Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind

In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. The finite difference method is used to reduce the equations obtained by our approach for a system of algebraic equations that can be efficiently solved. Numerical results obtained with our approach are presented and compared with the results obtained by other numerical methods. The numerical results show the efficiency of the proposed approach.

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[1] Agarwal P, Choi J, Paris RB. Extended Riemann-Liouville fractional derivative operator and its applications. J Nonlinear Sci Appl 2015; 8: 451-466.
[2] Azizi H, Loghmani GB. Numerical approximation for space fractional diffusion equations via Chebyshev finite difference method. J Fract Cal Appl 2013; 4: 303-311.
[3] Azizi H, Loghmani GB. A numerical method for space fractional diffusion equations using a semi-disrete scheme and Chebyshev collocation method. J Math Comput Sci 2014; 8: 226-235.
[4] Benson D, Wheatcraft SW, Meerschaert MM. Application of a fractional advection-dispersion equation. Water Resour Res 2000; 36: 1403-1412.
[5] Boyd JP. Chebyshev and Fourier Spectral Methods. 2nd ed. Mineola, NY, USA: Dover, 2001.
[6] Bhrawy AH, Baleanu D, Mallawi F. A new numerical technique for solving fractional sub-diffusion and reaction sub-diffusion equations with non-linear source term. Thermal Science 2015; 19: 25-34.
[7] Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA. New operational matrices for solving fractional differential equations on the half-line. PLoS ONE 2015; 10: e0126620.
[8] Bhrawy AH, Zaky MA, Baleanu D. New numerical approximations for space-time fractional Burgers equations via a Legendre spectral collocation method. Romanian Reports in Physics 2015; 67: 340-349.
[9] Canuto C, Quarteroni A, Hussaini MY, Zang TA. Spectral Methods: Fundamentals in Single Domains. Berlin, Germany: Springer-Verlag, 2006.
[10] Carreras BA, Lynch VE, Zaslavsky GM. Anomalous diffusion and exit time distribution of particle travers in plasma turbulence models. Phys Plasmas 2001; 8: 5096-5103.
[11] Cui M. Compact finite difference method for the fractional diffusion equation. J Comput Phys 2009; 228: 7792-7804.
[12] Dalir M, Bashour M. Applications of fractional calculus. Appl Math Sci 2010; 21: 1021-1032.
[13] Dehghan M, Saadatmandi A. Chebyshev finite difference method for Fredholm integro-differential equation. Int J Comput Math 2008; 85: 123-130.
[14] Diethelm K. The Analysis of Fractional Differential Equations. Berlin, Germany: Springer-Verlag, 2010.
[15] Elbarbary EM. Chebyshev finite difference approximation for the boundary value problems. Appl Math Comput 2003; 139: 513-523.
[16] Ervin VJ, Roop JP. Variational solution of fractional advection dispersion equations on bounded domains in R d . Numer Meth Part D E 2007; 23: 256-281.
[17] Gao G, Sun Z. A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 2011; 230: 586-595.
[18] Khader MM. On the numerical solutions for the fractional diffusion equation. Commun Nonlinear Sci 2011; 16: 2535-2542.
[19] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. San Diego, CA, USA: Elsevier, 2006.
[20] Mason JC, Handscomb DC. Chebyshev Polynomials. New York, NY, USA: Chapman and Hall, 2003.
[21] Meerschaert MM, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 2004; 172: 65-77.
[22] Meerschaert MM, Tadjeran C. Finite difference approximations for two-sided space fractional partial differential equations. Appl Numer Math 2006; 56: 80-90.
[23] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY, USA: Wiley, 1993.
[24] Nagy AM, Sweilam NH. An efficient method for solving fractional Hodgkin-Huxley model. Phys Lett A 2014; 378: 1980-1984.
[25] Oldham KB, Spanier J. The Fractional Calculus. New York, NY, USA: Academic Press, 1974.
[26] Podlubny I. Fractional Differential Equations. New York, NY, USA: Academic Press, 1999.
[27] Ross B. The development of fractional calculus 1695-1900. Hist Math 1977; 4: 75-89.
[28] Rossikhin YA, Shitikova MV. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 1997; 50: 15-67.
[29] Saadatmandi A, Dehghan M. Numerical solution of the one-dimensional wave equation with an integral condition . Numer Meth Part D E 2007; 23: 282-292.
[30] Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl 2010; 59: 1326-1336.
[31] Saadatmandi A, Dehghan M. A tau approach for solution of the space fractional diffusion equation. Comput Math Appl 2011; 62: 1135-1142.
[32] Salahshour S, Ahmadian A, Senu N, Baleanu N, Agarwal P. On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem. Entropy 2015; 17: 885-902.
[33] Scalas E, Gorenflo R, Mainardi F, Raberto M. Revisiting the derivation of the fractional order diffusion equation. Fractals 2003; 11: 281-289.
[34] Sousa E. Numerical approximations for fractional diffusion equation via splines. Comput Math Appl 2011; 62: 983-944.
[35] Srivastava HM, Agarwal P. Certain fractional integral operators and the generalized incomplete hypergeometric functions. Applications & Applied Mathematics 2013; 8: 333-345.
[36] Su L, Wang W, Xu Q. Finite difference methods for fractional dispersion equations. Appl Math Comput 2010; 216: 3329-3334.
[37] Sweilam NH, Khader MM, Mahdy AM. Crank-Nicolson FDM for solving time-fractional diffusion equation. J Fract Cal Appl 2012; 2: 1-9.
[38] Sweilam NH, Khader MM, Nagy AM. Numerical solution of two-sided space-fractional wave equation using finite difference method. J Comput Appl Math 2011; 235: 2832-2841.
[39] Sweilam NH, Nagy AM. Numerical solution of fractional wave equation using Crank-Nicholson method. World Appl Sci J 2011; 13: 71-75.
[40] Sweilam NH, Nagy AM, El-Sayed Adel A. Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation. Chaos Soliton Fract 2015; 73: 141-147.
[41] Sweilam NH, Nagy AM, El-Sayed Adel A. On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind. Journal of King Saud University-Science 2016; 28: 41-47.
[42] Tadjeran C, Meerschaert M, Scheffler HP. A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 2006; 213: 205-213.
[43] Tariboon J, Ntouyas SK, Agarwal P. New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv Differ Equ-Ny 2015; 2015: 18.
[44] Tatari M, Haghighi M. A generalized Laguerre-Legendre spectral collocation method for solving initial-boundary value problems. Appl Math Model 2014; 38: 1351-1364.
[45] Zaslavsky GM, Stevens D, Weitzner H. Self-similar transport in incomplete chaos. Phys Rev E 1993; 48: 1683-1694.