Investigation of a Transition from Steady Convection to Chaos in Porous Media Using Piecewise Variational Iteration Method
In this paper, a new dependable algorithm based on an adaptation of the standard variational iteration method (VIM) is used for analyzing the transition from steady convection to chaos for lowto-intermediate Rayleigh numbers convection in porous media. The solution trajectories show the transition from steady convection to chaos that occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. The VIM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the considered model and other dynamical systems. We shall call this technique as the piecewise VIM. Numerical comparisons between the piecewise VIM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the proposed technique is a promising tool for the nonlinear chaotic and nonchaotic systems.
[1] D. A. Nield and A. Bejan, Convection in porous media, Springer, New York, 2006.
[2] M. D. Graham and P. H. Steen, Plume formation and resonant bifurcations in porous-media convection, J. Fluid. Mech., vol. 272, pp.
67-89, 1994.
[3] A. S. M. Cherkaoui and W. S. D. Wilcock, Characteristics of high Rayleigh number two-dimensional convection in an open-top porous
layer heated from below, J. Fluid. Mech., vol. 394 pp. 241-260, 1999.
[4] L. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today,
vol.54, pp. 34-39, 2001.
[5] J. Otero et al., High-Rayleigh-number convection in a fluid-saturated
porous layer, J. Fluid Mech., vol. 500, pp. 263-281, 2004.
[6] E. N. Lorenz, Deterministic non-periodic flows, J. Atmos.Sci., vol. 20,
pp. 130-141,1963.
[7] P. Vadasz and S. Olek, Weak turbulence and chaos for low Prandtl
number gravity driven convection in porous media, Transport in Porous
Media, vol. 37, pp. 69-91,1999.
[8] J.-H. He, Approximate analytical solution for seepage flow with
fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng.,
vol. 167, pp. 57-68, 1998.
[9] J.-H. He, Approximate solution of nonlinear differential equations with
convolution product nonlinearities, Comput. Meth. Appl. Mech. Eng.,
vol. 167, pp. 69-73, 1998.
[10] J.-H. He, Variational iteration methodÔÇöa kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech., vol. 34, pp. 699-708, 1999.
[11] V. Marinca, An approximate solution for one-dimensional weakly
nonlinear oscillations, Int. J. Non-linear Sci. Numer. Simul., vol. 3, pp. 107-110, 2002.
[12] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger-s and coupled Burger-s equations, J. Comput. Appl.
Math., vol. 181(2), pp. 245-51, 2005.
[13] S. Abbasbandy, An approximation solution of a nonlinear equation with
Riemann-Liouville-s fractional derivatives by He-s variational iteration
method, J. Comput. Appl. Math., vol. 207(1), pp. 53-58, 2007.
[14] S. Abbasbandy, A new application of He-s variational iteration method
for quadratic Riccati differential equation by using Adomian-s
polynomials, J. Comput. Appl. Math., vol. 207(1), pp. 59-63, 2007.
[15] S. Momani and S. Abuasad, Application of He-s variational iteration
method to Helmholtz equation, Chaos, Solitons & Fractals, vol. 27(5),
pp. 1119-1123, 2006.
[16] Z. M. Odibat and S. Momani, Application of variational iteration method
to nonlinear differential equations of fractional order, Int. J. Non-linear
Sci. Numer. Simul., vol. 7(1), pp. 27-34, 2006.
[17] A.-M. Wazwaz, The variational iteration method for rational solutions
for KdV, K(2,2), Burgers, and cubic Boussinesq equations, J. Comput.
Appl. Math., vol. 207(1), pp. 18-23, 2007.
[18] M. Tatari and M. Dehghan, He-s variational iteration method for
computing a control parameter in a semi-linear inverse parabolic
equation, Chaos, Solitons & Fractals, vol. 33(2), pp. 671-677, 2007.
[19] A.-M. Wazwaz, The variational iteration method for exact solutions of
Laplace equation, Phys. Lett. A, vol. 363(4), pp. 260-262, 2007.
[20] E. Yusufoglu and A. Bekir, Numerical simulation of equal-width wave
equation, Comput. Math. Appl., vol. 54(7-8), pp. 1147-1153, 2007.
[21] N. H. Sweilam, Fourth order integro-differential equations using
variational iteration method, Comput. Math. Appl., vol. 54(7-8), pp.
1086-1091, 2007.
[22] J.-H. He, Variational iteration method-some recent results and new
interpretations, J. Comput. Appl. Math., vol. 207(1), pp. 3-17, 2007.
[23] J.-H. He and X.-H. Wu, Variational iteration method: new development
and applications, Comput. Math. Appl., vol. 54(7-8), pp. 881-894, 2007.
[24] A.-M. Wazwaz, A reliable algorithm for obtaining positive solutions for
nonlinear boundary value problems, Comput. Math. Appl., vol. 4, pp.
1237-1244, 2001.
[25] S. Strogatz, Nonlinear dynamics and chaos: with applications to
physics, biology, chemistry, and engineering, Addison-Wesley publishing company,1994.
[1] D. A. Nield and A. Bejan, Convection in porous media, Springer, New York, 2006.
