Analytical Solution Approach Research Articles

Analytical solution approach for nonlinear vibration of shear deformable imperfect FG-GPLR porous nanocomposite cylindrical shells

This study presents an analytical solution approach to examine the nonlinear vibration of geometrically imperfect functionally graded porous circular cylindrical shells reinforced with graphene platelets (GPL) surrounded on an elastic foundation. First-order shear deformation theory is employed to formulate the considered problem. Four porosity distributions and four GPLs dispersion patterns are considered which vary through the thickness direction. The effective mechanical properties of considered functionally graded graphene platelet-reinforced porous nanocomposites are characterized via a micromechanical model. Governing equations are derived by Hamilton’s principle and then were transformed into a set of ordinary differential equations using the Galerkin method. Afterward, the nonlinear frequency response curves are obtained with the use of the method of multiple scales. Numerical results are provided to explore the effect of parameters such as initial imperfection, geometry, porous distribution, porosity coefficient, and GPLs’ scheme and weight fraction on the nonlinear frequency-response curve.

A variational approach for analytical buckling solution of moderately thick microplate using strain gradient theory incorporating two-variable refined plate theory: a benchmark study

In this paper, two-variable refined plate theory and strain gradient theory (SGT) are adopted and termed (TV-SGT) for buckling analysis of moderately thick microplate. The TV-SGT embraces both shear and micro-effects simultaneously and, therefore, does not need the shear correction factor. Notably, the governing equations and new boundary conditions are derived using variational approach. Moreover, the boundary conditions are generalized for complicated problems. The constitutive equations are solved analytically as a benchmark problem for simply supported thick moderately microplate. The accuracy and simplicity of utilizing this theory for investigating stability analysis of thick microplate are assessed, and the critical buckling load of the microplate for uniaxial and biaxial loadings is studied. The impacts of thickness, length scale parameters and dimensions on the non-dimensionalized critical buckling load of the microplate are investigated. Consequently, increasing thickness leads to the rising gap between TV-SGT and classical theory results. Therefore, classical plate theory is ruled out in analysing of the moderately thick microplate, whereas TV-SGT is well defined to analyse the moderately thick microplate.

The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements

Physical systems in partial differential equations can be interpreted in a visual form using a wave simulation. In particular, the interpretation of the differential equations used is in the nonlinear hyperbolic model, but in its completion, there are some limitations to the stability requirements found. The aim of this study is to investigate the analytical and numerical analysis of a wave equation with a similar unit and fractal intervals using the Fourier coefficient. The method in this research is to use the analytical solution approach, the spectral method, and the finite difference method. The hyperbolic wave equation's analytical solution approach, illustrated in the Fourier analysis, uses a pulse triangle. The spectral method minimizes errors when there is the addition of the same sample grid points or the periodic domain's expansion with a trigonometric basis. Meanwhile, different ways offer a more efficient solution. Based on the research results, the information obtained is that the Fourier analysis illustrates the pulse triangle use to solve the solution. These results are also suitable for adding sample points to the same spectra. Fourier analysis requires a relatively long time to solve one pulse triangle graph to need another solution, namely the finite difference method. However, its use is still limited in terms of stability when faced with more complex problems.

Effects of local thinning defects and stepped thickness for free vibration of cylindrical shells using a symplectic exact solution approach

Based on the Reissner shell theory and a new symplectic method, an analytical exact solution approach is presented in this paper to deal with free vibration of stepped cylindrical shells with thickness variations in both the axial and circumferential directions. Such cylindrical shells are often applied for designing aeronautics and space structures and hence any local defects and local geometric variations will result in significant design consequences. For this reason, this paper intends to establish a cylindrical shell model with local wall defects including local wall thinning and wall thickness stepped variations. A new exact solution approach for free vibration of the model are obtained by superposing the eigensolutions of the perfect shell. To ensure validity of the present result and the universality of the method, the solutions for three forms of stepped shells are compared with published data and finite element analysis. The effects of geometric parameters of the local thinning defect on free vibration characteristics are also investigated and presented.

Dissection of Interaction Kinetics through Single-Molecule Interaction Simulation.

The ability to extract kinetic interaction parameters from single-molecule fluorescence resonance energy transfer trajectories without the need for solving complex single-molecule differential equations has the potential to address some of the critical biophysical questions. Here, we provide a noise-free single-molecule interaction simulation (SMIS) tool to give the expected dwell-time distributions and relative populations of each FRET level based on the assigned kinetic model and to dissect kinetic interaction parameters from single-molecule FRET trajectories. The method provides the expected dwell-time distributions, average transition rates, and relative populations of each FRET level based on the assigned kinetic model. By comparing with ground truth data and experimental data, we demonstrated that SMIS is useful to quantify the interaction kinetic rate constants without using the traditional single-molecule analytical solution approach.

