×

A new approach to space fractional differential equations based on fractional order Euler polynomials. (English) Zbl 1499.49067

Summary: The fractional order Euler polynomials are introduced to obtain the solution of the class of space fractional diffusion equations. This is an innovative method for solving space fractional differential equations among the fractional calculus. These properties are utilized to transform the partial differential equation to algebraic equations with unknown Euler coefficients. The fractional derivatives are described based on the Caputo sense by using Riemann-Liouville fractional integral operator. A new hybrid function approximation based on fractional Euler polynomials and the algebraic polynomial has been initiated. The solution obtained by our method coincides with the solution obtained through other methods mentioned in the literature. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
35R11 Fractional partial differential equations
Full Text: DOI

References:

[1] M. Abramowitz, I. A. Stegun,Handbook of Mathematical Functions, National Bureau of Standards, Washington D.C., 1964. · Zbl 0171.38503
[2] M. Aslefallah, D. Rostamy,A numerical scheme for solving space-fractional equation by finite differences theta-method, Int. J. Adv. Appl. Math. Mech.1(4) (2014), 1-9. · Zbl 1359.65146
[3] H. Azizi, G. B. Loghmani,Numerical approximation for space fractional diffusion equations via chebyshev finite difference method, J. Fract. Calc. Appl.4(2) (2013), 303-311. · Zbl 1488.65219
[4] E. Barkai, R. Metzler, J. Klafter,From continuous time random walks to the fractional fokker-planck equation, Phys. Rev. E (3)61(2000), 132-138.
[5] D. A. Benson, S. Wheatcraft, M. M. Meerschaert,Application of a fractional advection dispersion equation, Water Resource Research36(6) (2000), 1403-1412.
[6] A. H. Bhrawy,A new numerical algorithm for solving a class of fractional advection - dispersion equation with variable coefficients using Jacobi polynomials, Abstr. Appl. Anal.2013 (2013), Article ID 954983, 9 p. · Zbl 1470.65172
[7] E. H. Doha, A. H. Bhrawy, D. Baleanu, S. S. Ezz-Eldien,The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation, Adv. Difference Equ. 231(2014), 1-14. · Zbl 1343.65126
[8] D. Elliott,An asymptotic analysis of two algorithms for certain Hadamard finitepart integrals, IMA J. Numer. Anal.13(1993), 445-462. · Zbl 0780.65014
[9] V. D. Gejji, H. Jafari,Solving a multi-order fractional differential equation, Appl. Math. Comput.189(2007), 541-548. · Zbl 1122.65411
[10] M. Giona, H. E. Roman,A theory of transport phenomena in disordered systems, Chem. Engin. J.49(1992), 1-10.
[11] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto,Fractional Calculus and Continuous-Time FinanceIII:the Diffusion Limit; in: M. Kohlmann, S. Tang (eds.),Mathematical Finance, Birkhäuser, Basel, 2001, 171-180. · Zbl 1138.91444
[12] R. Hilfer,Exact solutions for a class of fractal time random walks, Fractals3(1995), 211-216. · Zbl 0881.60066
[13] S. Kazem, S. Abbasbandy, Sunil Kumar,Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Modelling37(2013), 5498-5510. · Zbl 1449.33012
[14] M. M. Khader,On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul.16(2011), 2535-2542. · Zbl 1221.65263
[15] Q. M. Luo, F. Qi,Generalizations of Euler numbers and polynomials, RGMIA Research Report Collection5(3) (2002), 1-8.
[16] J. T. Machado, A. C. J. Luo, R. S. Barbosa, M. F. Silva, L. B. Figueiredo,Nonlinear Science and Complexity, Springer, New York, 2011.
[17] R. Magin, M. D. Ortigueira, I. Podlubny, J. J. Trujillo,On the fractional signals and systems, Signal Process.91(2011), 350-371. · Zbl 1203.94041
[18] F. Mainardi,Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. · Zbl 1210.26004
[19] M. M. Meerschaert, D. A. Benson, H. P. Scheffler, P. Becker-Kern,Governing equations and solutions of anomalous random walk limits, Phys. Rev. E (3)66(2002), 102-105.
[20] R. Metzler, E. Barkai, J. Klafter,Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett.82(18) (1999), 3563-3567.
[21] S. Momani,Non-perturbative analytical solutions of the space and time fractional Burgers equations, Chaos Solitons Fractals28(4) (2006), 930-937. · Zbl 1099.35118
[22] S. Momani, S. Abuasad,Application of He’s variational iteration method to Helmholtz equation, Chaos Solitons Fractals27(5) (2006), 1119-1123. · Zbl 1086.65113
[23] S. Momani, Z. Odibat,Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett., A355(2006), 271-279. · Zbl 1378.76084
[24] Z. Odibat, S. Momani,Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals36(1) (2008), 167- 174. · Zbl 1152.34311
[25] I. Podlubny,Fractional Differential Equations, Academic Press, New York, 1999. · Zbl 0924.34008
[26] S. Z. Rida, A. M. Yousef,On the fractional order Rodrigues formula for the Legendre polynomials, Adv. Appl. Math. Sci.10(2011), 509-518. · Zbl 1239.26008
[27] A. Saadatmandi, M. Dehghan,A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl.62(2011), 1135-1142. · Zbl 1228.65203
[28] E. Sousa,Numerical approximations for fractional diffusion equation via splines, Comput. Math. Appl.62(2011), 938-944. · Zbl 1228.65153
[29] C. Tadjeran, M. M. Meerschaert,A second order accurate numerical method for the twodimensional fractional diffusion equation, J. Comput. Phys.220(2007), 813-823 · Zbl 1113.65124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.