An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation

Mathematical Methods in the Applied Sciences

Volumn 38, Issue 1, 2015, Pages 27-36

  Aslan, Ismail ^a  

^a IZMIR INSTITUTE OF TECHNOLOGY   (Turkey)

Author keywords

(G' G) expansion method; Differential difference equation; Fractional derivative

Indexed keywords

DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; HYPERBOLIC FUNCTIONS;

(G'/G) EXPANSION METHODS; ANALYTICAL ALGORITHMS; DIFFERENTIAL-DIFFERENCE EQUATIONS; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL; MODIFIED RIEMANN-LIOUVILLE DERIVATIVE; PHYSICAL INTERPRETATION; SCIENCE AND ENGINEERING;

RATIONAL FUNCTIONS;

EID: 84919399370     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.3047     Document Type: Article

References (46)

1. Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley: New York, 1993.
2. Oldham KB, Spanier J. The Fractional Calculus. Academic Press: New York, London, 1974.
3. Podlubny I. Fractional Differential Equations. Academic Press: San Diego, 1999.
4. Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach: Yverdon, 1993.
5. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Elsevier: San Diego, 2006.
6. Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 2006; 51: 1367-1376.
7. Cresson J. Non-differentiable variational principles. Journal ofMathematical Analysis and Applications 2005; 307: 48-64.
8. Kolwankar KM, Gangal AD. Local fractional Fokker-Planck equation. Physical Review Letters 1998; 80: 214-217.
9. Cui M. Compact finite difference method for the fractional diffusion equation. Journal of Computational Physics 2009; 228: 7792-7804.
10. Huang Q, Huang G, Zhan H. A finite element solution for the fractional advection-dispersion equation. Advances inWater Resources 1578; 31.
11. Odibat Z, Momani S. A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters 2008; 21: 194-199.
12. El-Sayed AMA, Gaber M. The Adomian decomposition method for solving partial differential equations of fractal order in finite domains. Physics Letters A 2006; 359: 175-182.
13. Odibat Z, Momani S. The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics. Computers & Mathematics with Applications 2009; 58: 2199-2208.
14. He JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000; 35: 37-43.
15. Lu B. The first integral method for some time fractional differential equations. Journal ofMathematical Analysis and Applications 2012; 395: 684-693.
16. Zhang S, Zhang HQ. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A 2011; 375: 1069-1073.
17. Lakestani M, Dehghan M, Irandoust-pakchin S. The construction of operational matrix of fractional derivatives using B-spline functions. Communications in Nonlinear Science and Numerical Simulation 2012; 17: 1149-1162.
18. Saadatmandi A, Dehghan M. A tau approach for solution of the space fractional diffusion equation. Computers & Mathematicswith Applications 2011; 62: 1135-1142.
19. Dehghan M, Manafian J, Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations 2010; 26: 448-479.
20. Dehghan M, Manafian J, Saadatmandi A. The solution of the linear fractional partial differential equations using the homotopy analysis method. Zeitschrift fur Naturforschung 2010; 65a: 935-949.
21. Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Computers & Mathematicswith Applications 2010; 59: 1326-1336.
22. Fermi E, Pasta J, Ulam S. Collected Papers of Enrico Fermi. University of Chicago Press: Chicago, 1965.
23. Toda M. Theory of Nonlinear Lattices. Springer-Verlag: New-York, 1989.
24. Dai CQ, Wang YY. Exact traveling wave solutions of the discrete nonlinear Schrodinger equation and the hybrid lattice equation obtained via the exp-function method. Physica Scripta 2008; 78: 015013.
25. Ma WX, You Y. Rational solutions of the Toda lattice equation in Casoratian form. Chaos, Solitons & Fractals 2004; 22: 395-406.
26. Zhu SD, Chu YM, Qiu SL. The homotopy perturbation method for discontinued problems arising in nanotechnology. Computers & Mathematics with Applications 2009; 58: 2398-2401.
27. Yang P, Chen Y, Li ZB. ADM-Padé technique for the nonlinear lattice equations. AppliedMathematics and Computation 2009; 210: 362-375.
28. Hu XB, Ma WX. Application of Hirota's bilinear formalism to the Toeplitz lattice-some special soliton-like solutions. Physics Letters A 2002; 293: 161-165.
29. Zhang S, Dong L, Ba JM, Sun YN. The (G'/G)-expansion method for nonlinear differential-difference equations. Physics Letters A 2009; 373: 905-910.
30. Saadatmandi A, Dehghan M. Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients. Computers & Mathematics with Applications 2010; 59: 2996-3004.
31. He JH, Wu GC, Austin F. The variational iteration method which should be followed. Nonlinear Science Letters A 2010; 1: 1-30.
32. Zhang S, Dong L, Ba JM, Sun YN. The (G'/G)-expansion method for a discrete nonlinear Schrodinger equation. Pramana - Journal of Physics 2010; 74: 207-218.
33. Bin Z. (G'/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoretical Physics 2012; 58: 623-630.
34. Gepreel KA, Omran S. Exact solutions for nonlinear partial fractional differential equations. Chinese Physics B 2012; 21: 110204.
35. Jumarie G. Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative. Applied Mathematics Letters 2009; 22: 1659-1664.
36. Wu GC, Lee EWM. Fractional variational iteration method and its application. Physics Letters A 2010; 374: 2506-2509.
37. Jumarie G. New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations. Mathematical and ComputerModelling 2006; 44: 231-254.
38. Li ZB, He JH. Fractional complex transformation for fractional differential equations. Computers & Mathematics with Applications 2010; 15: 970-973.
39. Aslan I. A note on the (G'/G)-expansion method again. AppliedMathematics and Computation 2010; 217: 937-938.
40. Narita K. New discrete modified KdV equation. Progress of Theoretical Physics 1991; 86: 817-824.
41. Narita K. N-soliton solution of a lattice equation related to the discrete MKdV equation. Journal of Mathematical Analysis and Applications 2011; 381: 963-965.
42. Narita K. Solutions for a system of difference-differential equations related to the self-dual network equation. Progress of Theoretical Physics 2001; 106: 1079-1096.
43. Dehghan M, Manafian J, Saadatmandi A. Analytical treatment of some partial differential equations arising in mathematical physics by using the exp-function method. International Journal of Modern Physics B 2011; 25: 2965-2981.
44. Dehghan M, Heris JM. Application of the exp-function method for solving a partial differential equation arising in biology and population genetics. International Journal for Numerical Methods in Heat and Fluid Flow 2011; 21: 736-753.
45. Aslan I., Marinakis V. Some remarks on Exp-function method and its applications. Communications in Theoretical Physics 2011; 56: 397-403.
46. Aslan I. Some remarks on Exp-function method and its applications-A supplement. Communications in Theoretical Physics 2013; 60: 521-525.