##
**Solution of fractional differential equations by using differential transform method.**
*(English)*
Zbl 1152.34306

Summary: We implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Riccati and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.

### MSC:

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

26A33 | Fractional derivatives and integrals |

45D05 | Volterra integral equations |

PDFBibTeX
XMLCite

\textit{A. Arikoglu} and \textit{I. Ozkol}, Chaos Solitons Fractals 34, No. 5, 1473--1481 (2007; Zbl 1152.34306)

Full Text:
DOI

### References:

[1] | Ahmad WM, El-Khazali R. Fractional-order dynamical models of love. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2006.01.098; Ahmad WM, El-Khazali R. Fractional-order dynamical models of love. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2006.01.098 · Zbl 1133.91539 |

[2] | Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), Academic Press: Academic Press New York · Zbl 0924.34008 |

[3] | Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J Roy Austral Soc, 13, 529-539 (1967) |

[4] | Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31, 1248-1255 (2007) · Zbl 1137.65450 |

[5] | Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl Math Comput, 131, 517-529 (2002) · Zbl 1029.34003 |

[6] | Momani S, Shawagfeh N. Decomposition method for solving fractional Riccati differential equations. Appl Math Comput, in press, doi:10.1016/j.amc.2006.05.008; Momani S, Shawagfeh N. Decomposition method for solving fractional Riccati differential equations. Appl Math Comput, in press, doi:10.1016/j.amc.2006.05.008 · Zbl 1107.65121 |

[7] | Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys Lett A, 355, 271-279 (2006) · Zbl 1378.76084 |

[8] | Jumarie, G., Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations, Chaos, Solitons & Fractals, 22, 907-925 (2004) · Zbl 1068.60053 |

[9] | Benghorbal MM. Power series solution of fractional differential equations and symbolic derivatives and integrals. PhD thesis, The University of Western Ontario, London, Ontorio, 2004.; Benghorbal MM. Power series solution of fractional differential equations and symbolic derivatives and integrals. PhD thesis, The University of Western Ontario, London, Ontorio, 2004. |

[10] | Arikoglu, A.; Ozkol, I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl Math Comput, 168, 1145-1158 (2005) · Zbl 1090.65145 |

[11] | Arikoglu, A.; Ozkol, I., Solution of difference equations by using differential transform method, Appl Math Comput, 174, 442-454 (2006) |

[12] | Arikoglu A, Ozkol I. Solution of differential-difference equations by using differential transform method. Appl Math Comput, in press, doi:10.1016/j.amc.2006.01.022; Arikoglu A, Ozkol I. Solution of differential-difference equations by using differential transform method. Appl Math Comput, in press, doi:10.1016/j.amc.2006.01.022 · Zbl 1148.65310 |

[13] | Diethelm, K.; Ford, N. J., Numerical solution of the Bagley-Torvik equation, Bit, 42, 490-507 (2004) · Zbl 1035.65067 |

[14] | El-Mesiry, A. E.M.; El-Sayed, A. M.A.; El-Saka, H. A.A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl Math Comput, 160, 683-699 (2005) · Zbl 1062.65073 |

[15] | Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J Math Anal Appl, 265, 229-248 (2002) · Zbl 1014.34003 |

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.