Chebyshev wavelet finite difference method: a new approach for solving initial and boundary value problems of fractional order. (English) Zbl 1470.65143
Summary: A new method based on a hybrid of Chebyshev wavelets and finite difference methods is introduced for solving linear and nonlinear fractional differential equations. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of linear or nonlinear algebraic equations. This method can be considered as a nonuniform finite difference method. Some examples are given to verify and illustrate the efficiency and simplicity of the proposed method.
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65T60 | Numerical methods for wavelets |
Software:
Wavelet ToolboxReferences:
| [1] | Butzer, P. L.; Westphal, U., An Introduction to Fractional Calculus (2000), Singapore: World Scientific, Singapore · Zbl 0987.26005 |
| [2] | Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002 |
| [3] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0 |
| [4] | Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027 |
| [5] | Ciesielski, M.; Jacek, L., Numerical simulations of anomalous diffusion |
| [6] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [7] | Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation, 14, 3, 674-684 (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014 |
| [8] | Abdulaziz, O.; Hashim, I.; Momani, S., Solving systems of fractional differential equations by homotopy-perturbation method, Physics Letters A, 372, 4, 451-459 (2008) · Zbl 1217.81080 · doi:10.1016/j.physleta.2007.07.059 |
| [9] | Elbeleze, A. A.; Kılıçman, A.; Taib, B. M., Applications of homotopy perturbation and variational iteration methods for fredholm integro-differential equation of fractional order, Abstract and Applied Analysis, 2012 (2012) · Zbl 1253.65201 · doi:10.1155/2012/763139 |
| [10] | Kadem, A.; Kılıçman, A., The approximate solution of fractional Fredholm integrodifferential equations by variational iteration and homotopy perturbation methods, Abstract and Applied Analysis, 2012 (2012) · Zbl 1242.65284 |
| [11] | Wu, G.-C.; Lee, E. W. M., Fractional variational iteration method and its application, Physics Letters A, 374, 25, 2506-2509 (2010) · Zbl 1237.34007 · doi:10.1016/j.physleta.2010.04.034 |
| [12] | Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 27-34 (2006) · Zbl 1401.65087 |
| [13] | Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling, 32, 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025 |
| [14] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1999), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0924.34008 |
| [15] | Yuste, S. B., Weighted average finite difference methods for fractional diffusion equations, Journal of Computational Physics, 216, 1, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006 |
| [16] | Odibat, Z., Approximations of fractional integrals and Caputo fractional derivatives, Applied Mathematics and Computation, 178, 2, 527-533 (2006) · Zbl 1101.65028 · doi:10.1016/j.amc.2005.11.072 |
| [17] | Odibat, Z. M., Computational algorithms for computing the fractional derivatives of functions, Mathematics and Computers in Simulation, 79, 7, 2013-2020 (2009) · Zbl 1161.65319 · doi:10.1016/j.matcom.2008.08.003 |
| [18] | Zhang, Y., A finite difference method for fractional partial differential equation, Applied Mathematics and Computation, 215, 2, 524-529 (2009) · Zbl 1177.65198 · doi:10.1016/j.amc.2009.05.018 |
| [19] | Daftardar-Gejji, V.; Jafari, H., Solving a multi-order fractional differential equation using Adomian decomposition, Applied Mathematics and Computation, 189, 1, 541-548 (2007) · Zbl 1122.65411 · doi:10.1016/j.amc.2006.11.129 |
| [20] | Momani, S.; Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Applied Mathematics and Computation, 177, 2, 488-494 (2006) · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025 |
| [21] | Momani, S.; Odibat, Z., Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics, 207, 1, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015 |
| [22] | Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling, 32, 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025 |
| [23] | Che Hussin, C. H.; Kılıçman, A., On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method, Mathematical Problems in Engineering, 2011 (2011) · Zbl 1213.65148 · doi:10.1155/2011/724927 |
| [24] | Odibat, Z.; Momani, S.; Erturk, V. S., Generalized differential transform method: application to differential equations of fractional order, Applied Mathematics and Computation, 197, 2, 467-477 (2008) · Zbl 1141.65092 · doi:10.1016/j.amc.2007.07.068 |
| [25] | Hussin, C. H. C.; Kılıçman, A., On the solution of fractional order nonlinear boundary value problems by using differential transformation method, European Journal of Pure and Applied Mathematics, 4, 2, 174-185 (2011) · Zbl 1389.35125 |
| [26] | Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 1-4, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341 |
| [27] | Lubich, Ch., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Mathematics of Computation, 45, 172, 463-469 (1985) · Zbl 0584.65090 · doi:10.2307/2008136 |
| [28] | Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numerical Algorithms, 16, 3-4, 231-253 (1997) · Zbl 0926.65070 · doi:10.1023/A:1019147432240 |
| [29] | Kadem, A.; Kılıçman, A., Note on transport equation and fractional Sumudu transform, Computers & Mathematics with Applications, 62, 8, 2995-3003 (2011) · Zbl 1232.44002 · doi:10.1016/j.camwa.2011.08.009 |
| [30] | Li, Y.; Sun, N., Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Computers & Mathematics with Applications, 62, 3, 1046-1054 (2011) · Zbl 1228.65135 · doi:10.1016/j.camwa.2011.03.032 |
