Operational Tau method for nonlinear multi-order FDEs

Iranian Journal of Numerical Analysis and Optimization

مقاله 3، دوره 4، شماره 2 - شماره پیاپی 6، بهمن 2014، صفحه 43-55   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v4i2.33577

نویسنده

P. Mokhtary^*

Department of Mathematics, Sahand University of Technology, Tabriz, Iran.

چکیده

This paper presents an operational formulation of the Tau method based upon orthogonal polynomials by using a reduced set of matrix operations for the numerical solution of nonlinear multi-order fractional differential equations(FDEs). The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of non-linear algebraic equations. Some numerical examples are provided to demonstrate the validity and applicability of the method.

کلیدواژه‌ها

Fractional differential equations(FDEs)؛ Caputo derivative؛ Operational Tau method

مراجع

1. Abbasbandy, S. and Taati, A. Numerical solution of the system of nonlin-ear Volterra integro-differential equations with nonlinear differential partby the operational Tau method and error estimation, J. Comput. Appl.Math., 231 (2009) 106-113.

2. Arikoglu, A. and Ozkol, I. Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals, 34 (2007) 1473-1481.

3. Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T. Spectral methodsfundamentals in single domains,Springer-Verlag, Berlin, 2006.

4. Cang, J., Tan, Y., Xu, H. and Liao, S. Series solutions of non-linear Riccatidifferential equations with fractional order, Chaos Solitons Fractals, 40 (2009) 1-9.

5. Diethelm, K. and Walz, G. Numerical solution of fractional order differ-ential equations by extrapolation, Numer. Algorithms, 16 (1997) 231-253.

6. Freilich, J. and Ortiz, E. Numerical solution of systems of ordinary differ-ential equations with the Tau method, an error analysis, Math. Comp., 39(1982) 467-479.

7. Galeone, L. and Garrappa, R. On multistep methods for differential equa-tions of fractional order, Mediterr. J. Math., 3 (2006) 565-580.

8. Ghorbani, A. Toward a new analytical method for solving nonlinear frac-tional differential equations,Comput. Methods Appl. Mech. Engrg., 197(2009) 4173-4179.

9. Ghoreishi, F. and Yazdani, S. An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. Math. Appl., 61 (2011) 30-43.

10. Ghoreishi, F. and Hadizadeh, M. Numerical computation of the Tau ap-proximation for the Volterra-Hammerstein integral equations, Numer. Algorithms,52 (2009) 541-559.

11. Gottlieb, D. and Orszag, S. Numerical analysis of spectral methods, the-ory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 26. SIAM, Philadelphia, 1997.

12. Jafari, H. and Sei, S. Solving a system of non-linear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 1962-1969.

13. Jafari, H. and Daftardar-Gejji, V. Solving a system of nonlinear frac-tional differential equations using Adomian decomposition, J. Comput.Appl. Math., 196 (2006) 644-651.

14. Kilbas, A., Srivastava, M. and Trujillo, J. Theory and applications of fractional differential equations, Elsevier, 2006.

15. Lanzos, C. Trigonometric interpolation of empirical and analytical func-tions, J. Math. Phys., 17 (1983) 123-199.

16. Lanczos, C. Introduction, tables of Chebyshev polynomials, Appl. Math. Ser. US Bur. Stand., 9, Government Printing Office Washington, 1952.

17. Lanczos, C. Applied analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956.

18. Lin, R. and Liu, F. Fractional high order methods for the nonlinear frac-tional ordinary differential equation, Nonlinear Anal., 66 (2007) 856-869.

19. Mokhtary, P. and Ghoreishi, F. The L2-convergence of the legendre spec-tral Tau matrix formulation for onlinear fractional integro differential equations, Numer. lgorithms, 58 (2011) 475-496.

20. Momani, S. and Odibat, Z. Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals,31 (2007) 1248-1255.

21. Momani, S. and Shawagfeh, N. Decomposition method for solving frac-tional Riccati differential quations, Appl. Math. Comput., 182 (2006)1083-1092.

22. Odibat, Z. and Momani, S. Modified Homotopy perturbation method: application to quadratic Ricatti differential equation of fractional order,Chaos Solitons Fractals, 36 (2008) 167-174.

23. Onumanyi, P. and Ortiz, E. Numerical solution of stiff and singularity perturbed boundary value problems with a segmented-adaptive formulation of the Tau Method, Math. Comp., 43 (1984) 189-203.

24. Ortiz, E. and Dinh, A. An error analysis of the Tau method for a class of singularity perturbed problems for differential equations, Math. Methods Appl. Sci., 6 (1984) 457-466.

25. Ortiz, E. and Dinh, A. On the convergence of the Tau method for non-linear differential equations of icatti's type, Nonlinear Anal., 9 (1985)53-60.

26. Ortiz, E. The Tau method, SIAM J. Numer. Anal, 6 (1969) 480-492.

27. Ortiz, E. and Samara, H. An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27(1981) 15-25.

28. Ortiz, E. and Samara, H. Numerical solution of differential eigenvalue problems with an operational approach to the Tau method, Computing, 31(1983) 95-103.

29. Ortiz, E. and Samara, H. Numerical solution of partial differential equa-tions with variable coefficients with an operational approach to the Tau method, Comput. Math. Appl., 10 (1984) 5-13.

30. Podlubny, I. Fractional differential equations, Academic Press, 1999.