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)
Solution of delay differential equation by means of homotopy analysis method. (English) Zbl 1187.34081
The paper contains an algorithm of approximate analytical solution of delay differential equations based on the homotopy analysis method and modified homotopy analysis method. Several examples of linear, nonlinear and systems of initial value problems of delay differential equations are solved by these algorithms. The convergence of the methods is proved.
MSC:
34K07Theoretical approximation of solutions of functional-differential equations
34K05General theory of functional-differential equations
34K28Numerical approximation of solutions of functional-differential equations
Software:
BVPh; DELSOL
References:
[1]Wikipedia A: Delay differential equation, http://en.wikipedia.org/wiki/Delay-differential-equation , 10 April 2008
[2]Ulsoy, A.G.: Analytical solution of a system of homogeneous delay differential equations via the Lambert function. In: Proceedings of the American Control Conference, Chicago, IL, June 2000
[3]Evans, D.J., Raslan, K.R.: The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82, 49–54 (2005) · Zbl 1069.65074 · doi:10.1080/00207160412331286815
[4]Adomian, G., Rach, R.: Nonlinear Stochastic differential delay equation. J. Math. Anal. Appl. 91, 94–101 (1983) · Zbl 0504.60067 · doi:10.1016/0022-247X(83)90094-X
[5]Adomian, G., Rach, R.: A nonlinear delay differential equation. J. Math. Anal. Appl. 91, 301–304 (1983) · Zbl 0504.60068 · doi:10.1016/0022-247X(83)90152-X
[6]Shakeri, F., Dehghan, M.: Solution of delay differential equations via a homotopy perturbation method. Math. Comput. Model. 48, 486–498 (2008). doi: 10.1016/j.mcm.2007.09.016 · Zbl 1145.34353 · doi:10.1016/j.mcm.2007.09.016
[7]Liao, S.J.: Homotopy analysis method and its application. Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai (1992)
[8]Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC. Boca Raton (2003)
[9]Liao, S.J.: Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169, 1186–1194 (2005) · Zbl 1082.65534 · doi:10.1016/j.amc.2004.10.058
[10]Liao, S.J.: An explicit totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlinear Mech. 34, 759–778 (1999) · Zbl 05137896 · doi:10.1016/S0020-7462(98)00056-0
[11]Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[12]Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Studies Appl. Math. 117, 239–263 (2006) · Zbl 1145.76352 · doi:10.1111/j.1467-9590.2006.00354.x
[13]Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007) · Zbl 1170.76307 · doi:10.1016/j.physleta.2006.09.060
[14]Hayat, T., Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005) · Zbl 1094.76005 · doi:10.1007/s11071-005-7346-z
[15]Hayat, T., Abbas, Z., Sajid, M.: Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction. Phys. Lett. A 372, 2400–2408 (2008)
[16]Hayat, T., Sajid, M., Ayub, M.: On explicit analytic solution for MHD pipe flow of a fourth grade fluid. Commun. Nonlinear Sci. Numer. Simul. 13, 745–751 (2008) · Zbl 1221.76221 · doi:10.1016/j.cnsns.2006.07.009
[17]Xu, H., Liao, S.J.: Dual solutions of boundary layer flow over an upstream moving plate. Commun. Nonlinear Sci. Numer. Simul. 13, 350–358 (2008) · Zbl 1131.35066 · doi:10.1016/j.cnsns.2006.04.008
[18]Abbasbandy, S.: Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method. Nonlinear Dyn. 52, 35–40 (2008) · Zbl 1173.35646 · doi:10.1007/s11071-007-9255-9
[19]Bataineh, A.S., Noorani, M.S.M., Hashim, I.: On a new reliable modification of homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 409–423 (2009). doi: 10.1016/j.cnsns.2007.10.007 · Zbl 1221.65195 · doi:10.1016/j.cnsns.2007.10.007
[20]Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006) · Zbl 1236.80010 · doi:10.1016/j.physleta.2006.07.065
[21]Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. 13, 539–546 (2008). doi: 10.1016/j.cnsns.2006.03.008 · Zbl 1132.34305 · doi:10.1016/j.cnsns.2006.06.006
[22]Alomari, A.K., Noorani, M.S.M., Nazar, R.: Solutions of heat-like and wave-like equations with variable coefficients by means of the homotopy analysis method. Chin. Phys. Lett. 25, 589–592 (2008) · doi:10.1088/0256-307X/25/2/064
[23]Alomari, A.K., Noorani, M.S.M., Nazar, R.: Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2008.01.008 (in press)
[24]Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007) · Zbl 05200826 · doi:10.1016/j.physleta.2006.09.105
[25]Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Modified homotopy analysis method for solving systems of second-order BVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 430–442 (2009). doi: 10.1016/j.cnsns.2007.09.012 · Zbl 1221.65196 · doi:10.1016/j.cnsns.2007.09.012
[26]Bataineh, A.S., Noorani, M.S.M., Hashim, I.: The homotopy analysis method for Cauchy reaction-diffusion problems. Phys. Lett. A 372, 613–618 (2008) · Zbl 1217.35101 · doi:10.1016/j.physleta.2007.07.069
[27]Wu, Y., Cheung, K.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion. doi: 10.1016/j.wavemoti.2008.07.002 (in press)
[28]Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Monthly 109(3), 217–234 (2002) · Zbl 1024.01010 · doi:10.2307/2695352
[29]Wille, D.R., Baker, C.T.H.: DELSOL-a numerical code for the solution of systems of delay–differential equations. Appl. Numer. Math. 9, 223–234 (1992) · Zbl 0747.65055 · doi:10.1016/0168-9274(92)90017-8