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)
A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. (English) Zbl 1231.65126
Summary: We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.
MSC:
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34A08Fractional differential equations
45J05Integro-ordinary differential equations
References:
[1]Magin, R. L.: Fractional calculus in bioengineering, (2006)
[2]Miller, K.; Ross, B.: An introduction to the fractional calaulus and fractional differential equations, (1993)
[3]Ortigueira, M.: Introduction to fraction linear systems. Part 2: discrete-time case, IEE proc., vis. Image signal process. 147, 71-78 (2000)
[4]Deng, J.; Ma, L.: Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. math. Lett. 23, 676-680 (2010) · Zbl 1201.34008 · doi:10.1016/j.aml.2010.02.007
[5]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[6]Ray, S. S.; Bera, R. K.: Solution of an extraordinary differential equation by Adomian decomposition method, J. appl. Math. 4, 331-338 (2004) · Zbl 1080.65069 · doi:10.1155/S1110757X04311010
[7]El-Sayed, A. M. A.; El-Kalla, I. L.; Ziada, E. A. A.: Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations, Appl. numer. Math. 60, 788-797 (2010) · Zbl 1192.65092 · doi:10.1016/j.apnum.2010.02.007
[8]Abdulaziz, O.; Hashim, I.; Momani, S.: Application of homotopy-perturbation method to fractional ivps, J. comput. Appl. math. 216, 574-584 (2008) · Zbl 1142.65104 · doi:10.1016/j.cam.2007.06.010
[9]Yanga, S.; Xiao, A.; Su, H.: Convergence of the variational iteration method for solving multi-order fractional differential equations, Comput. math. Appl. 60, 2871-2879 (2010) · Zbl 1207.65109 · doi:10.1016/j.camwa.2010.09.044
[10]Odibat, Z.; Momani, S.; Xu, H.: A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. math. Model. 34, 593-600 (2010) · Zbl 1185.65139 · doi:10.1016/j.apm.2009.06.025
[11]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[12]Kumer, P.; Agrawal, O. P.: An approximate method for numerical solution of fractional differential equations, Signal process. 84, 2602-2610 (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007
[13]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
[14]Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamics, (1988) · Zbl 0658.76001
[15]Doha, E. H.; Bhrawy, A. H.: Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. numer. Math. 58, 1224-1244 (2008) · Zbl 1152.65112 · doi:10.1016/j.apnum.2007.07.001
[16]Doha, E. H.; Bhrawy, A. H.: Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer. methods partial differential equations 25, 712-739 (2009) · Zbl 1170.65099 · doi:10.1002/num.20369
[17]Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.: A Jacobi–Jacobi dual-Petrov–Galerkin method for third- and fifth-order differential equations, Math. comput. Modelling 53, 1820-1832 (2011) · Zbl 1219.65077 · doi:10.1016/j.mcm.2011.01.002
[18]Doha, E. H.; Bhrawy, A. H.; Ezzeldeen, S. S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. math. Model. (2011)
[19]Bhrawy, A. H.; Alofi, A. S.; Ezzeldeen, S. S.: A quadrature tau method for variable coefficients fractional differential equations, Appl. math. Lett. (2011)
[20]Ghoreishi, F.; Yazdani, S.: An extension of the spectral tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. math. Appl. 61, 30-43 (2011) · Zbl 1207.65108 · doi:10.1016/j.camwa.2010.10.027
[21]Vanani, S. Karimi; Aminataei, A.: Tau approximate solution of fractional partial differential equations, Comput. math. Appl. (2011)
[22]Esmaeili, S.; Shamsi, M.: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. nonlinear sci. Numer. simul. 16, 3646-3654 (2011) · Zbl 1226.65062 · doi:10.1016/j.cnsns.2010.12.008
[23]Pedas, A.; Tamme, E.: On the convergence of spline collocation methods for solving fractional differential equations, J. comput. Appl. math. 235, 3502-3514 (2011) · Zbl 1217.65154 · doi:10.1016/j.cam.2010.10.054
[24]Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solutions, Appl. math. Comput. 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2
[25]Ford, N. J.; Connolly, J. A.: Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations, Comput. appl. Math. 229, 382-391 (2009) · Zbl 1166.65066 · doi:10.1016/j.cam.2008.04.003
[26]Mdallal, Q. M.; Syam, M. I.; Anwar, M. N.: A collocation-shooting method for solving fractional boundary value problems, Commun. nonlinear sci. Numer. simul. 15, 3814-3822 (2010) · Zbl 1222.65078 · doi:10.1016/j.cnsns.2010.01.020
[27]Jafari, H.; Das, S.; Tajadodi, H.: Solving a multi-order fractional differential equation using homotopy analysis method, J. King saud univ. Sci. 23, 151-155 (2011)