Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

Journal of Computational and Applied Mathematics

Volumn 236, Issue 13, 2012, Pages 3349-3359

  Pedas, Arvet ^a   Tamme, Enn ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Boundary value problem; Caputo derivative; Fractional differential equation; Graded grid; Spline collocation method

Indexed keywords

ATTAINABLE ORDER OF CONVERGENCE; CAPUTO DERIVATIVES; COLLOCATION POINTS; CONVERGENCE RATES; EXACT SOLUTION; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS; GRADED GRID; LINEAR BOUNDARY VALUE PROBLEM; NUMERICAL EXAMPLE; NUMERICAL SOLUTION; PIECEWISE POLYNOMIALS; REGULARITY PROPERTIES; SPLINE COLLOCATION METHOD; THEORETICAL RESULT;

ALGORITHMS;

BOUNDARY VALUE PROBLEMS;

EID: 84859537689     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2012.03.002     Document Type: Article

References (26)

1. K. Diethelm The Analysis of Fractional Differential Equations Lecture Notes in Mathematics vol. 2004 2010 Springer Berlin
2. H. Brunner, A. Pedas, and G. Vainikko Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels SIAM J. Numer. Anal. 39 2001 957 982
3. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies vol. 204 2006 Elsevier Amsterdam
4. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego
5. S.G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives. Theory and Applications 1993 Gordon and Breach Science Publishers Yverdon
6. R. Agarwal, M. Benchohra, and H. Hamani A survey on existence results for boundary value problems of nonlinear fractional equations and inclusion Acta Appl. Math. 109 2010 973 1033
7. R.P. Agarwal, and B. Ahmad Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions Comput. Math. Appl. 62 2011 1200 1214
8. B. Ahmad, J.J. Nieto, and J. Pimentel Some boundary value problems of fractional differential equations and inclusions Comput. Math. Appl. 62 2011 1238 1250
9. M. ur Rehman, R.A. Khan, and N.A. Asif Three point boundary value problems for nonlinear fractional differential equations Acta Math. Sci. 31B 2011 1337 1346
10. X. Yang, Z. Wei, and W. Dong Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations Commun. Nonlinear Sci. Numer. Simul. 17 2012 85 92
11. J.T. Edwards, N.J. Ford, and A.C. Simpson The numerical solution of linear multi-term fractional differential equations: systems of equations J. Comput. Appl. Math. 148 2002 401 418
12. K. Diethelm, and N.J. Ford Numerical solution of the Bagley-Torvik equation BIT 42 2002 490 507
13. K. Diethelm, N.J. Ford, and A.D. Freed Detailed error analysis for a fractional Adams method Numer. Algorithms 36 2004 31 52
14. N.J. Ford, and J.A. Connoly Systems-based decomposition schemes for approximate solution of multi-term fractional differential equations J. Comput. Appl. Math. 229 2009 382 391
15. R. Garrappa On some explicit Adams multistep methods for fractional differential equations J. Comput. Appl. Math. 229 2009 392 399
16. A. Pedas, and E. Tamme Spline collocation methods for linear multi-term fractional differential equations J. Comput. Appl. Math. 236 2011 167 176
17. N.J. Ford, and M.L. Morgado Fractional boundary value problems: analysis and numerical methods Fract. Calc. Appl. Anal 14 2011 554 567
18. E.H. Doha, A.H. Bhrawy, and S.S. Ezz-Eldien A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order Comput. Math. Appl. 62 2011 2364 2373
19. G.M. Al-Mdallal, M.I. Syam, and M.N. Anwar A collocation-shooting method for solving fractional boundary value problems Commun. Nonlinear Sci. Numer. Simul. 15 2010 3814 3822
20. M. ur Rehman, and R.A. Khan A Numerical method for solving boundary value problems for fractional differential equations Appl. Math. Model. 36 2012 894 907
21. G. Vainikko Multidimensional Weakly Singular Integral Equations Lecture Notes in Mathematics vol. 1549 1993 Springer Berlin
22. H. Brunner Collocation Methods for Volterra Integral and Related Functional Equations Cambridge Monographs on Applied and Computational Mathematics vol. 15 2004 Cambridge University Press Cambridge, UK
23. M. Kolk, A. Pedas, and G. Vainikko High-order methods for Volterra integral equations with general weak singularities Numer. Funct. Anal. Optim. 30 2009 1002 1024
24. A. Pedas, and E. Tamme Spline collocation method for integro-differential equations with weakly singular kernels J. Comput. Appl. Math. 197 2006 253 269
25. A. Pedas, and E. Tamme Product integration for weakly singular integro-differential equations Math. Model. Anal. 16 2011 153 172
26. A. Pedas, and E. Tamme On the convergence of spline collocation methods for solving fractional differential equations J. Comput. Appl. Math. 235 2011 3502 3514