[2] M. D. Graham and P. H. Steen, Plume formation and resonant bifurcations in porous-media convection, J. Fluid. Mech., vol. 272, pp.
67-89, 1994.
[3] A. S. M. Cherkaoui and W. S. D. Wilcock, Characteristics of high Rayleigh number two-dimensional convection in an open-top porous
layer heated from below, J. Fluid. Mech., vol. 394 pp. 241-260, 1999.
[4] L. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today,
vol.54, pp. 34-39, 2001.
[5] J. Otero et al., High-Rayleigh-number convection in a fluid-saturated
porous layer, J. Fluid Mech., vol. 500, pp. 263-281, 2004.
[6] E. N. Lorenz, Deterministic non-periodic flows, J. Atmos.Sci., vol. 20,
pp. 130-141,1963.
[7] P. Vadasz and S. Olek, Weak turbulence and chaos for low Prandtl
number gravity driven convection in porous media, Transport in Porous
Media, vol. 37, pp. 69-91,1999.
[8] J.-H. He, Approximate analytical solution for seepage flow with
fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng.,
vol. 167, pp. 57-68, 1998.
[9] J.-H. He, Approximate solution of nonlinear differential equations with
convolution product nonlinearities, Comput. Meth. Appl. Mech. Eng.,
vol. 167, pp. 69-73, 1998.
[10] J.-H. He, Variational iteration methodÔÇöa kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech., vol. 34, pp. 699-708, 1999.
[11] V. Marinca, An approximate solution for one-dimensional weakly
nonlinear oscillations, Int. J. Non-linear Sci. Numer. Simul., vol. 3, pp. 107-110, 2002.
[12] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger-s and coupled Burger-s equations, J. Comput. Appl.
Math., vol. 181(2), pp. 245-51, 2005.
[13] S. Abbasbandy, An approximation solution of a nonlinear equation with
Riemann-Liouville-s fractional derivatives by He-s variational iteration
method, J. Comput. Appl. Math., vol. 207(1), pp. 53-58, 2007.
[14] S. Abbasbandy, A new application of He-s variational iteration method
for quadratic Riccati differential equation by using Adomian-s
polynomials, J. Comput. Appl. Math., vol. 207(1), pp. 59-63, 2007.
[15] S. Momani and S. Abuasad, Application of He-s variational iteration
method to Helmholtz equation, Chaos, Solitons & Fractals, vol. 27(5),
pp. 1119-1123, 2006.
[16] Z. M. Odibat and S. Momani, Application of variational iteration method
to nonlinear differential equations of fractional order, Int. J. Non-linear
Sci. Numer. Simul., vol. 7(1), pp. 27-34, 2006.
[17] A.-M. Wazwaz, The variational iteration method for rational solutions
for KdV, K(2,2), Burgers, and cubic Boussinesq equations, J. Comput.
Appl. Math., vol. 207(1), pp. 18-23, 2007.
[18] M. Tatari and M. Dehghan, He-s variational iteration method for
computing a control parameter in a semi-linear inverse parabolic
equation, Chaos, Solitons & Fractals, vol. 33(2), pp. 671-677, 2007.
[19] A.-M. Wazwaz, The variational iteration method for exact solutions of
Laplace equation, Phys. Lett. A, vol. 363(4), pp. 260-262, 2007.
[20] E. Yusufoglu and A. Bekir, Numerical simulation of equal-width wave
equation, Comput. Math. Appl., vol. 54(7-8), pp. 1147-1153, 2007.
[21] N. H. Sweilam, Fourth order integro-differential equations using
variational iteration method, Comput. Math. Appl., vol. 54(7-8), pp.
1086-1091, 2007.
[22] J.-H. He, Variational iteration method-some recent results and new
interpretations, J. Comput. Appl. Math., vol. 207(1), pp. 3-17, 2007.
[23] J.-H. He and X.-H. Wu, Variational iteration method: new development
and applications, Comput. Math. Appl., vol. 54(7-8), pp. 881-894, 2007.
[24] A.-M. Wazwaz, A reliable algorithm for obtaining positive solutions for
nonlinear boundary value problems, Comput. Math. Appl., vol. 4, pp.
1237-1244, 2001.
[25] S. Strogatz, Nonlinear dynamics and chaos: with applications to
physics, biology, chemistry, and engineering, Addison-Wesley publishing company,1994.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55310", author = "Mohamed M. Mousa and Aidarkhan Kaltayev Shahwar F. Ragab", title = "Investigation of a Transition from Steady Convection to Chaos in Porous Media Using Piecewise Variational Iteration Method", abstract = "In this paper, a new dependable algorithm based on an adaptation of the standard variational iteration method (VIM) is used for analyzing the transition from steady convection to chaos for lowto-intermediate Rayleigh numbers convection in porous media. The solution trajectories show the transition from steady convection to chaos that occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. The VIM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the considered model and other dynamical systems. We shall call this technique as the piecewise VIM. Numerical comparisons between the piecewise VIM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the proposed technique is a promising tool for the nonlinear chaotic and nonchaotic systems.
", keywords = "Variational iteration method, free convection, Chaos, Lorenz equations.", volume = "3", number = "10", pages = "797-10", }
