Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Applied Mathematics and Computation

Volumn 216, Issue 8, 2010, Pages 2276-2285

  Li, Yuanlu ^a   Zhao, Weiwei ^a  

^a NANJING UNIVERSITY OF INFORMATION SCIENCE AND TECHNOLOGY   (China)

Author keywords

Fractional calculus; Fractional order differential equations; Haar wavelet; Operational matrix

Indexed keywords

FRACTIONAL CALCULUS; FRACTIONAL ORDER; FRACTIONAL ORDER DIFFERENTIAL EQUATIONS; HAAR WAVELETS; OPERATIONAL MATRICES;

CALCULATIONS; DIFFERENTIAL EQUATIONS; NUMERICAL ANALYSIS; WAVELET TRANSFORMS;

DIFFERENTIATION (CALCULUS);

EID: 77953135236     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2010.03.063     Document Type: Article

References (34)

1. Odibat Z., and Momani S. Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 32 (2008) 28-39
2. Momani S., and Odibat Z. Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207 (2007) 96-110
3. El-Wakil S.A., Elhanbaly A., and Abdou M.A. Adomian decomposition method for solving fractional nonlinear differential equations. Appl. Math. Comput. 182 (2006) 313-324
4. 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
5. Das S. Analytical solution of a fractional diffusion equation by variational iteration method. Comput. Math. Appl. 57 (2009) 483-487
6. Arikoglu A., and Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos, Solitons Fract. 34 (2007) 1473-1481
7. Arikoglu A., and Ozkol I. Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons Fract. 40 (2009) 521-529
8. Darania P., and Ebadian A. A method for the numerical solution of the integro-differential equations. Appl. Math. Comput. 188 (2007) 657-668
9. Erturk V.S., and Momani S. Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215 (2008) 142-151
10. Erturk V.S., Momani S., and Odibat Z. Application of generalized differential transform method to multi-order fractional differential equations. Comm. Nonlinear Sci. Numer. Simulat. 13 (2008) 1642-1654
11. Meerschaert M., and Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56 (2006) 80-90
12. Odibat Z., and Shawagfeh N. Generalized Taylor's formula. Appl. Math. Comput. 186 (2007) 286-293
13. I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, 1997, eprint, .
14. Wu J.L. A wavelet operational method for solving fractional partial differential equations numerically. Appl. Math. Comput. 214 (2009) 31-40
15. Lepik U. Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput. 214 (2009) 468-478
16. Chen C., and Hsiao C. Haar wavelet method for solving lumped and distributed-parameter systems. IEE P.-Contr. Theor. Ap. 144 (1997) 87-94
17. Bujurke N., Salimath C., and Shiralashetti S. Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series. Nonlinear Dynam. 51 (2008) 595-605
18. Karimi H., Lohmann B., Maralani P., and Moshiri B. A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets. Int. J. Comput. Math. 81 (2004) 1121-1132
19. Pawlak M., and Hasiewicz Z. Nonlinear system identification by the Haar multiresolution analysis. IEEE T. Circuits-I 45 (1998) 945-961
20. Hsiao C., and Wang W. Optimal control of linear time-varying systems via Haar wavelets. J. Optimiz. Theory App. 103 (1999)
21. Karimi H., Moshiri B., Lohmann B., and Maralani P. Haar wavelet-based approach for optimal control of second-order linear systems in time domain. J. Dyn. Control Syst. 11 (2005) 237-252
22. Babolian E., and Shahsavaran A. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets. J. Comput. Appl. Math. 225 (2009) 87-95
23. Bujurke N.M., Shiralashetti S.C., and Salimath C.S. An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations. J. Comput. Appl. Math. 227 (2009) 234-244
24. Guf J., and Jiang W. The Haar wavelets operational matrix of integration. Int. J. Syst. Sci. 27 (1996) 623-628
25. Maleknejad K., and Mirzaee F. Using rationalized Haar wavelet for solving linear integral equations. Appl. Math. Comput. 160 (2005) 579-587
26. Maleknejad K., Mirzaee F., and Abbasbandy S. Solving linear integro-differential equations system by using rationalized Haar functions method. Appl. Math. Comput. 155 (2004) 317-328
27. Ordokhani Y. Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Haar functions. Appl. Math. Comput. 180 (2006) 436-443
28. Razzaghi M., and Ordokhani Y. An application of rationalized Haar functions for variational problems. Appl. Math. Comput. 122 (2001) 353-364
29. Razzaghi M., and Yousefi S. The Legendre wavelets operational matrix of integration. Int. J. Syst. Sci. 32 (2001) 495-502
30. Reihani M.H., and Abadi Z. Rationalized Haar functions method for solving Fredholm and Volterra integral equations. J. Comput. Appl. Math. 200 (2007) 12-20
31. Oldham K., and Spanier J. The Fractional Calculus (1974), Academic Press, New York
32. Kilicman A., and Al Zhour Z.A.A. Kronecker operational matrices for fractional calculus and some applications. Appl. Math. Comput. 187 (2007) 250-265
33. Diethelm K., and Ford J. Numerical solution of the Bagley-Torvik equation. Bit. Numer. Math. 42 (2002) 490-507
34. El-Mesiry A., El-Sayed A., and El-Saka H. Numerical methods for multi-term fractional (arbitrary) orders differential equations. Appl. Math. Comput. 160 (2005) 683-699