Convergent optimal variational iteration method and applications to heat and fluid flow problems

Abstract

Purpose – In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method. Design/methodology/approach – A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method. Findings – The optimal variational iteration method is found to be useful for heat and fluid flow problems. Originality/value – The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

“…Equation (1) and other equations of a similar nature are sometimes solved by analytical and semi-analytical methods such as residual power series method [13] and optimal variational iteration method [14,15]. However, the most common way of solution is numerical integration [16], and numerous techniques have been put forth for this.…”

A New Numerical Approach for the Analysis of Variable Fractal and Fractional Order Differential Equations

“…Various approximation techniques are available to procure these nonlinear “ fractional differential equations ”. The numerical treatment of the operator may uses direct and indirect implementation techniques, such as the “ Homotopy Perturbation Method ” [ 5 ], “ Variational Iteration Method ” [ 6 ], “ Homotopy Perturbation Transform Method ” [ 7 ], “ Adomian Decomposition Method ” [ 8 ], “ Wavelet Method ” [ 9 ]], [ 10 ], Chaotic system [ 11 ] and “ Laplace Transform ” [ 12 ]. Further, Riemann–Liouville, Riesz, Grunwald–Letnikov, Weyl, and Liouville-Caputo have proposed some of the fractional order derivative concepts.…”

Thermal investigation of fully wet longitudinal porous fin of functionally graded material

“…The pin fin of different profiles has also been considered and the optimisation and efficiency in the presence of a fully wet condition by Vahabzadeh et al (2015) has been investigated. The variational iteration method has been used to study the heat and fluid flow problem by Turkyilmazoglu (2016). The spectral collocation method has been implemented and the thermal features of wetted longitudinal porous fin by Darvishi et al (2016) have been deliberated.…”