We analyze the convergence properties of the spectral method when used to approximate smooth solutions of delay differential or integral equations with two or more vanishing delays. It is shown that for the pantograph-type functional equations the spectral methods yield the familiar exponential order of convergence. Various numerical examples are used to illustrate these results.

An analytical solution of an infinite lubricated inclined slider bearing with nonNewtonian Powell-Eyring fluid as lubricant is presented. The homotopy perturbation method (HPM) is used and solutions for velocity components and pressure distribution in the slider bearing are derived with the method. The pressure distributions in the bearing are illustrated graphically for a range of non-Newtonian fluid material and bearing geometric parameters, and the effects of these parameters on the load carrying capacity of the bearing are discussed.

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The Internet of Things (IoT) industry is growing very fast to transform factories, homes, farms and practically everything else to make them efficient and intelligent. IoT is applied in different resilient scenarios and applications. IoT faces lots of challenges due to lack of computational power, battery and storage resources. Fortunately, the rise of blockchain technology facilitates IoT in many security solutions. Using blockchain, communication between IoT and emerging computing technologies is made efficient. In this work, we propose a secure service provisioning scheme with a fair payment system for Lightweight Clients (LCs) based on blockchain. Furthermore, an incentive mechanism based on reputation is proposed. We use consortium blockchain with the Proof of Authority (PoA) consensus mechanism. Furthermore, we use Smart Contracts (SCs) to validate the services provided by the Service Providers (SPs) to the LCs, transfer cryptocurrency to the SPs and maintain the reputation of the SPs. Moreover, the Keccak256 hashing algorithm is used for converting the data of arbitrary size to the hash of fixed size. AES128 encryption technique is used to encrypt service codes before sending to the LCs. The simulation results show that the LCs receive validated services from the SPs at an affordable cost. The results also depict that the participation rate of SPs is increased because of the incentive mechanism.

The analysis explored a numerical simulation of microorganisms, carbon nanotubes (CNTs) and ferric oxide water-based hybrid nanofluid flow induced by a wavy fluctuating spinning disc with energy propagation. In the presence of CNTs and magnetic nanoparticulates, the nanofluid is synthesized. The exceptional tensile strength, flexibility, and electrical and thermal conductivity of carbon nanotubes and iron nanoparticles have been extensively reported. The motive of the proposed analysis is to optimize thermal energy conveyance efficiency for a spectrum of industrial and biomedical applications. The phenomena have been expressed as a system of partial differential equations (PDEs) which contain the momentum, energy, concentration, and motile microorganism equations. The modeled equations have been diminished to the dimensionless system of nonlinear ODEs through a similarity framework. The Matlab built-in package boundary value solver has been utilized to solve the obtained system of ODEs. The findings are compared to the PCM technique for validity purposes. The results are illustrated graphically and discussed. The layout of a rotating disc has a positive effect on energy transition and velocity profile. The irregular rotating surface increases energy progression up to 15% relative to a smooth surface. The accumulation of nanocomposites (CNTs and magnetic nanoparticles) significantly enhanced the thermal capabilities of the liquid medium. When operating with a low distribution, it is more impactful.

In this article, the heat transfer is examined in spinning flow of the Casson nanofluid. The spinning motion is produced with the help of stretching and rotating cylinder. In current work two cases, both for constant and variant axial temperature over the exterior of the spinning cylinder are studied. For controlling the rate of heat transfer, some heat effects such as heat generation/absorption, Joule heating and thermal radiations have been considered for flow system. For investigation the influence of thermophoretic force and Brownian motion exerted by Casson nanofluid, the famous Buongiorno model has been employed. The modeled equations have converted to dimensionless form using a set of similar variables. To improve the solution convergence of flow equations a compressed variable is also employed. As the main outcome of this work, it has observed that, with growing values of Reynolds number and magnetic parameter the flow characteristics reduce in all directions. A growth in temperature profile is also observed for constant wall temperature (CWT), while a reduction is noticed in temperature profile for prescribed surface temperature (PST). Moreover, the concentration of fluid reduces with growing values of Prandtl and Lewis numbers.