Convergence Analysis of Modified Iterative Methods to Solve Linear Systems

In the past years, a growing interest to solve linear systems with modified iterative methods has been shown by researchers. Recently, Dehghan and Hajarian (J Vib Control, doi:10.1177/10775463124, 2012)based on preconditioned methods, introduced modified accelerated overrelaxation methods for solving linear systems. These authors stated that the best property of the mentioned methods is that they can be used under mild conditions than the Milaszewicz’s (Linear Algebra Appl 93:161–170, 1987) preconditioner. In this paper, we show that the Milaszewicz’s preconditioner is applicable under mild conditions and also, under these conditions, Milaszewicz’s preconditioner is superior to the Dehghan and Hajarian’s preconditioner. Numerical examples are also given to illustrate our results.     

Theoretical investigation on conformational preferences and structural properties of 2-lithio-1,3-diphosphinane and 2-lithio-1,3-dimethyl-1,3-diphosphinane

A new family of dialphosphacyclohexane is introduced which has three nucleophilic centers. The conformational stabilities and structural properties of 2-lithio-1,3-diphosphinane and 2-lithio-1,3-dimethyl-1,3-diphosphinane were investigated computationally by DFT calculations and NBO analyses at B3LYP/6-31+G(d,p). Relative energy trend in 1,3-diphosphinane and 1,3-dimethyl-1,3-diphosphinane conformations explored from steric and hyperconjugative point of view. The stability trend of 2-lithio derivatives based on calculated relative energies in 1,3-diphosphinane is 1d > 1f > 1b > 1a > 1c > 1e and for 1,3-dimethyl-1,3-diphosphinane derivatives is 2f > 2d > 2b > 2a > 2c > 2e. Calculated NBO atomic charges indicate that high positive charge at lithium and small C–Li Wiberg bond indexes in these derivatives are demonstrators of ionic nature of the C–Li bonds. Stereoelectronic interactions, polarizability of phosphorus, and chelate formation between each of phosphorous and lithium are determining factors in stability trend observed in these derivatives. Ease of lithiation in bis(dimethylphosphino)methane, 1,3-diphosphinane, and 1,3-dimethyl-1,3-diphosphinane derivatives was estimated and compared by isodesmic reaction.     

Hyperchaos control of the hyperchaotic Chen system by optimal control design

The aim of this paper is to study the chaos, optimal control, and adaptive control of the hyperchaotic Chen system. In this paper, applying the Pontryagin’s minimum principle (PMP), the optimal control inputs for the interested model are obtained with respect to the selected measure. A piecewise-spectral homotopy analysis method (PSHAM) is used for solving the hyperchaotic Chen system and the extreme conditions obtained from the PMP. Furthermore, an adaptive control approach and a parameter estimation update law are introduced for the hyperchaotic Chen system with completely unknown parameters. The control results are established using the Krasovskii–LaSalle principle. Finally, numerical simulations are included to demonstrate the effectiveness of the proposed control strategy.     

Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.