A new analytical approximate method for the solution of fractional differential equations

International Journal of Computer Mathematics

Volumn 85, Issue 1, 2008, Pages 131-142

  Oturanç, Galip ^a   Kurnaz, Aydin ^a   Keskin, Yildiray ^a  

^a SELCUK UNIVERSITY   (Turkey)

Author keywords

Adomian decomposition; Fractional differential equations; Fractional differential transformation; Non integer order; Series solution

Indexed keywords

COMPUTATIONAL METHODS; DIFFERENTIAL EQUATIONS; ITERATIVE METHODS; MATHEMATICAL TRANSFORMATIONS;

ADOMIAN DECOMPOSITIONS; APPROXIMATE METHODS; FRACTIONAL DIFFERENTIAL EQUATIONS;

APPROXIMATION THEORY;

EID: 37749035318     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/00207160701405477     Document Type: Article

References (31)

1. Oldham, K.B. and Spanier, J., 1974, The Fractional Calculus (New York: Academic Press).
2. Lu, J.G., 2006, Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos, Solitons &Fractals, 27, 519-525.
3. Lu, J.G. and Chen, G., 2006, A note on the fractional-order Chen system. Chaos, Solitons & Fractals, 27, 685-688.
4. He, J.H., 1998, Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computational Methods in Applied Mechanical Engineering, 167, 57-68.
5. Luchko, Y. and Gorenflo, R., 1998, The initial value problem for some fractional differential equations with the Caputo derivative. Preprint Series A08-98, Fachbereich Mathematik und Informatik, Frei Universität Berlin.
6. Sheu, L.J., Chen, H.K., Chen, J.H. and Tam, L.M., 2007, Chaotic dynamics of the fractionally damped Duffing equation, Chaos, Solitons & Fractals, 31, 1203-1212.
7. El-Sayed, A.M.A., 1992, On the fractional differential equations. Applied Mathematics and Computation, 49, 205-213.
8. El-Sayed, A.M.A., 1993, Linear differential equations of fractional orders. Applied Mathematics and Computation, 55, 1-12.
9. Kemple, S. and Beyer, H., 1997, Global and causal solutions of fractional differential equations. In Transform Methods and Special Functions: Varna96, Proceedings of 2nd International Workshop (SCTP), pp. 210-216.
10. Momani, S. and Odibat, Z., 2007, Numerical comparision of methods for solving linear differential equations of fractional order. Chaos, Solitons &Fractals, 31(5), 1248-1255.
11. Yan, J. and Li, C., 2007, On chaos synchronization of fractional differential equations. Chaos, Solitons & Fractals, 32, 725-735.
12. Luchko, Y. and Srivastava, H.M., 1995, The exact solution of certain differential equations of fractional order by using operational calculus. Computers and Mathematics with Applications, 29, 73-85.
13. Momani, S. and Abuasad, S., 2006, Application of He's variational-iteration method to Helmholtz equation. Chaos, Solitons & Fractals, 27, 1119-1123.
14. Shawagfeh, N.T., 2002, Analytical approximate solutions for nonlinear fractional differential equations. Applied Mathematics and Computation, 131, 517-529.
15. Bulut, H. and Evans, D.J., 2001, On solution of the Ricatti equation by the decompositions method. International Journal of Computer Mathematics, 80, 1-7.
16. Ray, S.S. and Bera, R.K., 2005, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Applied Mathematics and Computation, 167, 561-571.
17. George, A.J. and Chakrabarti, A., 1995, The Adomian method applied to some extraordinary differential equations. Applied Mathematics Letters, 8, 91-97.
18. Asil, V., Bulut, H., and Evans, D.J., 2005, The Adomian decomposition method for the approximate solution of homogeneous differential equations with dual variable and dual coefficients. International Journal of Computer Mathematics, 82, 977-986.
19. Odibat, Z.M. and Momani, S., 2006, Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 27-34.
20. Dragenescu, G.E., Cofan, N., and Rujan, D.L., 2005, Nonlinear vibrations of a nano sized sensor with fractional damping. Journal of Optoelectronics and Advanced Materials, 7, 877-884.
21. He, J.H., 2006, Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20, 1141-1199.
22. He, J.H., 2006, Non-Perturbative Methods for Strongly Nonlinear Problems (Berlin: Dissertation.de-Verlag im Internet).
23. Arikoglu, A. and Ozkol, I., 2007, Solution of fractional differential equations by using differential transform method. Chaos, Solitons & Fractals, 34, 1473-1481.
24. Chen, C.K. and Ju, S.P., 2004, Application of differential transformation to transient advective-dispersive transport equation. Applied Mathematics and Computation, 155, 25-38.
25. Abdel-Halim Hassan, I.H., 2002, Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation, 129, 183-201.
26. Kurnaz, A. and Oturanç, G., 2005,The differential transform approximation for the system of ordinary differential equations. International Journal of Computer Mathematics, 82, 709-719.
27. Bildik, N. and Konuralp, A., 2006, The use of variatonal iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 65-70.
28. Diethelm, K., Ford, N.J., Freed, A.D., and Luchko, Y., 2005, Algorithms for the fractional calculus: a selection of numerical methods. Computer Methods in Applied Mechanics and Engineering, 194, 743-773.
29. Gorenflo, R. and Mainardi, F., 1997, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi (Eds) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223-276 (Vienna: Springer).
30. Odibat, Z.M. and Shawagfeh, N.T., 2006, Generalized Taylor's formula. Applied Mathematics and Computation, 186, 286-293.
31. Oldham, K.B., A unified treatment of electrolysis at an expanding mercury electrode. Analytical, Chemistry, 41, 936.