##
**Numerical solution of fractional integro-differential equations by collocation method.**
*(English)*
Zbl 1100.65126

The paper concerns the numerical solution of fractional integro-differential equations of the type \(D^qy(t)=p(t)y(t)+f(t)+\int_0^tK(t,s)y(s)ds, t\int=[0,1]\) by polynomial spline functions. Specific existing literature relating to fractional differential equations is cited. A definition of the Riemann-Liouville fractional differential operator of order \(q, q \notin \mathbb{N},\) is given.

After introducing the polynomial spline space the author derives a collocation spline method. The solution of an \(m \times m\) system of linear equations on each of \(N\) subintervals is involved. A theorem by L. Blank [Nonlinear World 4, No. 4, 473–491 (1997)] is stated and used and a new theorem is stated and proved. Two numerical examples are presented to illustrate the results. Each concerns a fractional integro-differential equation with a given initial condition and a known exact solution. Computed absolute errors in the numerical solution are given for three values of \(t\).

After introducing the polynomial spline space the author derives a collocation spline method. The solution of an \(m \times m\) system of linear equations on each of \(N\) subintervals is involved. A theorem by L. Blank [Nonlinear World 4, No. 4, 473–491 (1997)] is stated and used and a new theorem is stated and proved. Two numerical examples are presented to illustrate the results. Each concerns a fractional integro-differential equation with a given initial condition and a known exact solution. Computed absolute errors in the numerical solution are given for three values of \(t\).

Reviewer: Pat Lumb (Chester)

### MSC:

65R20 | Numerical methods for integral equations |

45J05 | Integro-ordinary differential equations |

26A33 | Fractional derivatives and integrals |

PDFBibTeX
XMLCite

\textit{E. A. Rawashdeh}, Appl. Math. Comput. 176, No. 1, 1--6 (2006; Zbl 1100.65126)

Full Text:
DOI

### References:

[1] | Bagley, R. L.; Trovik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27, 201-210 (1983) · Zbl 0515.76012 |

[2] | L. Blank, Numerical treatment of differential equation of fractional order, Numerical Analysis Report 287. Manchester Center for Numerical Computational Mathematics, 1996.; L. Blank, Numerical treatment of differential equation of fractional order, Numerical Analysis Report 287. Manchester Center for Numerical Computational Mathematics, 1996. |

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

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

[5] | A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint series A08-98, Fachbreich Mathematik and Informatik, Freic Universitat Berlin, 1998.; A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint series A08-98, Fachbreich Mathematik and Informatik, Freic Universitat Berlin, 1998. |

[6] | Momani, S. M., Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18, 81-86 (2000) · Zbl 0967.45004 |

[7] | Oldham, K. B.; Spanier, J., The fractional calculus, Mathematics in Science and Engineering, vol. 111 (1974), Academic Press: Academic Press New York · Zbl 0428.26004 |

[8] | Olmstead, W. E.; Handelsman, R. A., Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev., 18, 275-291 (1976) · Zbl 0323.45008 |

[9] | Podlubny, I., Numerical solution of ordinary fractional differential equations by the fractional difference method, (Advances in Difference Equation (Veszprém, 1995) (1997), Gordon and Breach: Gordon and Breach Amsterdam), 507-515 · Zbl 0893.65051 |

[10] | Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, Fractional Differential Equations, Some Methods of their Solution and Some of their Applications. Fractional Differential Equations, Some Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press San Diego · Zbl 0924.34008 |

[11] | E.A. Rawashdeh, Numerical solution of semidifferential equations by collocation method, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.05.029; E.A. Rawashdeh, Numerical solution of semidifferential equations by collocation method, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.05.029 · Zbl 1090.65097 |

[12] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gorden and Breach: Gorden and Breach Yverdon · Zbl 0818.26003 |

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.