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)
Monotone iterative sequences for nonlinear integro-differential equations of second order. (English) Zbl 1231.45016
Summary: We present an efficient algorithm based on a monotone method for the solution of a class of nonlinear integro-differential equations of second order. This method is applied to derive two monotone sequences of upper and lower solutions which are uniformly convergent. Theorems which list the conditions for the existence of such sequences are presented. The numerical results demonstrate reliability and efficiency of the proposed algorithm.
MSC:
45J05Integro-ordinary differential equations
34K10Boundary value problems for functional-differential equations
References:
[1]Agarwal, R. P.: Boundary value problems for higher order integro-differential equations, Nonlinear anal. 7, 259-270 (1983) · Zbl 0505.45002 · doi:10.1016/0362-546X(83)90070-6
[2]Agarwal, R. P.: Boundary value problems for high ordinary differential equations, (1986)
[3]Wazwaz, A.: A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Appl. math. Comput. 181, 1703-1712 (2006) · Zbl 1105.65128 · doi:10.1016/j.amc.2006.03.023
[4]Maleknejad, K.; Hadizadeh, M.: A new computational method for Volterra–Fredholm integral equations, J. comput. Appl. math. 37, 1-8 (1999) · Zbl 0940.65151 · doi:10.1016/S0898-1221(99)00107-8
[5]Contea, D.; Preteb, I.: Fast collocation methods for Volterra integral equations of convolution type, J. comput. Appl. math. 196, 652-663 (2006) · Zbl 1104.65122 · doi:10.1016/j.cam.2005.10.018
[6]Saberi-Nadja, J.; Ghorbani, A.: Hes homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput. math. Appl. 58, 2379-2390 (2009) · Zbl 1189.65173 · doi:10.1016/j.camwa.2009.03.032
[7]Brunner, H.: On the numerical solution of nonlinear Volterra integro-differential equations, Bit 13, 381-390 (1973) · Zbl 0265.65056 · doi:10.1007/BF01933399
[8]Brunner, H.: On the numerical solution of nonlinear Volterra–Fredholm integral equation by collocation methods, SIAM J. Numer. anal. 27, No. 4, 987-1000 (1990) · Zbl 0702.65104 · doi:10.1137/0727057
[9]Lepik, O.: Haar wavelet method for nonlinear integro-differential equations, Appl. math. Comput. 176, 324-333 (2006) · Zbl 1093.65123 · doi:10.1016/j.amc.2005.09.021
[10]Ebadi, G.; Rahimi-Ardabili, M.; Shahmorad, S.: Numerical solution of the nonlinear Volterra integro-differential equations by the tau method, Appl. math. Comput. 188, 1580-1586 (2007) · Zbl 1119.65123 · doi:10.1016/j.amc.2006.11.024
[11]Maleknejad, K.; Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra Fredholm integro-differential equations, Appl. math. Comput. 145, 641-653 (2003) · Zbl 1032.65144 · doi:10.1016/S0096-3003(03)00152-8
[12]Darania, P.; Ivaz, K.: Numerical solution of nonlinear Volterra–Fredholm integro-differential equations, Comput. math. Appl. 56, 2197-2209 (2008) · Zbl 1165.65404 · doi:10.1016/j.camwa.2008.03.045
[13]Wazwaz, A. M.: The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. math. Comput. 216, 1304-1309 (2010) · Zbl 1190.65199 · doi:10.1016/j.amc.2010.02.023
[14]Liz, E.; Nieto, Juan J.: Boundary value problems for second order integro-differential equations of Fredholm type 1, J. comput. Appl. math. 72, 215-225 (1996) · Zbl 0857.45006 · doi:10.1016/0377-0427(95)00273-1
[15]Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations, (1984) · Zbl 0549.35002
[16]Al-Refai, M.; Syam, M.; Al-Mdallal, Q. M.: Analytical sequences of upper and lower solutions for a class of elliptic equations, J. math. Anal. appl. 374, 402-411 (2011) · Zbl 1202.35095 · doi:10.1016/j.jmaa.2010.09.034
[17]Al-Refai, M.: Existence, uniqueness and bounds for a problem in combustion theory, J. comput. Appl. math. 167, 255-269 (2004) · Zbl 1042.80005 · doi:10.1016/j.cam.2003.10.023
[18]Bartle, R. G.; Sherbert, D. R.: Introduction to real analysis, (2000)