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 comparative study of numerical methods for solving quadratic Riccati differential equations. (English) Zbl 1210.65131
Summary: We use a Legendre wavelet method for solving quadratic Riccati differential equations and perform a comparative study between the proposed method and other existing methods. Our results show that in comparison with other existing methods, the Legendre wavelet method provides a fast convergent series of easily computable components. The present study is illustrated by exploring two kinds of nonlinear Riccati differential equations that shows the pertinent features of the Legendre wavelet method.
MSC:
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
65T60Wavelets (numerical methods)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L20Stability and convergence of numerical methods for ODE
References:
[1]He, J. H.: Comparison of homotopy perturbation method and homotopy analysis method, Appl. math. Comput. 156, 527-539 (2004) · Zbl 1062.65074 · doi:10.1016/j.amc.2003.08.008
[2]Tan, Y.; Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation, Commun. non-linear sci. Numer. simulation 13, 539-546 (2008) · Zbl 1132.34305 · doi:10.1016/j.cnsns.2006.06.006
[3]Reid, W. T.: Riccati differential equations, (1972)
[4]El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A.: Solving Riccati differential equation using Adomian’s decomposition method, Appl. math. Comput. 157, 503-514 (2004) · Zbl 1054.65071 · doi:10.1016/j.amc.2003.08.049
[5]Geng, F.; Lin, Y.; Cui, M.: A piecewise variational iteration method for Riccati differential equations, Comput. math. Appl. 58, 2518-2522 (2009) · Zbl 1189.65164 · doi:10.1016/j.camwa.2009.03.063
[6]Abbasbandy, S.: Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. math. Comput. 172, 485-490 (2006) · Zbl 1088.65063 · doi:10.1016/j.amc.2005.02.014
[7]Abbasbandy, S.: A new application of he’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. comput. Appl. math. 207, 59-63 (2007) · Zbl 1120.65083 · doi:10.1016/j.cam.2006.07.012
[8]Abbasbandy, S.: Iterated he’s homotopy perturbation method for quadratic Riccati differential equation, Appl. math. Comput. 175, 581-589 (2006) · Zbl 1089.65072 · doi:10.1016/j.amc.2005.07.035
[9]Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T.: Spectral methods in fluid dynamics, (1988) · Zbl 0658.76001
[10]Razzaghi, M.; Yousefi, S.: Legendre wavelets operational matrix of integration, Int. J. Syst. sci. 32, No. 4, 495-502 (2001) · Zbl 1006.65151 · doi:10.1080/002077201300080910
[11]Mohammadi, F.; Hosseini, M. M.: Legendre wavelet method for solving linear stiff systems, J. adv. Res. differential equations 2, No. 1, 47-57 (2010)
[12]F. Mohammadi, M.M. Hosseini, S.T. Mohyud-Din, Legendre wavelet Galerkin method for solving ordinary differential equations with non analytic solution, Int. J. Syst. Sci., in press. · Zbl 1218.65078 · doi:10.1080/00207721003658194
[13]Khellat, F.; Yousefi, S.: The linear Legendre mother wavelets operational matrix of integration and its application, J. franklin inst. 343, 181-190 (2006) · Zbl 1127.65105 · doi:10.1016/j.jfranklin.2005.11.002
[14]Razzaghi, M.; Yousefi, S.: Legendre wavelets direct method for variational problems, Math. comput. Simulations 53, No. 3, 185-192 (2000)
[15]Yousefi, S.; Razzaghi, M.: Legendre wavelets method for the nonlinear Volterra Fredholm integral equations, Math. comput. Simulations 70, 1-8 (2005) · Zbl 1205.65342 · doi:10.1016/j.matcom.2005.02.035
[16]Tang, B. Q.; Li, X. F.: A new method for determining the solution of Riccati differential equations, Appl. math. Comput. 194, 431-440 (2007) · Zbl 1193.65116 · doi:10.1016/j.amc.2007.04.061