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)
A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems. (English) Zbl 0983.65090
Summary: We propose an algorithm for solving the nonliner two-point boundary value problem $$u''(x)+\lambda F(x,u(x))= 0,\quad 0< x< 1,\quad u(0)= u(1)= 0,$$ that has at least one positive solution for $\lambda$ in a compatible interval. Our method stems mainly from combining the decomposition series solution obtained by Adomian decomposition method with Padé approximates. The validity of the approach is verified through illustrative numerical examples.

65L10Boundary value problems for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Ha, K. S.; Lee, Y. H.: Existence of multiple positive solutions of boundary value problems. Nonlinear analysis 28, 1429-1438 (1997) · Zbl 0874.34016
[2] Agarwal, R. P.: Boundary value problems for high ordinary differential equations. (1986) · Zbl 0619.34019
[3] Agarwal, R. P.; O’regan, D.: Boundary value problems for superlinear second order ordinary delay differential equations. J. differential equations 130, 333-335 (1996)
[4] Agarwal, R. P.; Wong, F. H.; Lian, W. C.: Positive solutions for nonlinear singular boundary value problems. Appl. math. Lett. 12, No. 2, 115-120 (1999) · Zbl 0934.34015
[5] O’regan, D.: Theory of boundary value problems. (1994)
[6] O’regan, D.: Existence theory for ordinary differential equations. (1997)
[7] Adomian, G.: Nonlinear stochastic operator equations. (1986) · Zbl 0609.60072
[8] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[9] Adomian, G.: A review of the decomposition method in applied mathematics. J. math. Analysis and applications 135, 501-544 (1988) · Zbl 0671.34053
[10] Adomian, G.; Rach, R.: Analytic solution of nonlinear boundary-value problems in several dimensions. J. math. Analysis and applications 174, No. 1, 118-127 (1993) · Zbl 0796.35017
[11] Adomian, G.; Elrod, M.; Rach, R.: New approach to boundary value equations and applications to a generalization of Airy’s equation. J. math. Analysis and applications 140, No. 2, 554-568 (1989) · Zbl 0678.65057
[12] Adomian, G.: Explicit solutions of nonlinear partial differential equations. Appl. math. Comput. 88, No. 2/3, 117-126 (1997) · Zbl 0904.35077
[13] Adomian, G.: Delayed nonlinear dynamical systems. Mathl. comput. Modelling 22, No. 3, 77-80 (1995) · Zbl 0830.65072
[14] Adomian, G.: The diffusion-Brusselator equation. Computers math. Applic. 29, No. 5, 1-3 (1995) · Zbl 0827.35056
[15] Boyd, J.: Padé approximant algorithm for solving nonlinear ordinary differential equations. Comput. phys. 11, No. 3, 299-303 (1997)
[16] Venkatarangan, S. N.; Rajalashmi, K.: Modification of Adomian’s decomposition method to solve equations containing radicals. Computers math. Applic. 29, No. 6, 75-80 (1995) · Zbl 0818.34007
[17] Wazwaz, A. M.: Analytical approximations and Padé approximates for Volterra’s population model. Appl. math. Comput. 100, 13-25 (1999) · Zbl 0953.92026
[18] Wazwaz, A. M.: The modified decomposition method and Padé approximates for solving the Thomas-Fermi equation. Appl. math. Comput. 105, 11-19 (1999) · Zbl 0956.65064
[19] Deeba, E.; Khuri, S.; Xie, S.: An algorithm for solving boundary value problems. J. comput. Physics 159, No. 2, 125-138 (2000) · Zbl 0959.65091
[20] A.M. Wazwaz, A new algorithm for solving boundary value problems for higher-order integro-differential equations, Appl. Math. Comput. (to appear). · Zbl 1023.65150
[21] Wazwaz, A. M.: The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and 12th-order. International journal of nonlinear sciences and numerical simulation 1, 17-24 (2000) · Zbl 0966.65058
[22] A.M. Wazwaz, Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions, Appl. Math. Comput. (to appear). · Zbl 1027.35016
[23] Bellomo, N.; Monaco, R.: A comparison between Adomian’s decomposition method and perturbation techniques for nonlinear random differential equations. J. mathl. Anal. and applic. 110, 495-502 (1985) · Zbl 0575.60064
[24] Rach, R.: On the Adomian decomposition method and comparisons with picards method. J. mathl. Anal. and applic. 128, 480-483 (1987) · Zbl 0645.60067
[25] Wazwaz, A. M.: A first course in integral equations. (1997) · Zbl 0924.45001
[26] Wazwaz, A. M.: A comparison between Adomian decomposition method and Taylor series method in the series solution. Appl. math. Comput. 79, 37-44 (1998) · Zbl 0943.65084
[27] Hon, Y. C.: A decomposition method for the Thomas-Fermi equation. SEA bull. Math. 20, No. 3, 55-58 (1996) · Zbl 0858.34017
[28] He, J. H.: Variational iteration method--A kind of non-linear analytical technique. Intern. J. Non-linear mech. 34, 699-708 (1999) · Zbl 05137891
[29] Wazwaz, A. M.: The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. math. Comput. 110, 251-464 (2000) · Zbl 1023.65109
[30] Maleknejad, K.; Hadizadeh, M.: A new computational method for Volterra-Fredholm integral equations. Computers math. Applic. 37, No. 9, 1-8 (1999) · Zbl 0940.65151