zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Solving a nonlinear fractional differential equation using Chebyshev wavelets. (English) Zbl 1222.65087
Summary: Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.
65L99Numerical methods for ODE
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
45J05Integro-ordinary differential equations
[1]Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1999)
[2]Gaul, L.; Klein, P.; Kemple, S.: Damping description involving fractional operators, Mech syst signal pr 5, 81-88 (1991)
[3]Suarez, L.; 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
[4]Momani, S.: An algorithm for solving the fractional convection – diffusion equation with nonlinear source term, Commun nonlinear sci numer simul 12, No. 7, 1283-1290 (2007) · Zbl 1118.35301 · doi:10.1016/j.cnsns.2005.12.007
[5]Jafari, H.; Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun nonlinear sci numer simul 14, No. 5, 1962-1969 (2009) · Zbl 1221.35439 · doi:10.1016/j.cnsns.2008.06.019
[6]Sweilam, N. H.; Khader, M. M.; Al-Bar, R. F.: Numerical studies for a multi-order fractional differential equation, Phys lett A 371, No. 1 – 2, 26-33 (2007) · Zbl 1209.65116 · doi:10.1016/j.physleta.2007.06.016
[7]Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method, Comput math appl 57, No. 3, 483-487 (2009) · Zbl 1165.35398 · doi:10.1016/j.camwa.2008.09.045
[8]Arikoglu, A.; Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method, Chaos solitons fract 40, No. 2, 521-529 (2009) · Zbl 1197.45001 · doi:10.1016/j.chaos.2007.08.001
[9]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, No. 8, 1642-1654 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[10]Meerschaert, M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl numer math 56, No. 1, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[11]Odibat, Z.; Shawagfeh, N.: Generalized Taylor’s formula, Appl math comput 186, No. 1, 286-293 (2007) · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[12]Wu, J. L.: A wavelet operational method for solving fractional partial differential equations numerically, Appl math comput 214, No. 1, 31-40 (2009) · Zbl 1169.65127 · doi:10.1016/j.amc.2009.03.066
[13]Lepik: Solving fractional integral equations by the Haar wavelet method, Appl math comput 214, No. 2, 468-478 (2009) · Zbl 1170.65106 · doi:10.1016/j.amc.2009.04.015
[14]Chen, C.; Hsiao, C.: Haar wavelet method for solving lumped and distributed-parameter systems, IEE P-contr theor appl 144, No. 1, 87-94 (1997) · Zbl 0880.93014 · doi:10.1049/ip-cta:19970702
[15]Bujurke, N.; Salimath, C.; Shiralashetti, S.: Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series, Nonlinear dyn 51, No. 4, 595-605 (2008) · Zbl 1171.65407 · doi:10.1007/s11071-007-9248-8
[16]Karimi, H.; Lohmann, B.; Maralani, P.; Moshiri, B.: A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets, Int J comput math 81, No. 9, 1121-1132 (2004) · Zbl 1068.65088 · doi:10.1080/03057920412331272225
[17]Pawlak, M.; Hasiewicz, Z.: Nonlinear system identification by the Haar multiresolution analysis, IEEE trans circuits I 45, No. 9, 945-961 (1998) · Zbl 0952.93021 · doi:10.1109/81.721260
[18]Hsiao, C.; Wang, W.: Optimal control of linear time-varying systems via Haar wavelets, J optim theory appl 103, No. 3, 641-655 (1999) · Zbl 0941.49018 · doi:10.1023/A:1021740209084
[19]Karimi, H.; Moshiri, B.; Lohmann, B.; Maralani, P.: Haar wavelet-based approach for optimal control of second-order linear systems in time domain, J dyn control syst 11, No. 2, 237-252 (2005) · Zbl 1063.49002 · doi:10.1007/s10883-005-4172-z
[20]Sadek, I.; Abualrub, T.; Abukhaled, M.: A computational method for solving optimal control of a system of parallel beams using Legendre wavelets, Math comput model 45, No. 9 – 10, 1253-1264 (2007) · Zbl 1117.49026 · doi:10.1016/j.mcm.2006.10.008
[21]Bujurke, N. M.; Shiralashetti, S. C.; Salimath, C. S.: An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations, J comput appl math 227, No. 2, 234-244 (2009) · Zbl 1162.65040 · doi:10.1016/j.cam.2008.03.012
[22]Babolian, E.; Masouri, Z.; Hatamzadeh-Varmazyar, S.: Numerical solution of nonlinear Volterra – Fredholm integro-differential equations via direct method using triangular functions, Comput math appl 58, No. 2, 239-247 (2009) · Zbl 1189.65306 · doi:10.1016/j.camwa.2009.03.087
[23]Kajani, M.; Vencheh, A.: The Chebyshev wavelets operational matrix of integration and product operation matrix, Int J comput math 86, No. 7, 1118-1125 (2008) · Zbl 1169.65072 · doi:10.1080/00207160701736236
[24]Reihani, M. H.; Abadi, Z.: Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J comput appl math 200, No. 1, 12-20 (2007) · Zbl 1107.65122 · doi:10.1016/j.cam.2005.12.026
[25]Khellat, F.; Yousefi, S.: The linear Legendre mother wavelets operational matrix of integration and its application, J franklin inst 343, No. 2, 181-190 (2006) · Zbl 1127.65105 · doi:10.1016/j.jfranklin.2005.11.002
[26]Razzaghi, M.; Yousefi, S.: The Legendre wavelets operational matrix of integration, Int J syst sci 32, No. 4, 495-502 (2001) · Zbl 1006.65151 · doi:10.1080/002077201300080910
[27]Machado, J. A. Tenreiro: Fractional derivatives: probability interpretation and frequency response of rational approximations, Commun nonlinear sci numer simul 14, No. 9-10, 3492-3497 (2009)
[28]Kilicman, A.; Zhour, Z. A. A. Al: Kronecker operational matrices for fractional calculus and some applications, Appl math comput 187, No. 1, 250-265 (2007) · Zbl 1123.65063 · doi:10.1016/j.amc.2006.08.122
[29]Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos solitons fract 36, No. 1, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[30]El-Mesiry, A.; El-Sayed, A.; El-Saka, H.: Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl math comput 160, No. 3, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026