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)
Numerical comparison of methods for solving second-order ordinary initial value problems. (English) Zbl 1141.65369
Summary: We apply the Adomian decomposition method (ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the quintic $C^{2}$-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
Full Text: DOI
[1] Adomian, G.: A review of the decomposition method in applied mathematics. J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053
[2] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[3] Sallam, S.; Anwar, M. Naim: Quintic C2-spline integration methods for solving second-order ordinary initial value problems. J. comput. Appl. math. 115, No. 1 -- 2, 495-502 (2000) · Zbl 0974.65073
[4] Micala, Gh.: Approximate solution of differential equation $y(2)=f(x,y)$. Nonlinear anal. 33, 699-714 (1998)
[5] Kramarz, L.: Stability of collocation methods for the numerical solution of y″=$f(x,y)$. Bit 20, 215-222 (1980) · Zbl 0425.65043
[6] Semler, C.; Gentleman, W. C.; Paidoussis, M. P.: Numerical solutions of second order implicit non-linear ordinary differential equations. Sound vibr. 195, No. 4, 553-574 (1996) · Zbl 1235.65003
[7] Vigo-Aguiar, Jesus; Ramos, Higinio: Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations. Appl. math. Comput. 158, No. 1, 187-211 (2003) · Zbl 1042.65053
[8] Wazwaz, A. M.: A new method for solving singular initial value problems in the second-order ordinary differential equations. Appl. math. Comput. 128, No. 1, 45-57 (2002) · Zbl 1030.34004
[9] Wazwaz, A. M.: The numerical solution of special fourth-order boundary value problems by the modified decomposition method. Int. J. Comput. math. 79, No. 3, 345-356 (2002) · Zbl 0995.65082
[10] Cherrualt, Y.: Convergence of Adomian’s method. Kybernetes 18, 31-38 (1989) · Zbl 0697.65051
[11] Cherrualt, Y.; Adomian, G.: Decomposition methods: a new proof of convergence. Math. comput. Model. 18, 103-106 (1993) · Zbl 0805.65057
[12] Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. math. Comput. 111, 53-69 (2000) · Zbl 1023.65108
[13] Venkatarangan, S. N.; Rajalakshmi, K.: A modification of Adomian’s solution for nonlinear oscillatory systems. Comput. math. Appl. 29, No. 6, 67-73 (1995) · Zbl 0818.34006
[14] Baker, G. A.: Essentials of Padé approximants. (1975) · Zbl 0315.41014
[15] Jiao, Y. C.; Yamamoto, Y.; Dang, C.; Hao, Y.: An after treatment technique for improving the accuracy of Adomian’s decomposition method. Comput. math. Appl. 43, No. 6 -- 7, 783-798 (2002) · Zbl 1005.34006
[16] Nagle, R. K.; Saff, E. B.: Fundamentals of differential equations. (1994) · Zbl 0773.34003