Hostname: page-component-6bf8c574d5-vmclg Total loading time: 0 Render date: 2025-02-16T17:26:19.463Z Has data issue: false hasContentIssue false
  • English
  • Français

A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

Part of: Real functions

Published online by Cambridge University Press:  04 November 2010

N. H. SWEILAM  and
M. M. KHADER
N. H. SWEILAM*
Affiliation:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt (email: [email protected])
M. M. KHADER
Affiliation:
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt (email: [email protected])
*
✉For correspondence; e-mail: n˙[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results.

Keywords

Chebyshev pseudo-spectral methodfractional-order integro-differential equations of Volterra type

MSC classification

Secondary: 26A33: Fractional derivatives and integrals
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 4 , April 2010 , pp. 464 - 475
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Arikoglu, A. and Ozkol, I., “Solution of fractional integro-differential equations by using fractional differential transform method”, Chaos Solitons Fractals 34 (2009) 521–529.CrossRefGoogle Scholar
[2]Bagley, R. L. and Torvik, P. J., “A theoretical basis for the application of fractional calculus to viscoelasticity”, J. Rheol. 27 (1983) 201–210.CrossRefGoogle Scholar
[3]Baillie, R. T., “Long memory processes and fractional integration in econometrics”, J. Econometrics 73 (1996) 5–59.CrossRefGoogle Scholar
[4]Caputo, M., “Linear models of dissipation whose Q is almost frequency independent. Part II”, Geophys. J. R. Astr. Soc. 13 (1967) 529–539.CrossRefGoogle Scholar
[5]Chow, T. S., “Fractional dynamics of interfaces between soft-nanoparticles and rough substrates”, Phys. Lett. A 342 (2005) 148–155.CrossRefGoogle Scholar
[6]Constantinides, A., Applied numerical methods with personal computers (McGraw-Hill, New York, 1987).Google Scholar
[7]Das, S., Functional fractional calculus for system identification and controls (Springer, New York, 2008).Google Scholar
[8]Diethelm, K., Ford, N. J. and Luchko, Yu., “Algorithms for the fractional calculus: a selection of numerical methods”, Comput. Methods Appl. Mech. Engrg. 194 (2005) 743–773.CrossRefGoogle Scholar
[9]Gorenflo, R. and Vessella, S., Abel integral equations (Springer, Berlin, 1991).CrossRefGoogle Scholar
[10]Hashim, I., Abdulaziz, O. and Momani, S., “Homotopy analysis method for fractional IVPs”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 674–684.CrossRefGoogle Scholar
[11]Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral methods in fluid dynamic (Prentice-Hall, Englewood Cliffs, NJ, 1988).Google Scholar
[12]Inc, M., “The approximate and exact solutions of the space-and time-fractional Burger’s equations with initial conditions by variational iteration method”, J. Math. Anal. Appl. 345 (2008) 476–484.CrossRefGoogle Scholar
[13]Kadem, A. and Baleanu, D., “Fractional radiative transfer equation within Chebyshev spectral approach”, Comput. Math. Appl. 59 (2010) 1865–1873.CrossRefGoogle Scholar
[14]Kadem, A. and Baleanu, D., “Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation”, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 491–501.CrossRefGoogle Scholar
[15]Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations (Elsevier, San Diego, 2006).Google Scholar
[16]Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations (John Wiley and Sons, New York, 1993).Google Scholar
[17]Momani, S. and Noor, M. A., “Numerical methods for fourth-order fractional integro-differential equations”, Appl. Math. Comput. 182 (2006) 754–760.Google Scholar
[18]Oldham, K. B. and Spanier, J., Fractional calculus: theory and applications, differentiation and integration to arbitrary order (Academic Press, New York, 1974).Google Scholar
[19]Podlubny, I., Fractional differential equations (Academic Press, San Diego, 1999).Google Scholar
[20]Rawashdeh, E. A., “Numerical solution of fractional integro-differential equations by collocation method”, Appl. Math. Comput. 176 (2006) 1–6.Google Scholar
[21]Saadatmandi, A. and Dehghan, M., “Numerical solution of the one-dimensional wave equation with an integral condition”, Numer. Methods Partial Differential Equations 23 (2007) 282–292.CrossRefGoogle Scholar
[22]Saadatmandi, A. and Dehghan, M., “Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method”, Comm. Numer. Methods Engrg. 24 (2008) 1467–1474.CrossRefGoogle Scholar
[23]Snyder, M.A., Chebyshev methods in numerical approximation (Prentice-Hall, Englewood Cliffs, NJ, 1966).Google Scholar
[24]Sweilam, N. H., Khader, M. M. and Al-Bar, R. F., “Numerical studies for a multi-order fractional differential equation”, Phys. Lett. A 371 (2007) 26–33.CrossRefGoogle Scholar
[25]Yuanlu, L., “Solving a nonlinear fractional differential equation using Chebyshev wavelets”, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 2284–2292.Google Scholar