Variational iteration method for one-dimensional nonlinear thermoelasticity. (English) Zbl 1131.74018

Sweilam, N. H.; Khader, M. M.

Chaos Solitons Fractals 32, No. 1, 145-149 (2007).

Summary: This paper applies the variational iteration method to solve Cauchy problem arising in one-dimensional nonlinear thermoelasticity. The advantage of this method is to overcome the difficulty of calculation of Adomian’s polynomials in Adomian’s decomposition method. The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that the variational iteration method is a powerful mathematical tool for solving nonlinear problems.

MSC:

74H15	Numerical approximation of solutions of dynamical problems in solid mechanics
74F05	Thermal effects in solid mechanics
74S30	Other numerical methods in solid mechanics (MSC2010)

Keywords:

Cauchy problem; Adomian’s polynomials

References:

[1]	Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2005.08.044, in press. · Zbl 1101.82018
[2]	Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burger’s and coupled burger’s equations, J comput appl math, 181, 2, 245-251, (2005) · Zbl 1072.65127
[3]	Draˇgaˇnnescu, GhE.; Cofan, N.; Rujan, D.L., Nonlinear vibrations a nano sized sensor with fractional damping, J optoelectron adv mater, 7, 2, 877-884, (2005)
[4]	Draˇgaˇnnescu, GhE.; Caˇpaˇlnaˇsan, V., Nonlinear relaxation phenomena in polycrystalline solids, Int J nonlinear sci numer simulat, 4, 219-225, (2003)
[5]	Jiang, S.; Bonn, Numerical solution for the Cauchy problem in nonlinear 1-D-thermoelasticity, Computing, 44, 147-158, (1990) · Zbl 0701.73001
[6]	He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput methods appl mech eng, 167, 1-2, 57-68, (1998) · Zbl 0942.76077
[7]	He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput methods appl mech eng, 167, 1-2, 69-73, (1998) · Zbl 0932.65143
[8]	He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Int J non-linear mech, 34, 699-708, (1999) · Zbl 1342.34005
[9]	He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl math comput, 114, 2-3, 115-123, (2000) · Zbl 1027.34009
[10]	He, J.H., A simple perturbation approach to Blasius equation, Appl math comput, 140, 2-3, 217-222, (2003) · Zbl 1028.65085
[11]	He, J.H.; Wan, Y.Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Int J circ theory appl, 32, 6, 629-632, (2004) · Zbl 1169.94352
[12]	He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, solitons & fractals, 19, 4, 847-851, (2004) · Zbl 1135.35303
[13]	He, J.H., A generalized variational principle in micromorphic thermoelasticity, Mech res commun, 32, 1, 93-98, (2005) · Zbl 1091.74012
[14]	Liu, H.M., Variational approach to nonlinear electrochemical system, Int J nonlinear sci numer simulat, 5, 1, 95-96, (2004)
[15]	Liu, H.M., Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method, Chaos, solitons & fractals, 23, 2, 573-576, (2005) · Zbl 1135.76597
[16]	Moura, D.C.A., A linear uncoupling numerical scheme for the nonlinear coupled thermodynamics equations, (), 204-211
[17]	Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons & fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113
[18]	Slemrod, M., Global existence, uniqueness and asymptotic stability of classical solutions in one-dimensional non-linear thermoelasticity, Arch rat anal, 76, 2, 97-134, (1981) · Zbl 0481.73009
[19]	Soliman AA. A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2005.08.054, in press. · Zbl 1099.35521
[20]	Wazwaz, A.M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput math appl, 41, 1237-1244, (2001) · Zbl 0983.65090