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Numerical solution of fractional differential equations using the generalized block pulse operational matrix. (English) Zbl 1228.65135
Summary: The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.

MSC:
65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
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[1] Laroche, E.; Knittel, D., An improved linear fractional model for robustness analysis of a winding system, Control eng. pract., 13, 659-666, (2005)
[2] Calderon, A.; Vinagre, B.; Feliu, V., Fractional order control strategies for power electronic buck converters, Signal process., 86, 2803-2819, (2006) · Zbl 1172.94377
[3] Sabatier, J.; Aoun, M.; Oustaloup, A.; Grgoire, G.; Ragot, F.; Roy, P., Fractional system identification for lead acid battery state of charge estimation, Signal process., 86, 2645-2657, (2006) · Zbl 1172.93399
[4] Feliu-Batlle, V.; Perez, R.; Rodriguez, L., Fractional robust control of main irrigation canals with variable dynamic parameters, Control eng. pract., 15, 673-686, (2007)
[5] Vinagre, B.; Monje, C.; Calderon, A.; Suarez, J., Fractional PID controllers for industry application. A brief introduction, J. vib. control, 13, 1419-1430, (2007) · Zbl 1171.70012
[6] Monje, C.; Vinagre, B.; Feliu, V.; Chen, Y., Tuning and auto-tuning of fractional order controllers for industry applications, Control eng. pract., 16, 798-812, (2008)
[7] Tavazoei, M.; Haeri, M., Chaos control via a simple fractional-order controller, Phys. lett. A, 372, 798-807, (2008) · Zbl 1217.70022
[8] Tavazoei, M.; Haeri, M.; Jafari, S.; Bolouki, S.; Siami, M., Some applications of fractional calculus in suppression of chaotic oscillations, IEEE trans. ind. electron., 55, 4094-4101, (2008)
[9] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 131, 517-529, (2002) · Zbl 1029.34003
[10] Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. appl., 57, 483-487, (2009) · Zbl 1165.35398
[11] Arikoglu, A.; Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons fractals, 34, 1473-1481, (2007) · Zbl 1152.34306
[12] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional order differential equations, Comput. math. appl., 59, 1326-1336, (2010) · Zbl 1189.65151
[13] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. methods partial differential equations, 26, 448-479, (2010) · Zbl 1185.65187
[14] Dehghan, M.; Manafian Herris, J.; Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. naturforsch., 65a, 935-949, (2010)
[15] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fractals, 31, 1248-1255, (2007) · Zbl 1137.65450
[16] Odibat, Z.; Shawagfeh, N., Generalized taylor’s formula, Appl. math. comput., 186, 286-293, (2007) · Zbl 1122.26006
[17] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, 1997, eprint arXiv:funct-an/9710005. · Zbl 0893.65051
[18] Chen, C.; Tsay, Y.; Wu, T., Walsh operational matrices for fractional calculus and their application to distributed systems, J. franklin inst., 303, 267-284, (1977) · Zbl 0377.42004
[19] Hwang, C.; Shih, Y., Laguerre operational matrices for fractional calculus and applications, Internat. J. control, 34, 577-584, (1981) · Zbl 0469.93033
[20] Maione, G., Inverting fractional order transfer functions through Laguerre approximation, Systems control lett., 52, 387-393, (2004) · Zbl 1157.26305
[21] Maione, G., A digital, noninteger order, differentiator using Laguerre orthogonal sequences, Int. J. intell. syst., 11, 77-81, (2006)
[22] Chen, C.; Hsiao, C., A state-space approach to Walsh series solution of linear systems, Int. J. syst. sci., 6, 833-858, (1975) · Zbl 0311.93015
[23] Glabisz, W., Direct Walsh-wavelet packet method for variational problems, Appl. math. comput., 159, 769-781, (2004) · Zbl 1063.65052
[24] Chyi, H.; Yen-Ping, S., Solution of population balance equations via block pulse functions, Chem. eng. J., 25, 39-45, (1982)
[25] Deb, A.; Sarkar, G.; Bhattacharjee, M.; Sen, S.K., All-integrator approach to linear SISO control system analysis using block pulse functions (BPF), J. franklin inst., 334, 319-335, (1997) · Zbl 0874.93050
[26] Maleknejad, K.; Shahrezaee, M.; Khatami, H., Numerical solution of integral equations system of the second kind by block-pulse functions, Appl. math. comput., 166, 15-24, (2005) · Zbl 1073.65149
[27] Babolian, E.; Masouri, Z., Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Comput. math. appl., 220, 51-57, (2008) · Zbl 1146.65082
[28] Hwang, C.; Chen, M., Analysis and optimal control of time-varying linear systems via shifted Legendre polynomials, Internat. J. control, 41, 1317-1330, (1985) · Zbl 0562.93035
[29] Paraskevopoulos, P., Legendre series approach to identification and analysis of linear systems, IEEE trans. automat. control, 30, 585-589, (1985) · Zbl 0559.93041
[30] Wang, X.T., Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials, Appl. math. comput., 184, 849-856, (2007) · Zbl 1114.65076
[31] Maleknejad, K.; Sohrabi, S.; Rostami, Y., Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. math. comput., 188, 123-128, (2007) · Zbl 1114.65370
[32] Hwang, C.; Shih, Y., Parameter identification via Laguerre polynomials, Int. J. syst. sci., 13, 209-217, (1982) · Zbl 0475.93033
[33] Zaman, S.; Jha, A., Parameter identification of non-linear systems using Laguerre operational matrices, Int. J. syst. sci., 16, 625-631, (1985) · Zbl 0567.93025
[34] Razzaghi, M.; Lin, S.D., Identification of time-varying linear and bilinear systems via Fourier series, Comput. electron. eng., 17, 237-244, (1991) · Zbl 0757.93018
[35] Razzaghi, M.; Arabshahi, A.; Lin, S.D., Identification of nonlinear differential equations via Fourier series operational matrix for repeated integration, Appl. math. comput., 68, 189-198, (1995) · Zbl 0821.65050
[36] Deb, A.; Dasgupta, A.; Sarkar, G., A new set of orthogonal functions and its application to the analysis of dynamic systems, J. franklin inst., 343, 1-26, (2006) · Zbl 1173.33306
[37] Deb, A.; Sarkar, G.; Sen, S., Block pulse functions, the most fundamental of all piecewise constant basis functions, Int. J. syst. sci., 25, 351-363, (1994) · Zbl 0834.42016
[38] El-Mesiry, A.; El-Sayed, A.; El-Saka, H., Numerical methods for multiterm fractional (arbitrary) orders differential equations, Appl. math. comput., 160, 683-699, (2005) · Zbl 1062.65073
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