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The main purpose of this paper is to find convenient methods to solve the differential‐algebraic equations which have great importance in various fields of science and engineering.

The paper applies a semi‐analytical approach, using both the homotopy analysis method (HAM) and the modified homotopy analysis method (MHAM) for finding the solution of linear and nonlinear DAEs.

The results show that the new modification can effectively reduce computational costs and accelerates the rapid convergence of the series solution.

Some high index DAEs are investigated to present a comparative study between the HAM and the MHAM.

This paper aims to explore and review the potential of robotic rehabilitation as a treatment approach for Alzheimer’s disease (AD) and its impact on the health and quality of life of AD patients.

The present discourse endeavors to provide a comprehensive overview of extant scholarly inquiries that have examined the salience of inhibitory mechanisms vis-à-vis robotic interventions and their impact on patients with AD. Specifically, this review aims to explicate the contemporary state of affairs in this realm by furnishing a detailed explication of ongoing research endeavors. With the objective of elucidating the significance of inhibitory processes in robotic therapies for individuals with AD, this analysis offers a critical appraisal of extant literature that probes the intersection of cognitive mechanisms and assistive technologies. Through a meticulous analysis of diverse scholarly contributions, this review advances a nuanced understanding of the intricate interplay between inhibitory processes and robotic interventions in the context of AD.

According to the review papers, it appears that implementing robot-assisted rehabilitation can serve as a pragmatic and effective solution for enhancing the well-being and overall quality of life of patients and families engaged with AD. Besides, this new feature in the robotic area is anticipated to have a critical role in the success of this innovative approach.

Due to the nascent nature of this cutting-edge technology and the constrained configuration of the mechanized entity in question, further protracted analysis is imperative to ascertain the advantages and drawbacks of robotic rehabilitation vis-à-vis individuals afflicted with Alzheimer’s ailment.

The potential for robots to serve as indispensable assets in the provision of care for individuals afflicted with AD is significant; however, their efficacy and appropriateness for utilization by caregivers of AD patients must be subjected to further rigorous scrutiny.

This paper reviews the current robotic method and compares the current state of the art for the AD patient.

The purpose of this paper is to investigate a three-dimensional boundary layer flow with considering heat and mass transfer on a nonlinearly stretching sheet by using a novel operational-matrix-based method.

The partial differential equations that governing the problem are converted into the system of nonlinear ordinary differential equations (ODEs) with considering suitable similarity transformations. A direct numerical method based on the operational matrices of integration and product for the linear barycentric rational basic functions is used to solve the nonlinear system of ODEs.

Graphical and tabular results are provided to illustrate the effect of various parameters involved in the problem on the velocity profiles, temperature distribution, nanoparticle volume fraction, Nusselt and Sherwood number and skin friction coefficient. Comparison between the obtained results, numerical results based on the Maple's dsolve (type = numeric) command and previous existing results affirms the efficiency and accuracy of the proposed method.

The motivation of the present study is to provide an effective computational method based on the operational matrices of the barycentric cardinal functions for solving the problem of three-dimensional nanofluid flow with heat and mass transfer. The convergence analysis of the presented scheme is discussed. The benefit of the proposed method (PM) is that, without using any collocation points, the governing equations are converted to the system of algebraic equations.

This paper aims to discuss the approximate solution of the nonlinear thin film flow problems. A new analytic approximate technique for addressing nonlinear problems, namely, the optimal homotopy asymptotic method (OHAM), is proposed and used in an application to the nonlinear thin film flow problems.

This approach does not depend upon any small/large parameters. This method provides a convenient way to control the convergence of approximation series and to adjust convergence regions when necessary.

The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. The results reveal that the method is explicit. By applying the method to nonlinear thin film flow problems, it was found to be simpler in applicability, and more convenient to control convergence. Therefore, the method shows its validity and great potential for the solution of nonlinear problems in science and engineering.

The proposed method is tested upon nonlinear thin film flow equation from the literature and the results are compared with the available approximate solutions including Adomian decomposition method (ADM), homotopy perturbation method, modified homotopy perturbation method and HAM. Moreover, the exact solution is compared with the available numerical solutions. The graphical representation of the solution is given by Maple and is physically interpreted.

The purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).

