Variational Iteration Method for Singular Perturbation Initial Value Problems with Delays

Abstract

The variational iteration method (VIM) is applied to solve singular perturbation initial value problems with delays (SPIVPDs). Some convergence results of VIM for solving SPIVPDs are given. The obtained sequence of iterates is based on the use of general Lagrange multipliers; the multipliers in the functionals can be identified by the variational theory. Moreover, the numerical examples show the efficiency of the method.

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.…”

Convergence of Variational Iteration Method for Fractional Delay Integrodifferential‐Algebraic Equations

“…The variational iteration method (VIM) is one of the important methods used to obtain approximate analytical solutions [22][23][24][25] and possesses some good properties, such as flexibility, convenience, accuracy, and less storage. In particular, this method was used to solve pantograph equations [26,27], differential (integral) equations [28][29][30], fractional differential (integral) equations [31][32][33][34], delay differentialalgebraic equations [35] and fractional differential-algebraic equations [36], and so forth.…”

A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line

“…The models involving fractional derivatives and integrals have memory; therefore it has proven to be very suitable for the description of memory and hereditary properties of various processes [3][4][5][6][7][8][9][10][11]. Interested readers can also see [12][13][14][15][16][17][18][19][20][21].…”