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 a nonlinear time-delay model in biology via semi-analytical approaches. (English) Zbl 1219.65062
Summary: The delay logistic equations have been extensively used as models in biology and other sciences, with particular emphasis on population dynamics. In this work, the variational iteration and Adomian decomposition methods are applied to solve the delay logistic equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. On the other hand, the Adomian decomposition method approximates the solution as an infinite series and usually converges to the accurate solution. Moreover, these techniques reduce the volume of calculations because they have no need of discretization of the variables, linearization or small perturbations. Illustrative examples are included to demonstrate the validity and applicability of the presented methods.
MSC:
65L03Functional-differential equations (numerical methods)
34K38Functional-differential inequalities
92D25Population dynamics (general)
References:
[1]Abbasbandy, S.: Numerical method for non-linear wave and diffusion equations by the variational iteration method, Int. J. Numer. meth. Engng. 78, 1836-1843 (2008) · Zbl 1159.76372 · doi:10.1002/nme.2150
[2]Abbasbandy, S.: An approximation solution of a nonlinear equation with Riemann – Liouville’s fractional derivatives by he’s variational iteration method, J. comput. Appl. math. 207, 53-58 (2007) · Zbl 1120.65133 · doi:10.1016/j.cam.2006.07.011
[3]Adomian, G.: A review of the decomposition method in applied mathematics, J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[4]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[5]Baker, C. T. H.; Paul, C. A.; Wille, D. R.: Issues in the numerical solution of evolutionary delay differential equations, Adv. comp. Math. 3, 171-196 (1995) · Zbl 0832.65064 · doi:10.1007/BF02988625
[6]Banks, H. T.; Lamm, P. K. D.: Estimation of delays and other parameters in nonlinear functional differential equation, SIAM J. Control. optim. 21, 895-915 (1983) · Zbl 0526.93015 · doi:10.1137/0321054
[7]Bellen, A.: One-step collocation for delay differential equations, J. comp. Appl. math. 10, 275-283 (1984) · Zbl 0538.65047 · doi:10.1016/0377-0427(84)90039-6
[8]Bellen, A.; Zennaro, M.: Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. numer. Math. 9, 321-346 (1992) · Zbl 0749.65042 · doi:10.1016/0168-9274(92)90025-9
[9]Bogacki, P.; Shampine, L. F.: A 3(2) pair of Runge – Kutta formulas, Appl. math. Lett. 2, 321-325 (1989) · Zbl 0705.65055 · doi:10.1016/0893-9659(89)90079-7
[10]Busenbrg, S.; Martell, M.: Delay differential equations and dynamical systems, (1999)
[11]Carvalho, L. A. V.; Cooke, K. L.: A nonlinear equation with piecewise continuous argument, Diff. int. Equat. 1, 359-367 (1988) · Zbl 0723.34061
[12]Cohen, D. S.; Rosenblat, S. A.: A delay logistic equation with variable growth rate, SIAM J. Appl. math. 42, 608-624 (1982) · Zbl 0494.45010 · doi:10.1137/0142043
[13]Dehghan, M.: Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, Appl. math. Comput. 157, 549-560 (2004) · Zbl 1054.65105 · doi:10.1016/j.amc.2003.08.098
[14]Dehghan, M.; Tatari, M.: Identifying an unknown function in a parabolic equation with overspecified data via he’s variational iteration method, Chaos solitons fractals 36, 157-166 (2008) · Zbl 1152.35390 · doi:10.1016/j.chaos.2006.06.023
[15]Dehghan, M.; Shakeri, F.: Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via he’s variational iteration technique, Commun. numer. Meth. engng. (2008)
