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Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method. (English) Zbl 1250.65094

Summary: Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A08 Fractional ordinary differential equations
34A30 Linear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65T60 Numerical methods for wavelets
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[1] Pedas, Arvet; Tamme, Enn, Spline collocation methods for linear multi-term fractional differential equations, J Comput Appl Math, 236, 167-176 (2011) · Zbl 1229.65138
[2] Pedas, Arvet; Tamme, Enn, On the convergence of spline collocation methods for solving fractional differential equations, J Comput Appl Math, 235, 3502-3514 (2011) · Zbl 1217.65154
[3] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput Math Appl, 59, 1326-1336 (2010) · Zbl 1189.65151
[4] Hall, C. A.; Meryer, W. W., Optimal error bounds for cubic spline interpolation, J Approx Theory, 16, 105-122 (1976) · Zbl 0316.41007
[5] Hwang, C.; Shih, Y. P., Laguerre operational matrices for fractional calculus and applications, Int J Control, 34, 3, 577-584 (1981) · Zbl 0469.93033
[6] Wang, C. H., On the generalization of block-pulse operational matrices for fractional and operational calculus, J Franklin Inst, 315, 2, 91-102 (1983) · Zbl 0544.44006
[7] Cui, C. K., An introduction to wavelets (1992), Academic Press: Academic Press San Diego, CA
[8] Rawashdeh, E. A., Numerical solution of fractional integro-differential equations by collocation method, Appl Math Comput, 176, 1-6 (2006) · Zbl 1100.65126
[9] Wu, G.; Lee, E. W.M., Fractional variational iteration method and its application, Phys Lett A, 374, 2506-2509 (2010) · Zbl 1237.34007
[10] Saeedi, H.; Mohseni Moghadam, M., Commun Nonlinear Sci Numer Simul, 16, 1154-1163 (2011) · Zbl 1221.65354
[11] Podlubny, I., Fractional differential equations (1999), Academic Press · Zbl 0918.34010
[12] Wu, J. L., A wavelet operational method for solving fractional partial differential equations numerically, Appl Math Comput, 214, 31-40 (2009) · Zbl 1169.65127
[13] Wang, J. Z., Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation, Appl Comput Harmonic Anal, 3, 154-163 (1996) · Zbl 0855.41006
[14] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron Trans Numer Anal, 5, 1-6 (1997) · Zbl 0890.65071
[15] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer Alg, 36, 31-52 (2004) · Zbl 1055.65098
[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)
[17] Blank, L., Numerical treatment of differential equations of fractional order, Manchester Centre Comput Math Numer Anal Rep, 287 (1996)
[18] Yuan, L.; Agrawal, O. P., A numerical scheme for dynamic systems containing fractional derivatives, Vibr Acoust, 124, 321-324 (2002)
[19] Ciesielski M, Leszczynski Jacek. Numerical simulations of anomalous diffusion. In: Computer Methods Mech, Conference Gliwice Wisla Poland, 2003.; Ciesielski M, Leszczynski Jacek. Numerical simulations of anomalous diffusion. In: Computer Methods Mech, Conference Gliwice Wisla Poland, 2003. · Zbl 1057.65507
[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, 18, 5, 931-943 (1987) · Zbl 0625.44003
[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, 2, 1, 5-12 (2011)
[22] ur Rehman; Khan, Rahmat Ali, A numerical method for solving boundary value problems for fractional differential equations, Appl Math Model (2011) · Zbl 1221.34018
[23] Ye, M., Optimal error bounds for the cubic spline interpolation of lower smooth function (II), Appl Math JCU, 13, 223-230 (1998) · Zbl 0915.41005
[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, 229, 382-391 (2009) · Zbl 1166.65066
[25] Ford, N. J.; Charles Simpson, A., The numerical solution of fractional differential equations: speed versus accuracy, Numer Alg, 26, 333-346 (2001) · Zbl 0976.65062
[26] Sweilam, N. H.; Khader, M. M.; Al-Bar, R. F., Numerical studies for a multi-order fractional differential equation, Phys Lett A, 371, 26-33 (2007) · Zbl 1209.65116
[27] Kumar, P.; Agrawal, O. P., An approximate method for numerical solution of fractional differential equations, Signal Process, 86, 2602-2610 (2006) · Zbl 1172.94436
[28] Kumar P. New numerical schemes for the solution of fractional differential equations. Southern Illinois University at Carbondale, vol. 134, 2006.; 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., 1, 178-185 (2006)
[30] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics aproach, Phys Rep, 339, 1-77 (2000) · Zbl 0984.82032
[31] Chang, R. Y.; Chen, K. C.; Wang, M. L., Modified Laguerre operational matrices for fractional calculus and applications, Int J Syst Sci, 16, 9, 1163-1172 (1985) · Zbl 0579.44005
[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; 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
[33] Yuste, S. B., Weighted average finite difference methods for fractional diffusion equations, J Comput Phys, 216, 264-274 (2006) · Zbl 1094.65085
[34] Momani, S.; Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by adomian decomposition method, Appl Math Comput, 177, 488-494 (2006) · Zbl 1096.65131
[35] Momani, S.; Odibat, Z., Numerical approach to differential equations of fractional order, J Comput Appl Math, 96-110 (2007) · Zbl 1119.65127
[36] Atanackovic, T. M.; Stankovic, B., On a numerical scheme for solving differential equations of fractional order, Mech Res Commun, 429-438 (2008) · Zbl 1258.65103
[37] Lucas, T. R., Error bounds for interpolationg cubic splines under various end conditions, SIAM J Numer Anal, 569-584 (1974) · Zbl 0286.65004
[38] Cai, W.; Wang, J., Adaptive multi-resolution collocation methods for Initial boundary value problems of nonlinear PDEs, SIAM J Numer Anal, 937-970 (1996) · Zbl 0856.65115
[39] Li, Yuanlu; Sun, Ning, Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Comput Math Appl, 216, 1046-1054 (2010) · Zbl 1228.65135
[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, 216, 2276-2285 (2010) · Zbl 1193.65114
[41] Odibat, Z.; Momani, S., Application of variational iteration method to equations of fractional order, Int J Nonlinear Sci Numer Simul, 7, 271-279 (2006) · Zbl 1378.76084
[42] Odibat, Z., Approximations of fractional integrals and Caputo fractional derivatives, Appl Math Comput, 527-533 (2006) · Zbl 1101.65028
[43] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Appl Math Model, 32, 28-39 (2008) · Zbl 1133.65116
[44] Odibat, Z. M., Computational algorithms for computing the fractional derivatives of functions, Math Comput Simul, 79 (2009)
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