Variational iteration method for singular perturbation initial value problems

A Comparison of Explicit Semi-Analytical Numerical Integration Methods for Solving Stiff ODE Systems

“…In addition, for a fixed order in NMsDTM or a fixed number of iterations in E I -4MsDTM, as the step size decreases the numerical solution converges to the exact one. While, for a fixed step size h, the convergence of (NMsDTM, E I -4MsDTM) only begins when (N, I) are (O(Kh), O(Lh)), respectively, where for Vulanovic, 1989, Theorem 1]; (El-Zahar and EL-Kabeir, 2013, Lemma 3.2); Zhao and Xiao, 2010;Zhao et al, 2014). Table 4, show us that, for the same step size h, the (I+4) MsDTM has the lowest CPU usage and the E I -4MsDTM has the highest CPU usage in solving (32).…”

“…Solving stiff problems is one of the most recent applications of these methods; see for example (Mahmood et al, 2005;Guzel and Bayram, 2005;Darvishi et al, 2007;Hassan, 2008;Zhao and Xiao, 2010;Aminikhah and Hemmatnezhad, 2011;Aminikhah, 2012;Zou et al, 2012;Atay and Kilic, 2013;Zhao et al, 2014;El-Zahar et al, 2014b). However, for some important classes of problems such as stiff ODE problems, singularly perturbed problems, chaotic and non-chaotic problems and nonlinear oscillators and for the sake of large convergence region, accuracy and efficiency, it is necessary to treat each of the above mentioned semi-analytical numerical methods as an algorithm in a sequence of time intervals.…”

On the Closed Form Solutions of Linear and Nonlinear Cauchy Reaction-Diffusion Equations Using the Hybrid of Fourier Transform and Variational Iteration Method

“…However, the solutions of these problems are valid only in onedirectional problem domain either in time or space problem domain. In other words, the unsatisfied boundary conditions in the solutions of the VIM and other semi-analytical methods play no role in the final results (Kasozi et al, 2011;Gupta and Singh, 2011;Othman et al, 2010;Gepreel, 2011;Aslanov, 2011;Madani et al, 2011;He et al, 2010;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011;Afshari et al, 2009;Aruchunan and Sulaiman, 2010). Therefore, there has been a deficiency as a built in short comes in the solutions using the VIM and other semi-analytical methods.…”

“…Moreover, due to the non-linear nature and variable coefficients of these differential equations, attentions are devoted to the approximate solutions obtained by semi analytical methods such as the Homotopy Perturbation Method (HPM) (He, 2009;Kasozi et al, 2011;Gupta and Singh, 2011;Othman et al, 2010;Gepreel, 2011;Aslanov, 2011) and Variational Iteration Method (VIM) (He et al, 2010;Shakeri, et al, 2009;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011). The VIM is developed by employing a correction functional and a general Lagrange multiplier for the differential equation.…”

“…Then by the variational principle the optimum value is obtained for the correction functional and it is solved using the iteration method (He et al, 2010;Shakeri, et al, 2009;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011). The variational iteration method has been applied to so many engineering real-life problems so far (He et al, 2010;Shakeri et al, 2009;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011). However, the solutions of these problems are valid only in onedirectional problem domain either in time or space problem domain.…”