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)
Approximate solutions to boundary value problems of higher order by the modified decomposition method. (English) Zbl 0959.65090
The author studies boundary value problems such as the following $$f^{(2m)}(x)= f(x,y),\quad 0\prec x\prec 1$$ with boundary conditions $$y^{(2j)}(0)= \alpha_{2j},\quad y^{(2j)}(1)= \beta_{2j},\quad j= 0,1,\dots,(m- 1).$$ The solution is found by using the decomposition method of Adomian searching $y(x)$ as a series $\sum^\infty_{n=0} y_n(x)$ and decomposing the nonlinear function by an infinite series of (Adomian) polynomials: $$f(x, y)= \sum^\infty_{n=0} A_n.$$ Numerical examples are treated and they prove the high accuracy of the Adomian method.

65L10Boundary value problems for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Siddiqi, S. S.; Twizell, E. H.: Spline solutions of linear eighth-order boundary-value problems. Comput. methods appl. Mech. engrg. 131, 309-325 (1996) · Zbl 0881.65076
[2] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. (1981) · Zbl 0142.44103
[3] Boutayeb, A.; Twizell, E. H.: Finite-difference methods for the solution of special eighth-order boundary-value problems. Intern. J. Computer math. 48, 63-75 (1993) · Zbl 0820.65046
[4] Djidjeli, K.; Twizell, E. H.; Boutayeb, A.: Numerical methods for special nonlinear boundary-value problems of order 2m. Journal of computational and applied mathematics 47, 35-45 (1993) · Zbl 0780.65046
[5] Bishop, R. E. D.; Cannon, S. M.; Miao, S.: On coupled bending and torsional vibration of uniform beams. J. sound and vibration 131, 457-464 (1989) · Zbl 1235.74158
[6] Agarwal, R. P.: Boundary value problems for high ordinary differential equations. (1986) · Zbl 0619.34019
[7] Baldwin, P.: Asymptotic estimates of the eigenvalues of a sixth-order boundary value problem obtained by using global phase-integral methods. Phil. trans. R. soc. Lond. A 322, 281-305 (1987) · Zbl 0625.76043
[8] Chawla, M. M.; Katti, C. P.: Finite difference methods for two-point boundary value problems involving higher order differential equations. Bit 19, 27-33 (1979) · Zbl 0401.65053
[9] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[10] Adomian, G.: A review of the decomposition method in applied mathematics. J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053
[11] Wazwaz, A. M.: A first course in integral equations. (1997) · Zbl 0924.45001
[12] Wazwaz, A. M.: Analytical approximations and Padé approximants for Volterra’s population model. Appl. math. And comput. 100, 13-25 (1999) · Zbl 0953.92026
[13] Wazwaz, A. M.: A reliable modification of Adomian’s decomposition method. Appl. math. And comput. 92, 1-7 (1998)
[14] Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. math. And comput. 111, No. 1, 33-51 (2000) · Zbl 1023.65108
[15] Cherruault, Y.: Convergence of Adomian’s method. Kybernetes 18, No. 2, 31-38 (1989) · Zbl 0697.65051
[16] Cherruault, Y.; Saccomandi, G.; Some, B.: New results for convergence of Adomian’s method applied to integral equations. Mathl. comput. Modelling 16, No. 2, 85-93 (1992) · Zbl 0756.65083
[17] Abbaoui, K.; Cherruault, Y.: Convergence of Adomian’s method applied to differential equations. Computers math. Applic. 28, No. 5, 103-109 (1994) · Zbl 0809.65073
[18] Mavoungou, T.; Cherruault, Y.: Convergence of Adomian’s method and applications to non-linear partial differential equations. Kybernetes 21, No. 6, 13-25 (1992) · Zbl 0801.35007
[19] Eugene, Y.: Application of the decomposition method to the solution method of the reaction-convection-diffusion equation. Appl. math. Comput. 56, 1-27 (1993) · Zbl 0773.76055
[20] Van Tonningen, S.: Adomian’s decomposition method: A powerful technique for solving engineering equations by computer. Computers education J. 5, No. 4, 30-34 (1995)
[21] Bellomo, N.; Monaco, R.: A comparison between Adomian’s decomposition methods and perturbation techniques for nonlinear random differential equations. J. of mathl. Anal. and applic. 110, 495-502 (1985) · Zbl 0575.60064
[22] Rach, R.: On the Adomian decomposition method and comparisons with picards method. J. math. Anal. and applic. 128, 480-483 (1987) · Zbl 0645.60067
[23] Wazwaz, A. M.: A comparison between Adomian decomposition method and Taylor series method in the series solutions. Appl. math. And comput. 79, 37-44 (1998) · Zbl 0943.65084