| [31] | Kılıçman, A.; Al Zhour, Z. A. A., Kronecker operational matrices for fractional calculus and some applications, Applied Mathematics and Computation, 187, 1, 250-265 (2007) · Zbl 1123.65063 · doi:10.1016/j.amc.2006.08.122 |
| [32] | Misiti, M.; Misiti, Y.; Oppenheim, G.; Poggi, J.-M., Wavelets Toolbox Users Guide (2000), The MathWorks |
| [33] | Li, Y., Solving a nonlinear fractional differential equation using Chebyshev wavelets, Communications in Nonlinear Science and Numerical Simulation, 15, 9, 2284-2292 (2010) · Zbl 1222.65087 · doi:10.1016/j.cnsns.2009.09.020 |
| [34] | Jafari, H.; Yousefi, S. A.; Firoozjaee, M. A.; Momani, S.; Khalique, C. M., Application of Legendre wavelets for solving fractional differential equations, Computers & Mathematics with Applications, 62, 3, 1038-1045 (2011) · Zbl 1228.65253 · doi:10.1016/j.camwa.2011.04.024 |
| [35] | Li, Y. L.; Zhao, W. W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Applied Mathematics and Computation, 216, 8, 2276-2285 (2010) · Zbl 1193.65114 · doi:10.1016/j.amc.2010.03.063 |
| [36] | Saeedi, H.; Mohseni Moghadam, M.; Mollahasani, N.; Chuev, G. N., A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1154-1163 (2011) · Zbl 1221.65354 · doi:10.1016/j.cnsns.2010.05.036 |
| [37] | Saeedi, H.; Moghadam, M. M., Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1216-1226 (2011) · Zbl 1221.65140 · doi:10.1016/j.cnsns.2010.07.017 |
| [38] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach · Zbl 0818.26003 |
| [39] | Hilfer, R., Applications of Fractional Calculus in Physics (2000), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0998.26002 |
| [40] | Tenreiro Machado, J. A., Fractional derivatives: probability interpretation and frequency response of rational approximations, Communications in Nonlinear Science and Numerical Simulation, 14, 9-10, 3492-3497 (2009) · doi:10.1016/j.cnsns.2009.02.004 |
| [41] | Mason, J. C.; Handscomb, D. C., Chebyshev Polynomials (2003), Chapman & Hall · Zbl 1015.33001 |
| [42] | Daubechies, I., Ten Lectures on Wavelets. Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61 (1992), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA · Zbl 0776.42018 · doi:10.1137/1.9781611970104 |
| [43] | Fan, Q. B., Wavelet Analysis (2008), Wuhan, China: Wuhan University Press, Wuhan, China |
| [44] | Derili, H.; Sohrabi, S., Numerical solution of singular integral equations using orthogonal functions, Mathematical Sciences, 2, 3, 261-272 (2008) · Zbl 1206.65259 |
| [45] | Clenshaw, C. W.; Curtis, A. R., A method for numerical integration on an automatic computer, Numerische Mathematik, 2, 197-205 (1960) · Zbl 0093.14006 · doi:10.1007/BF01386223 |
| [46] | Elbarbary, E. M. E.; El-Kady, M., Chebyshev finite difference approximation for the boundary value problems, Applied Mathematics and Computation, 139, 2-3, 513-523 (2003) · Zbl 1027.65098 · doi:10.1016/S0096-3003(02)00214-X |
| [47] | Adibi, H.; Assari, P., Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind, Mathematical Problems in Engineering, 2010 (2010) · Zbl 1192.65154 · doi:10.1155/2010/138408 |
| [48] | Davis, P. J.; Rabinowitz, P., Method of Numerical Integration (1984), London, UK: Academic Press, London, UK · Zbl 0537.65020 |
| [49] | Kumar, P.; Agrawal, O. P., An approximate method for numerical solution of fractional differential equations, Signal Processing, 86, 10, 2602-2610 (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007 |
| [50] | Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Computers & Mathematics with Applications, 59, 3, 1326-1336 (2010) · Zbl 1189.65151 · doi:10.1016/j.camwa.2009.07.006 |
| [51] | Wang, Y.; Fan, Q., The second kind Chebyshev wavelet method for solving fractional differential equations, Applied Mathematics and Computation, 218, 17, 8592-8601 (2012) · Zbl 1245.65090 · doi:10.1016/j.amc.2012.02.022 |
| [52] | El-Mesiry, A. E. M.; El-Sayed, A. M. A.; El-Saka, H. A. A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Applied Mathematics and Computation, 160, 3, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026 |
| [53] | Al-Mdallal, Q. M.; Syam, M. I.; Anwar, M. N., A collocation-shooting method for solving fractional boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 15, 12, 3814-3822 (2010) · Zbl 1222.65078 · doi:10.1016/j.cnsns.2010.01.020 |
| [54] | Mohammadi, F.; Hosseini, M. M.; Mohyud-Din, S. T., A new operational matrix for legendre wavelets and its applications for solving fractional order boundary values problems, International Journal of Physical Sciences, 6, 32, 7371-7378 (2011) · doi:10.5897/IJPS11.376 |
| [55] | ur Rehman, M.; Khan, R. A., A numerical method for solving boundary value problems for fractional differential equations, Applied Mathematical Modelling, 36, 3, 894-907 (2012) · Zbl 1243.65095 · doi:10.1016/j.apm.2011.07.045 |
| [56] | Wang, Y.-G.; Song, H.-F.; Li, D., Solving two-point boundary value problems using combined homotopy perturbation method and Green’s function method, Applied Mathematics and Computation, 212, 2, 366-376 (2009) · Zbl 1166.65362 · doi:10.1016/j.amc.2009.02.036 |
| [57] | ur Rehman, M.; Ali Khan, R., The Legendre wavelet method for solving fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16, 11, 4163-4173 (2011) · Zbl 1222.65063 · doi:10.1016/j.cnsns.2011.01.014 |
| [58] | Lakestani, M.; Dehghan, M.; Irandoust-pakchin, S., The construction of operational matrix of fractional derivatives using B-spline functions, Communications in Nonlinear Science and Numerical Simulation, 17, 3, 1149-1162 (2012) · Zbl 1276.65015 · doi:10.1016/j.cnsns.2011.07.018 |
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