On the convergence of spline collocation methods for solving fractional differential equations

Journal of Computational and Applied Mathematics

Volumn 235, Issue 12, 2011, Pages 3502-3514

  Pedas, Arvet ^a   Tamme, Enn ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Caputo derivative; Convergence analysis; Fractional differential equation; Graded grid; Spline collocation method; Volterra integral equation

Indexed keywords

CAPUTO DERIVATIVES; CONVERGENCE ANALYSIS; FRACTIONAL DIFFERENTIAL EQUATIONS; GRADED GRID; SPLINE COLLOCATION METHOD; VOLTERRA INTEGRAL EQUATIONS;

DIFFERENTIAL EQUATIONS; INITIAL VALUE PROBLEMS; POLYNOMIAL APPROXIMATION; SPLINES;

INTEGRAL EQUATIONS;

EID: 79953828683     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2010.10.054     Document Type: Article

References (28)

1. 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
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. S.G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives. Theory and Applications 1993 Gordon and Breach Science Publishers Yverdon
4. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego
5. mE methods Computing 71 2003 305 319
6. K. Diethelm, and N.J. Ford Numerical solution of the BagleyTorvik equation BIT 42 2002 490 507
7. K. Diethelm, and N.J. Ford Multi-order fractional differential equations and their numerical solution Appl. Math. Comput. 154 2004 621 640
8. 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 (Pubitemid 36409200)
9. A.E.M. El-Mesiry, A.M.A. El-Sayed, and H.A.A. El-Saka Numerical methods for multi-term fractional (arbitrary) orders differential equations Appl. Math. Comput. 160 2005 683 699
10. 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
11. R. Garrappa On some explicit Adams multistep methods for fractional differential equations J. Comput. Appl. Math. 229 2009 392 399
12. J.T. Katsikadelis Numerical solution of multi-term fractional differential equations Z. Angew. Math. Math. 89 2009 593 608
13. H. Brunner, and P.J. van der Houven The Numerical Solution of Volterra Equations 1986 North-Holland Amsterdam
14. C. Lubich Fractional linear multistep methods for Abel-Volterra integral equations of the second kind Math. Comp. 45 1985 463 469
15. Y. enesiz, Y. Keskin, and A. Kurnaz The solution of the BagleyTorvik equation with the generalized Taylor collocation method J. Franklin Inst. 347 2010 452 466
16. V. Daftardar-Gejji, and H. Jafari Solving a multi-order fractional differential equation using Adomian decomposition Appl. Math. Comput. 189 2007 541 548 (Pubitemid 46734260)
17. S. Momani, and Z. Obibat Numerical comparison of methods for solving linear differential equations of fractional order Chaos Solitons Fractals 31 2007 1248 1255 (Pubitemid 44345225)
18. L. Blank Numerical treatment of differential equations of fractional order Nonlinear World 4 1997 473 491
19. F. Dubois, and S. Mengu Mixed collocation for fractional differential equations Numer. Algorithms 34 2003 303 311 (Pubitemid 39063573)
20. E. Rawashdeh Numerical solution of semidifferential equations by collocation method Appl. Math. Comput. 174 2006 869 876
21. E. Rawashdeh Numerical solution of fractional integro-differential equations by collocation method Appl. Math. Comput. 176 2006 1 6
22. Lepik Solving fractional integral equations by the Haar wavelet method Appl. Math. Comput. 214 2009 468 478
23. K.E. Atkinson The Numerical Solution of Integral Equations of the Second Kind Cambridge Monographs on Applied and Computational Mathematics vol. 4 1997 Cambridge University Press Cambridge, UK
24. 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
25. A. Pedas, and E. Tamme Spline collocation method for integro-differential equations with weakly singular kernels J. Comput. Appl. Math. 197 2006 253 269 (Pubitemid 44176254)
26. G. Vainikko Multidimensional Weakly Singular Integral Equations Lecture Notes in Mathematics vol. 1549 1993 Springer Berlin
27. A. Pedas, and E. Tamme Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels J. Comput. Appl. Math. 213 2006 111 126
28. I. Parts, Piecewise polynomial collocation methods for solving weakly singular integro-differential equations, Ph.D. Thesis, Tartu, 2005. Available from the Internet: http://dspace.utlib.ee/bitstream/10062/851/5/parts.pdf.