[16]Dehghan, M.; Shakeri, F.: Application of he’s variational iteration method for solving the Cauchy reaction – diffusion problem, J. comput. Appl. math. 214, 435-446 (2008) · Zbl 1135.65381 · doi:10.1016/j.cam.2007.03.006
[17]Dehghan, M.; Shakeri, F.: The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. scr. 78, 1-11 (2008) · Zbl 1159.78319 · doi:10.1088/0031-8949/78/06/065004
[18]Dehghan, M.; Saadatmandi, A.: Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos solitons fractals 41, 1448-1453 (2009) · Zbl 1198.65202 · doi:10.1016/j.chaos.2008.06.009
[19]Dehghan, M.; Hashemi, B.: Solution of the fully fuzzy linear systems using the decomposition procedure, Appl. math. Comput. 182, 1568-1580 (2006) · Zbl 1111.65040 · doi:10.1016/j.amc.2006.05.043
[20]Dehghan, M.; Shakourifar, M.; Hamidi, A.: The solution of linear and nonlinear systems of Volterra functional equations using Adomian – Padé technique, Chaos solitons fractals 39, 2509-2521 (2009) · Zbl 1197.65223 · doi:10.1016/j.chaos.2007.07.028
[21]Dehghan, M.: On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. methods partial differential eq. 21, 24-40 (2005) · Zbl 1059.65072 · doi:10.1002/num.20019
[22]Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. Simulation 71, 16-30 (2006) · Zbl 1089.65085 · doi:10.1016/j.matcom.2005.10.001
[23]Dehghan, M.: Parameter determination in a partial differential equation from the overspecified data, Math. comput. Model. 41, 196-213 (2005) · Zbl 1080.35174 · doi:10.1016/j.mcm.2004.07.010
[24]Dehghan, M.; Hamidi, A.; Shakourifar, M.: The solution of coupled Burgers equations using Adomian – Padé technique, Appl. math. Comput. 189, 1034-1047 (2007) · Zbl 1122.65388 · doi:10.1016/j.amc.2006.11.179
[25]Dehghan, M.; Salehi, R.: A semi-numeric approach for solution of the eikonal partial differential equation and its applications, Numer. methods partial differential eq. 26, 702-722 (2010) · Zbl 1189.65237 · doi:10.1002/num.20482
[26]Fowler, A. C.: Asymptotic method for delay equations, J. eng. Math. 53, 271-290 (2005) · Zbl 1122.34343 · doi:10.1007/s10665-005-9016-z
[27]Fowler, A. C.: An asymptotic analysis of the delayed logistic equation when the delay is large, IMA J. Appl. math. 28, 41-49 (1982) · Zbl 0488.34075 · doi:10.1093/imamat/28.1.41
[28]Gordon, M. S.: A decomposition method for a semilinear boundary value problem with a quadratic nonlinearity, Int. J. Math. math. Sci. 6, 855-861 (2005) · Zbl 1077.35066 · doi:10.1155/IJMMS.2005.855
[29]Hutchinson, G. E.: Circular causal system in ecology, Ann. N.Y. Acad. sci. 50, 221-246 (1948)
[30]He, J. H.: Variational iteration method – a kind of nonlinear analytical technique: some examples, Internat. J. Nonlinear mech. 34, 699-708 (1999)
[31]He, J. H.: Variational iteration method — some recent results and new interpretation, J. comput. Appl. math. 207, 3-17 (2007) · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009
[32]He, J. H.: Variational iteration method for autonomous ordinary differential systems, Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[33]He, J. H.: Approximate analytical solution of Blasius equation, Commun. nonlinear sci. Numer. simul. 4, 75-78 (1999) · Zbl 0932.34005 · doi:10.1016/S1007-5704(99)90063-1
[34]He, J. H.: Variational iteration method for delay differential equations, Commun. nonlinear sci. Numer. simul. 2, 235-236 (1997) · Zbl 0924.34063
[35]He, J. H.: Variational iteration method — a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear mech. 34, 699-708 (1999)
