In this work, multi-point boundary value problems are considered. These problems have important roles in the modelling of various problems in physics and engineering. Although numerous works have been carried out on the existence and uniqueness to the solution of these problems, the numerical or analytical methods are not established for solving them. In this paper the well-known He's variational iteration method is applied for solving the multi-point boundary value problems. The method is modified and the results are shown using some test problems. These results show the efficiency of the new approach.
AbdouAMSolimanAA (2005) Variational iteration method for solving Burger's and coupled Burger's equations. Journal of Computational and Applied Mathematics181: 245–251.
2.
AftabizadehARGuptaCPXuJM (1989) Existence and uniqueness theorems for three-point boundary value problems. SIAM Journal on Mathematical Analysis20: 716–726.
3.
BaiCFangJ (2003) Existence of multiple positive solutions for nonlinear m-point boundary value problems. Journal of Mathematical Analysis and Applications281: 76–85.
4.
CaglarHCaglarNElfaituriK (2006) B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Applied Mathematics and Computation175: 72–79.
5.
DehghanM (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Mathematics and Computers in Simulation71: 16–30.
6.
DehghanMManafianJSaadatmandiA (2010a) Application of semi-analytic methods for the Fitzhugh-Nagumo equation which models the transmission of nerve impulses. Mathematical Methods in the Applied Sciences33: 1384–1398.
7.
DehghanMManafianJSaadatmandiA (2010b) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations26: 448–479.
8.
DehghanMSaadatmandiA (2009) Variational iteration method for solving the wave equation subject to an integral conservation condition. Chaos, Solitons and Fractals41: 1448–1453.
9.
DehghanMSalehiR (2010a) A semi-numeric approach for solution of the Eikonal partial differential equation and its applications. Numerical Methods for Partial Differential Equations26: 702–722.
10.
DehghanMSalehiR (2010b) Solution of a nonlinear time-delay model in biology via semi-analytical approaches. Computer Physics Communications181: 1255–1265.
11.
DehghanMShakeriF (2008a) Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem. Journal of Computational and Applied Mathematics214: 435–446.
12.
DehghanMSalehiR (2008b) Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astronomy13: 53–59.
13.
DehghanMShakeriF (2008c) Solution of a model describing biological species living together using the variational iteration method. Mathematical and Computer Modelling48: 685–699.
14.
DehghanMSalehiR (2008d) The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Physica Scripta78: 065004–065004.
15.
DehghanMShakeriF (2010) Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He's variational iteration technique. International Journal for Numerical Methods in Biomedical Engineering26: 705–715.
16.
DehghanMShakeriF (2011) A semi-numerical technique for solving the multi-point boundary value problems and engineering applications. International Journal for Numerical Methods in Heat and Fluid Flow. in press.
17.
DehghanMTatariM (2008) Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method. Chaos, Solitons and Fractals36: 157–166.
18.
FangQTsuchiyaTYamamotoT (2002) Finite difference, finite element and finite volume methods applied to two-point boundary value problems. Journal of Computational and Applied Mathematics139: 9–19.
19.
GhorbaniASaberi-NadjafiJ (2009) An effective modification of He's variational iteration method. Nonlinear Analysis: Real World Applications10: 2828–2833.
20.
HaSN (2001) A nonlinear shooting method for two-point boundary value problems. Computers & Mathematics with Applications42: 1411–1420.
21.
HaSNLeeCR (2002) Numerical study for two-point boundary value problems using Green's functions. Computers & Mathematics with Applications44: 1599–1608.
22.
HaqueMBaluchMHMohsenMFN (1986) Solution of multiple point, nonlinear boundary value problems by method of weighted residuals. International Journal of Computer Mathematics19: 69–84.
23.
HeJH (1997) Variational iteration method for delay differential equations. Communications in Nonlinear Science and Numerical Simulation2: 235–235.
24.
HeJH (2000) Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation114: 115–123.
25.
HeJH (2006) Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B20: 1141–1199.
26.
HeJHWuXH (2007) Variational iteration method: New development and applications. Computers & Mathematics with Applications54: 881–894.
27.
HendersonJ (2005) Five-point boundary value problems for third-order differential equations by solution matching. Mathematical and Computer Modelling42: 133–137.
28.
IncMEvansDJ (2003) The decomposition method for solving of a class of singular two–point boundary value problems. International Journal of Computer Mathematics80: 869–882.