For the system in the considered problem, the control variable is a function of the state variables and their derivatives. State variables in the problem are approximated by Legendre expansions as functions of time t. A constant matrix is given to express the derivatives of state variables. Therefore, control variables can be described as functions of time t. After that, the OCP is converted to an unconstrained optimization problem whose decision variables are the unknown coefficients in the Legendre expansions.

The convergence of the proposed algorithm is proved. Experimental results, which contain the controlled Duffing oscillator problem demonstrate that the proposed technique is faster than existing methods.

Experimental results, which contained the controlled Duffing oscillator problem demonstrate that the proposed technique can be faster while securing exactness.

Applying discrete linear optimal control to robot manipulators faces two challenging problems, namely nonlinearity and uncertainty. This paper aims to overcome nonlinearity and uncertainty to design the discrete optimal control for electrically driven robot manipulators.

Two novel discrete optimal control approaches are presented. In the first approach, a control-oriented model is applied for the discrete linear quadratic control while modeling error is estimated and compensated by a robust time-delay controller. Instead of the torque control strategy, the voltage control strategy is used for obtaining an optimal control that is free from the manipulator dynamics. In the second approach, a discrete optimal controller is designed by using a particle swarm optimization algorithm.

The first controller can overcome uncertainties, guarantee stability and provide a good tracking performance by using an online optimal algorithm whereas the second controller is an off-line optimal algorithm. The first control approach is verified by stability analysis. A comparison through simulations on a three-link electrically driven robot manipulator shows superiority of the first approach over the second approach. Another comparison shows that the first approach is superior to a bounded torque control approach in the presence of uncertainties.

The originality of this paper is to present two novel optimal control approaches for tracking control of electrically driven robot manipulators with considering the actuator dynamics. The novelty is that the proposed control approaches are free from the robot's model by using the voltage control strategy. The first approach is a novel discrete linear quadratic control design supported by a time-delay uncertainty compensator. The second approach is an off-line optimal design by using the particle swarm optimization.

Given the importance of logistics operations in business-to-consumer (B2C) e-commerce and growing interest in the related environmental effects, the purpose of this paper is to offer an up-to-date literature review on the topic of B2C e-commerce environmental sustainability, specifically from a logistics perspective.

The analysis focussed on a set of 56 papers published from 2001 to 2014 in 38 peer-reviewed international journals. The papers were analyzed and categorized according to the main features of the paper, the research method(s) adopted and the themes tackled.

There is a growing interest in sustainability issues. In the last 14 years, the focus has progressively shifted from the mere identification of the wide-ranging environmental effects of e-commerce to the need for a quantitative evaluation of their impact, although much remains to be done in this regard. Some industries, such as books and grocery, have largely been addressed, however, promising sectors in the e-commerce field, such as clothing and consumer electronics, have only been considered to a certain degree. Moreover, despite the emerging role of multichannel strategies, the environmental implications of the related logistics activities have not yet been studied in detail.

B2C e-commerce has grown in popularity, and its environmental implications are currently of key interest. This paper contributes to the understanding of the existing body of knowledge on this topic, presenting an up-to-date classification of articles and highlighting themes for further research activities. From a managerial perspective, this paper helps supply chain managers develop a clear understanding of both the logistics areas with the most impact on environmental sustainability and the KPIs used to quantify the environmental implications of e-commerce logistics operations comprehensively and effectively.

The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical solutions of the fractional KdV‐Burgers‐Kuramoto equation.

The authors used Maple Package to calculate the functions obtained from the HPM.

The fractional derivatives are described in the Caputo sense. HPM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.

The paper describes how the HPM has been successfully applied to find the solution of fractional KdV‐Burgers‐Kuramoto equation.

This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.

The realistic numerical solutions were obtained in a form of rapidly convergent series with easily computable components.

The figures show the effectiveness and good accuracy of the proposed method.

The paper obtains realistic numerical solutions in a form of rapidly convergent series with easily computable components. It shows the effectiveness and good accuracy of the proposed method.

The purpose of this paper is to apply He's variational iteration method (VIM) coupled with an auxiliary parameter, which proves very effective to control the convergence region of approximate solution.

The proposed algorithm is tested on generalized Hirota‐Satsuma coupled KdV equation.

Numerical results explicitly reveal the complete reliability, efficiency and accuracy of the suggested technique.

It is observed that the approach may be implemented on other nonlinear models of physical nature.