[36]Inokuti, M.; Sekine, H.; Mur, T.: General use of the Lagrange multiplier in nonlinear mathematical physics, , 156-162 (1978)
[37]Jafari, H.; Dehghan, M.; Sayevand, K.: Solving a fourth-order fractional diffusion – wave equation in a bounded domain by decomposition method, Numer. methods partial differential eq. 24, 1115-1126 (2008) · Zbl 1145.65115 · doi:10.1002/num.20308
[38]Al-Khaled, K.; Kaya, D.; Noor, M.: Numerical comparison of methods for solving parabolic equations, Appl. math. Comput. 157, 735-743 (2004) · Zbl 1061.65098 · doi:10.1016/j.amc.2003.08.079
[39]Khan, H.; Liao, S. -J.; Mohapatra, R. N.; Vajravelu, K.: An analytical solution for a nonlinear time-delay model in biology, Commun. nonlinear sci. Numer. simul. 14, 3141-3148 (2009) · Zbl 1221.65204 · doi:10.1016/j.cnsns.2008.11.003
[40]Kuang, Y.: Delay differential equations in population dynamics, (1993) · Zbl 0777.34002
[41]Kainhofer, R.: QMC methods for the solution of the delay differential equations, J. comput. Appl. math. 155, 239-252 (2003) · Zbl 1034.65001 · doi:10.1016/S0377-0427(02)00867-1
[42]Sen, A.; Mukherjee, D.: Chaos in the delay logistic equation with discontinuous delays, Chaos solitons fractals 40, 2126-2132 (2009) · Zbl 1198.93088 · doi:10.1016/j.chaos.2007.10.019
[43]Lesnic, D.: The decomposition method for initial value problems, Appl. math. Comput. 181, 206-213 (2006) · Zbl 1148.65081 · doi:10.1016/j.amc.2006.01.025
[44]Lesnic, D.: A nonlinear reaction – diffusion process using the Adomian decomposition method, Int. commun. Heat mass trans. 34, 129-135 (2007)
[45]Saadatmandi, A.; Dehghan, M.: Variational iteration method for solving a generalized pantograph equation, Comput. math. Appl. 58, 2190-2196 (2009) · Zbl 1189.65172 · doi:10.1016/j.camwa.2009.03.017
[46]Shakeri, F.; Dehghan, M.: Numerical solution of the Klein – Gordon equation via he’s variational iteration method, Nonlinear dyn. 51, 89-97 (2008) · Zbl 1179.81064 · doi:10.1007/s11071-006-9194-x
[47]Shakeri, F.; Dehghan, M.: Numerical solution of a biological population model using he’s variational iteration method, Comput. math. Appl. 54, 1197-1209 (2007) · Zbl 1137.92033 · doi:10.1016/j.camwa.2006.12.076
[48]Shakeri, F.; Dehghan, M.: Solution of a model describing biological species living together sing the variational iteration method, Math. comput. Model. 48, 685-699 (2008) · Zbl 1156.92332 · doi:10.1016/j.mcm.2007.11.012
[49]Shampine, L. F.; Thompson, S.: Solving ddes in Matlab, Appl. numer. Math. 37, 441-458 (2001) · Zbl 0983.65079 · doi:10.1016/S0168-9274(00)00055-6
[50]Shampine, L. F.: Interpolation for Runge – Kutta methods, SIAM J. Numer. anal. 22, 1014-1027 (1985) · Zbl 0592.65041 · doi:10.1137/0722060
[51]Sun, C.; Han, M.; Lin, Y.: Analysis of stability and Hopf bifurcation for a delayed logistic equation, Chaos solitons fractals 31, 672-682 (2007)
[52]Tatari, M.; Dehghan, M.: Solution of problems in calculus of variations via he’s variational iteration method, Phys. lett. A 362, 401-406 (2007) · Zbl 1197.65112 · doi:10.1016/j.physleta.2006.09.101
[53]Tatari, M.; Dehghan, M.: On the convergence of he’s variational iteration method, J. comput. Appl. math. 207, 121-128 (2007) · Zbl 1120.65112 · doi:10.1016/j.cam.2006.07.017
[54]Tatari, M.; Dehghan, M.; Razzaghi, M.: Application of the Adomian decomposition method for the Fokker – Planck equation, Math. comput. Model. 45, 639-650 (2007) · Zbl 1165.65397 · doi:10.1016/j.mcm.2006.07.010
