 2018.

N. A. Sheikh et al., "Comparison and analysis of the Atangana??Baleanu and Caputo??Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction," Results in physics, vol. 7, pp. 789-800, 2017.

N. A. Sheikh, F. Ali, I. Khan, M. Gohar, and M. Saqib, "On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models," The European Physical Journal Plus, vol. 132, no. 12, 2017.

A. Atangana and N. Bildik, "The use of fractional order derivative to predict the groundwater flow," Mathematical Problems in Engineering, vol. 2013, 2013.

K. A. Abro, A. D. Chandio, I. A. Abro, and I. Khan, "Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo??Fabrizio and Atangana??Baleanu fractional derivatives embedded in porous medium," Journal of Thermal Analysis and Calorimetry, vol. 135, no. 4, pp. 2197-2207, 2018.

A. Khan, D. Khan, I. Khan, F. Ali, F. U. Karim, and M. Imran, "MHD Flow of Sodium Alginate-Based Casson Type Nanofluid Passing Through A Porous Medium With Newtonian Heating," Sci Rep, vol. 8, no. 1, p. 8645, Jun 5 2018.

A. Khalid, I. Khan, A. Khan, and S. Shafie, "Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium," Engineering Science and Technology, an International Journal, vol. 18, no. 3, pp. 309-317, 2015.

DOI: https://doi.org/10.33959/cuijca.v3i2.28


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