29.
JangB (2008) Two-point boundary value problems by the extended Adomian decomposition method. Journal of Computational and Applied Mathematics219: 253–262.
30.
KrajcinovicD (1972) Sandwich beam analysis. Journal of Applied Mechanics39: 773–778.
31.
LiuGRWuTY (2002) Multipoint boundary value problems by differential quadrature method. Mathematical and Computer Modelling35: 215–227.
32.
MaR (1997) Existence theorems for a second-order three-point boundary value problem. Journal of Mathematical Analysis and Applications212: 430–442.
33.
MohsenAEl-GamelM (2008) On the Galerkin and collocation methods for two-point boundary value problems using sinc bases. Computers & Mathematics with Applications56: 930–941.
34.
MomaniSAbuasadS (2006) Application of He's variational iteration method to Helmholtz equation. Chaos, Solitons and Fractals27: 1119–1123.
35.
MomaniSOdibatZM (2006) Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Physics Letters A355: 271–279.
36.
OdibatZMMomaniS (2007) Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation7: 27–34.
37.
OzerH (2007) Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics. International Journal of Nonlinear Sciences and Numerical Simulation8: 513–518.
38.
SaadatmandiADehghanM (2009a) He's variational iteration method for solving a partial differential equation arising in modelling of the water waves. Zeitschrift für Naturforschung A64a: 783–787.
39.
SaadatmandiADehghanM (2009b) Variational iteration method for solving a generalized pantograph equation. Computers & Mathematics with Applications58: 2190–2196.
40.
Saadatmandi A and Dehghan M (2011) The use of sinc-collocation method for solving multi-point boundary value problems. Communications in Nonlinear Science and Numerical Simulation (in press).
41.
Saberi-NadjafiJTamamgarMR (2008) The variational iteration method: A highly promising method for solving the system of integro-differential equations. Computers & Mathematics with Applications56: 346–351.
42.
ShakeriFDehghanM (2007) Numerical solution of a biological population model using He's variational iteration method. Computers & Mathematics with Applications54: 1197–1209.
43.
ShakeriFDehghanM (2008a) Numerical solution of the Klein-Gordon equation via He's variational iteration method. Nonlinear Dynamics51: 89–97.
44.
ShakeriFDehghanM (2008b) Solution of the delay differential equations via homotopy perturbation method. Mathematical and Computer Modelling48: 486–498.
45.
ShakeriFDehghanM (2010) Application of the decomposition method of Adomian for solving the pantograph equation of order m. Zeitschrift für Naturforschung A65a: 453–460.
46.
SolimanAA (2006) A numerical simulation and explicit solutions of KdV–Burger's and Lax's seventh-order KdV equations. Chaos, Solitons and Fractals29: 294–302.
47.
TatariMDehghanM (2007a) He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation. Chaos, Solitons and Fractals33: 671–677.
48.
TatariMDehghanM (2007) On the convergence of He's variational iteration method. Journal of Computational and Applied Mathematics207: 121–128.
49.
TaiwoOA (2002) Exponential fitting for the solution of two–point boundary value problems with cubic spline collocation tau-method. International Journal of Computer Mathematics79: 299–306.
50.
TimoshenkoS (1961) Theory of Elastic Stability, New York: McGraw-Hill.
51.
WazwazAM (2007a) A comparison between the variational iteration method and Adomian decomposition method. Journal of Computational and Applied Mathematics207: 129–136.
52.
WazwazAM (2007b) The variational iteration method for a reliable treatment of the linear and the nonlinear Goursat problem. Applied Mathematics and Computation193: 455–462.
53.
WazwazAM (2007c) The variational iteration method for solutions for KdV K(2, 2), Burgers and cubic Boussinesq equations. Journal of Computational and Applied Mathematics207: 18–23.
54.
YaoQ (2005) Successive iteration and positive solution for nonlinear second-order three-point boundary value problems. Computers & Mathematics with Applications50: 433–444.
55.
YousefiSADehghanM (2010) The use of He's variational iteration method for solving variational problems. International Journal of Computer Mathematics87: 1299–1314.
56.
ZahraWK (2009) Exponential spline solutions for a class of two point boundary value problems over a semi-infinite range. Numerical Algorithms52: 561–573.
