zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The second kind Chebyshev wavelet method for solving fractional differential equations. (English) Zbl 1245.65090
Summary: The second kind Chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations. We first construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional order integration. The operational matrix of fractional order integration is utilized to reduce the fractional differential equations to system of algebraic equations. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

MSC:
65L05Initial value problems for ODE (numerical methods)
34A08Fractional differential equations
65T60Wavelets (numerical methods)
WorldCat.org
Full Text: DOI
References:
[1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of the Fractional Differential Equations, in: Math. Studies, vol. 204, Elsevier (North-Holland), Amsterdam, 2006. · Zbl 1092.45003
[2] Machado, J. Tenreiro; Kiryakova, Virginia; Mainardi, Francesco: Recent history of fractional calculus, Commun. nonlinear sci. Numer. simul. 3, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[3] Oldham, K. B.: Fractional differential equations in electrochemistry, Adv. eng. Soft. 41, 9-12 (2010) · Zbl 1177.78041 · doi:10.1016/j.advengsoft.2008.12.012
[4] Hilfer, R.: Applications of fractional calculus in physics, (2000) · Zbl 0998.26002
[5] Suarez, L. E.; Shokooh, A.: An eigenvector expansion method for the solution of motion containing fractional derivatives, J. appl. Mech. 64, 629-635 (1997) · Zbl 0905.73022 · doi:10.1115/1.2788939
[6] Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order, Appl. math. Model. 32, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[7] Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[8] El-Wakil, S. A.; Elhanbaly, A.; Abdou, M. A.: Adomian decomposition method for solving fractional nonlinear differential equations, Appl. math. Comput. 182, 313-324 (2006) · Zbl 1106.65115 · doi:10.1016/j.amc.2006.02.055
[9] Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos solitons fract. 34, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[10] Darania, P.; Ebadian, A.: A method for the numerical solution of the integro-differential equations, Appl. math. Comput. 188, 657-668 (2007) · Zbl 1121.65127 · doi:10.1016/j.amc.2006.10.046
[11] Erturk, V. S.; Momani, S.: Solving systems of fractional differential equations using differential transform method, J. comput. Appl. math. 215, 142-151 (2008) · Zbl 1141.65088 · doi:10.1016/j.cam.2007.03.029
[12] Erturk, V. S.; Momani, S.; Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations, Commun. nonlinear sci. Numer. simul. 13, 1642-1654 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[13] Li, Y. L.; Sun, N.: Numerical solution of fractional differential equations using the generalized block pulse operational matrix, Comput. math. Appl. 62, No. 3, 1046-1054 (2011) · Zbl 1228.65135 · doi:10.1016/j.camwa.2011.03.032
[14] Li, Y. L.; Zhao, W. W.: Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. math. Comput. 216, No. 8, 2276-2285 (2010) · Zbl 1193.65114 · doi:10.1016/j.amc.2010.03.063
[15] Jafari, H.; Yousefi, S. A.; Firoozjaee, M. A.; Momani, S.; Khalique, C. M.: Application of Legendre wavelets for solving fractional differential equations, Comput. math. Appl. 62, No. 3, 1038-1045 (2011) · Zbl 1228.65253 · doi:10.1016/j.camwa.2011.04.024
[16] Li, Y. L.: Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. nonlinear sci. Numer. simul. 15, 2284-2292 (2010) · Zbl 1222.65087 · doi:10.1016/j.cnsns.2009.09.020
[17] Saeedi, H.; Moghadam, M. Mohseni; Mollahasani, N.; Chuev, G. N.: A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun. nonlinear sci. Numer. simul. 16, 1154-1163 (2011) · Zbl 1221.65354 · doi:10.1016/j.cnsns.2010.05.036
[18] Saeedi, H.; Moghadam, M. Mohseni: Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Appl. math. Comput. 16, 1216-1226 (2011) · Zbl 1221.65140 · doi:10.1016/j.cnsns.2010.07.017
[19] Machado, J. A. Tenreiro: Fractional derivatives: probability interpretation and frequency response of rational approximations, Commun. nonlinear sci. Numer. simul. 14, 3492-3497 (2009)
[20] Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1999) · Zbl 0924.34008
[21] Khana, N. A.; Ara, A.; Jamil, M.: An efficient approach for solving the Riccati equation with fractional orders, Comput. math. Appl. 61, 2683-2689 (2011) · Zbl 1221.65205 · doi:10.1016/j.camwa.2011.03.017
[22] Q.B. Fan, Wavelet Analysis, Wuhan University Press, Wuhan, 2008.
[23] Kajani, M. Tavassoli; Vencheh, A. Hadi; Ghasemi, M.: The Chebyshev wavelets operational matrix of integration and product operation matrix, Int. J. Comput. math. 86, 1118-1125 (2008) · Zbl 1169.65072 · doi:10.1080/00207160701736236
[24] L. Zhu, Y.X. Wang, Q.B. Fan, Numerical computation method in solving integral equation by using the second Chebyshev wavelets, in: The 2011 International Conference on Scientific Computing, Las Vegas, USA, 2011, pp. 126 -- 130.
[25] Zhu, L.; Fan, Q. B.: Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. nonlinear sci. Numer. simul. 17, No. 6, 2333-2341 (2012) · Zbl 06056883
[26] Kilicman, A.: Kronecker operational matrices for fractional calculus and some applications, Appl. math. Comput. 187, 250-265 (2007) · Zbl 1123.65063 · doi:10.1016/j.amc.2006.08.122
[27] Mujeeb, U. R.; Rahmat, A. K.: The Legendre wavelet method for solving fractional differential equations, Commun nonlinear sci. Numer. simulat. 16, No. 11, 4163-4173 (2011) · Zbl 1222.65063 · doi:10.1016/j.cnsns.2011.01.014
[28] Diethelm, K.; Ford, N. J.: Numerical solution of the bagley -- torvik equation, Bit 42, 490-507 (2002) · Zbl 1035.65067
[29] Saadatmandi, A.; Dehghan, M.: A new operational matrix for solving fractional-order differential equations, Comput. math. Appl. 59, 1326-1336 (2010) · Zbl 1189.65151 · doi:10.1016/j.camwa.2009.07.006
[30] Hashim, I.; Abdulaziz, O.; Momani, S.: Homotopy analysis method for fractional ivps, Commun. nonlinear sci. Numer. simul. 14, 674-684 (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[31] Kumar, P.; Agrawal, O. P.: An approximate method for numerical solution of fractional differential equations, Signal process. 86, 2602-2610 (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007
[32] Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos soliton fract. 36, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041