Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method

Communications in Nonlinear Science and Numerical Simulation

Volumn 17, Issue 10, 2012, Pages 3934-3946

  Li, Xinxiu ^a  

^a SOUTHEAST UNIVERSITY   (China)

Author keywords

Caputo derivative; Cubic B spline function; Interpolating condition; Wavelet collocation method

Indexed keywords

DIFFERENTIAL EQUATIONS; INTERPOLATION; NUMERICAL METHODS;

B-SPLINE WAVELET; CAPUTO DERIVATIVES; CUBIC B -SPLINE; CUBIC B-SPLINE FUNCTION; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS; INTERPOLATING CONDITION; NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS; PHYSICAL PROCESS; WAVELET COLLOCATION METHOD;

COMPUTER PROGRAMMING;

EID: 84862819055     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2012.02.009     Document Type: Article

References (44)

1. Pedas Arvet, Tamme Enn Spline collocation methods for linear multi-term fractional differential equations. J Comput Appl Math 2011, 236:167-176.
2. Pedas Arvet, Tamme Enn On the convergence of spline collocation methods for solving fractional differential equations. J Comput Appl Math 2011, 235:3502-3514.
3. Saadatmandi A., Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl 2010, 59:1326-1336.
4. Hall C.A., Meryer W.W. Optimal error bounds for cubic spline interpolation. J Approx Theory 1976, 16:105-122.
5. Hwang C., Shih Y.P. Laguerre operational matrices for fractional calculus and applications. Int J Control 1981, 34(3):577-584.
6. Wang C.H. On the generalization of block-pulse operational matrices for fractional and operational calculus. J Franklin Inst 1983, 315(2):91-102.
7. Cui C.K. An introduction to wavelets 1992, Academic Press, San Diego, CA.
8. Rawashdeh E.A. Numerical solution of fractional integro-differential equations by collocation method. Appl Math Comput 2006, 176:1-6.
9. Wu G., Lee E.W.M. Fractional variational iteration method and its application. Phys Lett A 2010, 374:2506-2509.
10. Saeedi H., Mohseni Moghadam M., et al. Commun Nonlinear Sci Numer Simul 2011, 16:1154-1163.
11. Podlubny I. Fractional differential equations 1999, Academic Press.
12. Wu J.L. A wavelet operational method for solving fractional partial differential equations numerically. Appl Math Comput 2009, 214:31-40.
13. Wang J.Z. Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation. Appl Comput Harmonic Anal 1996, 3:154-163.
14. Diethelm K. An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal 1997, 5:1-6.
15. Diethelm K., Ford N.J., Freed A.D. Detailed error analysis for a fractional Adams method. Numer Alg 2004, 36:31-52.
16. Lakestani M., Dehghan M., Irandoust-pakchin S. The construction of operational matrix of fractional derivatives using B-spline functions. Commun Nonlinear Sci Numer Simul 2011, 10.1016/j.cnsns.2011.07.018.
17. Blank L. Numerical treatment of differential equations of fractional order. Manchester Centre Comput Math Numer Anal Rep 1996, 287.
18. Yuan L., Agrawal O.P. A numerical scheme for dynamic systems containing fractional derivatives. Vibr Acoust 2002, 124:321-324.
19. Ciesielski M, Leszczynski Jacek. Numerical simulations of anomalous diffusion. In: Computer Methods Mech, Conference Gliwice Wisla Poland, 2003.
20. Wang M.L., Chang R.Y., Yang S.Y. Generalization of generalized orthogonal polynomial operational matrices for fractional and operational calculus. Int J Syst Sci 1987, 18(5):931-943.
21. Khader M.M. Numerical solution of nonlinear multi-order fractioanl defferential equations by implementation of the operatioanl matrix of fractional derivative. Stud Nonlinear Sci 2011, 2(1):5-12.
22. ur Rehman, Khan Rahmat Ali A numerical method for solving boundary value problems for fractional differential equations. Appl Math Model 2011, 10.1016/j.apm.2011.07.045.
23. Ye M. Optimal error bounds for the cubic spline interpolation of lower smooth function (II). Appl Math JCU 1998, 13:223-230.
24. Ford N.J., Joseph Connolly A. Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. J Comput Appl Math 2009, 229:382-391.
25. Ford N.J., Charles Simpson A. The numerical solution of fractional differential equations: speed versus accuracy. Numer Alg 2001, 26:333-346.
26. Sweilam N.H., Khader M.M., Al-Bar R.F. Numerical studies for a multi-order fractional differential equation. Phys Lett A 2007, 371:26-33.
27. Kumar P., Agrawal O.P. An approximate method for numerical solution of fractional differential equations. Signal Process 2006, 86:2602-2610.
28. Kumar P. New numerical schemes for the solution of fractional differential equations. Southern Illinois University at Carbondale, vol. 134, 2006.
29. Kumar P., Agrawal O.P. Numerical scheme for the solution of fractional differential equations of order greater than one. J Comput Nonlinear Dyn. 2006, 1:178-185.
30. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics aproach. Phys Rep 2000, 339:1-77.
31. Chang R.Y., Chen K.C., Wang M.L. Modified Laguerre operational matrices for fractional calculus and applications. Int J Syst Sci 1985, 16(9):1163-1172.
32. Yousefi SA, Lotfi A, Dehghan M. The use of Legendre multiwavelet collocation method for solving the faractional optimal control problems. J Vibr Control [in press]. doi:10.1177/1077546311399950. doi:10.1177/1077546311399950.
33. Yuste S.B. Weighted average finite difference methods for fractional diffusion equations. J Comput Phys 2006, 216:264-274.
34. Momani S., Odibat Z. Analytical solution of a time-fractional Navier-Stokes equation by adomian decomposition method. Appl Math Comput 2006, 177:488-494.
35. Momani S., Odibat Z. Numerical approach to differential equations of fractional order. J Comput Appl Math 2007, 96-110.
36. Atanackovic T.M., Stankovic B. On a numerical scheme for solving differential equations of fractional order. Mech Res Commun 2008, 429-438.
37. Lucas T.R. Error bounds for interpolationg cubic splines under various end conditions. SIAM J Numer Anal 1974, 569-584.
38. Cai W., Wang J. Adaptive multi-resolution collocation methods for Initial boundary value problems of nonlinear PDEs. SIAM J Numer Anal 1996, 937-970.
39. Li Yuanlu, Sun Ning Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput Math Appl 2010, 216:1046-1054.
40. Li Yuanlu, Zhao Weiwei Haar wavelet operational matrix of fractional order integration and its applications in solving the fractioanl order differntial equations. Appl Math Comput 2010, 216:2276-2285.
41. Odibat Z., Momani S. Application of variational iteration method to equations of fractional order. Int J Nonlinear Sci Numer Simul 2006, 7:271-279.
42. Odibat Z. Approximations of fractional integrals and Caputo fractional derivatives. Appl Math Comput 2006, 527-533.
43. Odibat Z., Momani S. Numerical methods for nonlinear partial differential equations of fractional order. Appl Math Model 2008, 32:28-39.
44. Odibat Z.M. Computational algorithms for computing the fractional derivatives of functions. Math Comput Simul 2009, 79.