Modelling heterogeneous future wireless cellular networks: An analytical study for interaction of 5G femtocells and macro-cells

The unprecedented growth in the number of mobile nodes, connected devices, and data traffic lead to the dense deployment of fifth generation (5G) networks. In such networks, small cells or femtocells are becoming quite popular mainly because of their features such as consistent coverage regardless of the location, transmit power adjustment, energy efficiency, and their ability of providing higher data rates compared to traditional cellular networks. Although the number of femtocells in a heterogeneous cellular network environment can be increased with an aim of satisfying high data traffic in next-generation 5G networks, higher numbers of small cell deployments may result in unnecessary, and frequent handovers. Therefore, the need for improvement and optimization of femtocell handover paradigms as well as their coexistence with other technologies should be investigated in the era of 5G cellular networks. This study proposes an analytical model which can be used to analyse integrated heterogeneous wireless cellular networks which consist femtocells. Integration of fourth and fifth generation wireless cellular systems is considered for this purpose. The state diagram of the proposed model can be represented by a two dimensional Markov Chain. A new, novel decomposition approach is employed for solving the system for state probabilities and to obtain performance measures such as mean queue length, throughput and response time. The main advantage of the new solution method is its capability of considering interaction of two systems with large queuing capacities unlike the existing matrix based solution approaches such as spectral expansion and matrix geometric methods. The proposed analytical solution approach has been validated using simulation and a system of simultaneous equations.

A Novel Fast Convergence Control Scheme for a Class of 3D Chaotic Systems with Uncertain Parameters and External Disturbances

This paper studies the control of a class of 3D chaotic systems with uncertain parameters and external disturbances. A new method which is referred as the analytical solution approach is firstly proposed for constructing Lyapunov function. Then, for suppressing the trajectories of the 3D chaotic system to its equilibrium point 00,0,0, a novel fast convergence controller containing parameter λ which determines the convergence rate of the system is presented. By using the designed Lyapunov function, the stability of the closed-loop system is proved via the Lyapunov stability theorem. Computer simulations are employed to a new chaotic system to illustrate the effectiveness of the theoretical results.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

The price-setting newsvendor with Poisson demand

The price-setting newsvendor (PSN) model has received considerable attention since it was first introduced by Whitin (1955). However, the existing publications that study this model consistently assume the existence of a continuous density function of demand. In this paper, we study the PSN model with Poisson demand — that is, a discrete demand distribution without density function. The Poisson PSN has an important property, it combines price-dependency of variance and coefficient of variation of the (standard) additive and multiplicative models: demand variance decreases and the coefficient of variation increases in the selling price. We develop an analytical solution approach that covers a broad class of demand models, including linear and logit demand, explain how to apply our approach to more general demand functions via piece-wise linear approximation, and develop analytical and numerical insights. We characterize the behavior of the optimal price and we analyze the performance gap of different price-setting heuristics. Among other insights, we observe some instances in which a significant share of profits would be lost if the discrete nature of demand were not modeled explicitly. To help companies overcome this risk, we present an easily applicable decision rule with which to determine when to use simple heuristics and when to solve the associated discrete optimization problem.

Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory

In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.

Photoacoustics of core\u2013shell nanospheres using comprehensive modeling and analytical solution approach

Photoacoustic visualization of nanoparticles is capable of high contrast imaging at depth greater than that of traditional optical imaging techniques. Identifying the impact of various parameters on the photoacoustic signal is crucial in the design of effective medical imaging and diagnostics. Here, we develop a complete model of Fourier heat conduction incorporating the interfacial thermal resistance and photoacoustic equation for core-shell nanospheres in a fluid under nanosecond pulsed laser illumination. An analytical solution is obtained, elucidating the contribution of each region (core, shell, or the fluid) in the generation of the photoacoustic signal. The model reveals that the sharper the laser pulse temporal waveform is, the higher the sensitivity of the generated photoacoustic signal will be to the interfacial thermal resistance, and, thus, the higher the possibility of photoacoustic signal amplification will be using silica-coating. The comprehensive model and adopted analytical solution reveal the underlying physics of the photoacoustic signal generation form core-shell nanosphere systems.

Analytical solution and numerical approaches of the generalized Levèque equation to predict the thermal boundary layer

In this paper, the assumptions implicit in Leveque's approximation are re-examined, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. By defining a similarity variable the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge-Kutta fourth order (RK4) method. Finally, others important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail.

Postbuckling analysis of functionally graded graphene platelet-reinforced polymer composite cylindrical shells using an analytical solution approach

An analytical approach is proposed to study the postbuckling of circular cylindrical shells subject to axial compression and lateral pressure made of functionally graded graphene platelet-reinforced polymer composite (FG-GPL-RPC). The governing equations are obtained in the context of the classical Donnell shell theory by the von Karman nonlinear relations. Then, based on the Ritz energy method, an analytical solution approach is used to trace the nonlinear postbuckling path of the shell. The effects of several parameters such as the weight fraction of the graphene platelet (GPL), the geometrical properties, and distribution patterns of the GPL on the postbuckling characteristics of the FG-GPL-RPC shell are analyzed.

