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Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.

In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.

The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.

A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

This article aims to describe the effect of an endoscope and heat transfer on the peristaltic flow of a Jeffrey fluid through the gap between concentric uniform tubes.

The mathematical model of the present problem is carried out under long wavelength and low Reynolds number approximations. Analytical solutions for the velocity, temperature profiles, pressure gradient and volume flow rate are obtained.

The results indicate that the effect of the wave amplitude, radius ratio, Grashof number, the ratio of relaxation to retardation times and the radius are very pronounced in the phenomena. Also, a comparison of obtaining an analytical solution against previous literatures shows satisfactory agreement.

Analytical solutions for the velocity, temperature profiles, pressure gradient and volume flow rate are obtained. Numerical integration is performed to analyze the pressure rise and frictional forces on the inner and outer tubes.

The human resource (HR) allocation problem is one of the critical dimensions of the project management process. Due to this nature of the problem, researchers are continually optimizing one or more critical scheduling and allocation challenges in different ways. This study aims to optimize two goals, increasing customer satisfaction and reducing costs using the imperialist competitive algorithm.

Cloud-based e-commerce applications are preferred to conventional systems because they can save money in many areas, including resource use, running expenses, capital costs, maintenance and operation costs. In web applications, its core functionality of performance enhancement and automated device recovery is important. HR knowledge, expertise and competencies are becoming increasingly valuable carriers for organizational competitive advantage. As a result, HR management is becoming more relevant, as it seeks to channel all of the workers’ energy into meeting the organizational strategic objectives. The allocation of resources to maximize benefit or minimize cost is known as the resource allocation problem. Since discovering solutions in polynomial time is complicated, HR allocation in cloud-based e-commerce is an Nondeterministic Polynomial time (NP)-hard problem. In this paper, to promote the respective strengths and minimize the weaknesses, the imperialist competitive algorithm is suggested to solve these issues. The imperialist competitive algorithm is tested by comparing it to the literature’s novel algorithms using a simulation.

Empirical outcomes have illustrated that the suggested hybrid method achieves higher performance in discovering the appropriate HR allocation than some modern techniques.

The paper presents a useful method for improving HR allocation methods. The MATLAB-based simulation results have indicated that costs and waiting time have been improved compared to other algorithms, which cause the high application of this method in practical projects.

The main novelty of this paper is using an imperialist competitive algorithm for finding the best solution to the HR allocation problem in cloud-based e-commerce.

The purpose of this paper is to demonstrate the eligibility of the weighted residual methods (WRMs) applied to Jeffery-Hamel Flow. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research.

Three analytical methods (collocation, Galerkin and least square method) have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the forth order Runge-Kutta numerical method.

The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations.

It could be considered as a first endeavor to use the solution of the Jeffery-Hamel flow using these kind of analytical methods along with the numerical approach.

The purpose of this study is to determine the impact of two-phase power law nanofluid on a curved arterial blood flow under the presence of ovelapped stenosis. Over the past couple of decades, the percentage of deaths associated with blood vessel diseases has risen sharply to nearly one third of all fatalities. For vascular disease to be stopped in its tracks, it is essential to understand the vascular geometry and blood flow within the artery. In recent scenarios, because of higher thermal properties and the ability to move across stenosis and tumor cells, nanoparticles are becoming a more common and effective approach in treating cardiovascular diseases and cancer cells.

The present mathematical study investigates the blood flow behavior in the overlapped stenosed curved artery with cylinder shape catheter. The induced magnetic field and entropy generation for blood flow in the presence of a heat source, magnetic field and nanoparticle (Fe₃O₄) have been analyzed numerically. Blood is considered in artery as two-phases: core and plasma region. Power-law fluid has been considered for core region fluid, whereas Newtonian fluid is considered in the plasma region. Strongly implicit Stone’s method has been considered to solve the system of nonlinear partial differential equations (PDE’s) with 10–6 tolerance error.

The influence of various parameters has been discussed graphically. This study concludes that arterial curvature increases the probability of atherosclerosis deposition, while using an external heating source flow temperature and entropy production. In addition, if the thermal treatment procedure is carried out inside a magnetic field, it will aid in controlling blood flow velocity.

The findings of this computational analysis hold great significance for clinical researchers and biologists, as they offer the ability to anticipate the occurrence of endothelial cell injury and plaque accumulation in curved arteries with specific wall shear stress patterns. Consequently, these insights may contribute to the potential alleviation of the severity of these illnesses. Furthermore, the application of nanoparticles and external heat sources in the discipline of blood circulation has potential in the medically healing of illness conditions such as stenosis, cancer cells and muscular discomfort through the usage of beneficial effects.