[55]Verhulst, P. F.: Notice sur la loi Lu population parsuit dans son accroissment, Corresp. math. Phys. 10, 113-121 (1838)
[56]Vermiglio, A.: A one-step subregion method for delay differential equations, Calcolo 22, 429-455 (1986) · Zbl 0625.65079 · doi:10.1007/BF02575897
[57]Wazwaz, A. M.: A study on linear and nonlinear Schrödinger equations by the variational iteration method, Chaos solitons fractals 37, 1136-1142 (2008) · Zbl 1148.35353 · doi:10.1016/j.chaos.2006.10.009
[58]Wazwaz, A. M.: The variational iteration method for a reliable treatment of the linear and the nonlinear Goursat problem, Appl. math. Comput. 193, 455-462 (2007) · Zbl 1193.65185 · doi:10.1016/j.amc.2007.03.083
[59]Wazwaz, A. M.: The decomposition method applied to systems of partial differential equations and to the reaction – diffusion Brusselator model, Appl. math. Comput. 110, 251-264 (2000) · Zbl 1023.65109 · doi:10.1016/S0096-3003(99)00131-9
[60]Wright, K. E. M.: A nonlinear difference differential equation, J. reine angew. Math. 194, 66-87 (1955) · Zbl 0064.34203 · doi:10.1515/crll.1955.194.66 · doi:crelle:GDZPPN002177102
[61]Yildirim, A.; Öziş, T.: Solutions of singular ivps of Lane – Emden type by the variational iteration method, Nonlinear anal. 70, 2480-2484 (2009) · Zbl 1162.34005 · doi:10.1016/j.na.2008.03.012
[62]Yildirim, A.: Variational iteration method for modified Camassa – Holm and Degasperis – Procesi equations, Commun. numer. Meth. engng. (2008)
[63]Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. numer. Math. 52, 39-62 (2005) · Zbl 1063.65079 · doi:10.1016/j.apnum.2004.02.002
[64]Tatari, M.; Dehghan, M.: On the reconstruction of the first term in the variational iteration method for solving differential equations, Z. naturforsch. 65a, 203-208 (2010)
[65]M. Dehghan, S.A. Yousefi, A. Lotfi, The use of He’s variational iteration method for solving the telegraph and fractional telegraph equations, Commun. Numer. Meth. Engng. (2009), in press, doi:10.1002/cnm.1293
[66]M. Dehghan, R. Salehi, The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem, Commun. Numer. Meth. Engng. (2009), in press, doi:10.1002/cnm.1315
[67]Dehghan, M.; Tatari, M.: The use of Adomian decomposition method for solving problems in calculus of variations, Math. prob. Engng. 2006, 1-12 (2006) · Zbl 1200.65050 · doi:10.1155/MPE/2006/65379
[68]Dehghan, M.; Shakeri, F.: The numerical solution of the second Painlevé equation, Numer. methods partial differential eq. 25, 1238-1259 (2009) · Zbl 1172.65037 · doi:10.1002/num.20416
[69]Tatari, M.; Dehghan, M.: The use of the Adomian decomposition method for solving multipoint boundary value problems, Phys. scr. 73, 672-676 (2006)
[70]Dehghan, M.; Tatari, M.: Solution of a parabolic equation with a time-dependent coefficient and an extra measurement using the decomposition procedure of Adomian, Phys. scr. 72, 425-431 (2005) · Zbl 1102.65127 · doi:10.1088/0031-8949/72/6/001
[71]F. Shakeri, M. Dehghan, Application of the decomposition method of Adomian for solving the pantograph equation of order m, Z. Naturforsch. 65a (2010), in press
[72]Saadatmandi, A.; Dehghan, M.: He’s variational iteration method for solving a partial differential equation arising in modelling of the water waves, Z. naturforsch. 64a, 783-787 (2009)
[73]Tatari, M.; Dehghan, M.: Numerical solution of Laplace equation in a disk using the Adomian decomposition technique, Phys. scr. 72, 345-348 (2005) · Zbl 1128.65311 · doi:10.1238/Physica.Regular.072a00345