A New Approach for the Approximate Analytical Solution of Space-Time Fractional Differential Equations by the Homotopy Analysis Method

The motivation of this study is to construct the truncated solution of space-time fractional differential equations by the homotopy analysis method (HAM). The first space-time fractional differential equation is transformed into a space fractional differential equation or a time fractional differential equation before the HAM. Then the power series solution is constructed by the HAM. Finally, taking the illustrative examples into consideration we reach the conclusion that the HAM is applicable and powerful technique to construct the solution of space-time fractional differential equations.

Closed-Form Structural Stress Solutions for Spot Welds in Square Plates under Central Bending Conditions

<div class="section abstract"><div class="htmlview paragraph">A new closed-form structural stress solution for a spot weld in a square thin plate under central bending conditions is derived based on the thin plate theory. The spot weld is treated as a rigid inclusion and the plate is treated as a thin plate. The boundary conditions follow those of the published solution for a rigid inclusion in a square plate under counter bending conditions. The new closed-form solution indicates that structural stress solution near the rigid inclusion on the surface of the plate along the symmetry plane is larger than those for a rigid inclusion in an infinite plate and a finite circular plate with pinned and clamped outer boundaries under central bending conditions. When the radius distance becomes large and approaches to the outer boundary, the new analytical stress solution approaches to the reference stress whereas the other analytical solutions do not. The Mode II stress intensity factor solutions based on the new structural stress solution for a rigid inclusion in a square plate under central bending conditions are improved by up to 3% for similar spot welds in lap-shear specimens of equal thickness when compared with those obtained from the corresponding finite element analyses.</div></div>

Analytical solution and finite element approach to the dense re-entrant unit cells of auxetic structures

Chiral, star honeycomb, and re-entrant structures are among the most important structures of auxetic materials. In this study, a dense re-entrant unit cell is introduced for making a 3D auxetic structure to be used in high stiffness applications. A re-entrant structure is chosen due to its fundamental characteristics underlying the main characteristics of auxetic structures. The energy methods of solid mechanics along with numerical methods are used to study the fundamental concept of auxetic structures. Understanding the characteristics of the re-entrant structure will lead to the better comprehension of other structures of auxetic materials, which will eventually contribute to the advance of research in this new class of materials.

Pursuit differential-difference games with pure time-lag

The analytical approach for solution of pursuit differential-difference games with pure time-lag is considered. For the pursuit local problem with the fixed time the scheme of the method of resolving functions and Pontryagin's first direct method are developed. The integral presentation of game solution based on the time-delay exponential is proposed at first time. The guaranteed times of the game termination are found, and corresponding control laws are constructed. Comparison of the times of approach by the method of resolving functions and Pontryagin's first direct method for the initial problem are made.

Growth-induced instabilities of an elastic film on a viscoelastic substrate: analytical solution and computational approach via eigenvalue analysis

Growth-induced instabilities of an elastic film on a viscoelastic substrate: analytical solution and computational approach via eigenvalue analysis

Chemical diffusion coefficient of oxygen in thoria based fuel with 3 and 8 weight % urania

The chemical diffusion coefficients D˜ of oxygen in (Th1-yUy)O2+x cuboid specimens with 3 and 8 wt% UO2 were measured using the thermogravimetry method in the temperature range of 1316 K ≤ T ≤ 1428 K. The magnitude of the measured diffusion coefficients D˜ in these two specimens was as low as 5.3 × 10−11 ±2 × 10−11 m2 s−1 and as high as 4.5 × 10−10 ±2.2 × 10−12 m2 s−1 that generally increased with increasing uranium content. The activation energies for interstitial oxygen diffusion in the specimen thoria-urania matrix were found to range from 150.4 to 174.2 kJ mol−1, which was slightly higher than previously reported. An analytical solution approach with weight measurements was the technique used to assess the chemical diffusion coefficient in the specimens.

Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading

Levy's analytical solution approach is extended for analysis of rectangular strain gradient elastic plates under static loading for the first time with different boundary conditions at the edges using the method of superposition. The governing equation of equilibrium and the corresponding classical/non-classical boundary conditions for strain gradient flexural Kirchhoff plate under static loading are considered. Numerical examples on static bending of Kirchhoff nanoplates involving five different combinations of simply supported, clamped and free edge boundary conditions are presented. The effect of negative strain gradient terms is of hardening nature thus resulting in decrease in the deflection. Plates with geometry comparable to the microstructural length scale show significant size effect and this size dependency diminishes with the increase in the plate size.