Analytical Solution Technique Research Articles

Solutions to a two-phase mass flow model with generalized drag

Drag plays a dominant role in the interfacial momentum exchange in mixture mass flows. In this study, we examine a general two-phase mass flow model formulated by Pudasaini (2012), which incorporates drag. This model describes the mass flow comprising a mixture of solid particles and viscous fluid moving downhill under the influence of gravity. We construct explicit, analytical, and numerical solutions to the model using the Lie symmetry method. These new solutions disclose the role of generalized drag in the dynamics of both solid particles and viscous fluid. The solutions show that solid and fluid phases undergo nonlinear evolution in a coupled manner. Additionally, the solutions demonstrate that increased drag results in a tighter binding between solid and fluid components. We also analyze the role of pressure gradients. The solutions reveal that when solid pressure dominates fluid pressure, solid velocity increases faster than fluid velocity. These findings align with our expectations, emphasizing the importance of analytical solution techniques in understanding the complex process of mixture mass transport in mountain slopes and valleys, thereby enhancing our understanding.

Solitary wave solutions for the (1+1)-dimensional nonlinear Kakutani–Matsuuchi model of internal gravity waves

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.

Two‐Dimensional Analytical Solution for Transient Flow in Unsaturated Soils Considering Hydromechanical Coupling

AbstractThis paper presents a two‐dimensional solution for unsaturated transient seepage flow in a trapezoidal region using analytical solutions for flow in a series of rectangular regions. The coupled formulation considers the interaction among fluid flow due to differences in head and displacement of soil particles. The solution in each rectangular region is derived using Laplace Transform and Discrete Fourier Series for two top boundary conditions: specified pressure head and specified flow rate. The accuracy of the proposed solution is verified by comparing the results against those attained from the finite difference method. The proposed solution is applied to two example problems, demonstrating its efficacy and accuracy. The findings show a significant difference in the coupled and uncoupled results for fine‐grained soils possessing a small Gardner's coefficient. Additionally, the uncoupled pressure heads advanced quicker to a steady state than the coupled ones for soils with a small Gardner's coefficient. The coupling effect becomes less prominent as Gardner's coefficient increases. This study is the first attempt in the literature to provide a solution for unsaturated flow in a trapezoidal or non‐rectangular domain using analytical solution techniques, which can better represent complex real‐world applications (e.g., slopes, embankments, dams).

Vibration of smart sandwich plate with an auxetic core and dual-FG nanocomposite layers integrated with piezoceramic actuators

In the current research, a thorough investigation of the free vibrational characteristics of a novel smart sandwich plate has been performed utilizing two different analytical and numerical solution techniques. The proposed structure is a sandwich plate with an auxetic core and super stiff dual-FG nanocomposite layers integrated with piezoceramic actuators. The nanocomposite layers of the structure are made of FG metal-ceramic matrix to which carbon nanotubes (CNTs) are added through various FG distribution patterns. Besides, two thin piezoceramic actuators are attached to the nanocomposite layers, which causes the structure to be controllable by applying an electrical voltage. Using such super stiff dual-FG nanocomposite layers together with piezoceramic actuators and ultra-light auxetic core with negative Poisson’s ratio results in the light and smart sandwich plate with high structural stiffness, which its natural frequency responses will be analyzed in this study. For this purpose, the governing equations are derived according to the first-order shear deformation theory (FSDT) and then solved using two different strategies, including numerical and analytical solution methods. The responses obtained by these solution methods by the name of generalized differential quadrature (GDQ) and Navier are compared and verified with each other and other reputed references in the literature. Accordingly, detailed parametric studies in the framework of tabulated and graphical results are performed to reveal the impacts of all involved parameters on the natural frequency responses of the proposed smart sandwich plate.

A new approach to analyzing wellbore pressure dynamically in well-killing

Casing pressure during well-killing is both an unknown variable that must be addressed and a time-dependent function. Therefore, the calculation of the dynamic reaction of wellbore pressure during well-killing is a dynamic indeterminate boundary problem. Due to the intricacy of the analytical solution technique, finding an analytical solution for the dynamic issue of uncertain boundaries is nearly impossible. Current numerical simulation software is only capable of resolving problems with defined bounds (such as flow or pressure) and is incapable of continuously calculating the wellbore pressure, which varies dynamically with well-killing time. Developing a dynamic reaction computation method for wellbore pressure during well-killing can solve this issue. Based on the mathematical model and definite solution conditions of the wellbore pressure dynamic response under well-killing conditions, this paper applies the finite difference principle and constructs the calculation method of choke and kill wellbore pressure dynamic response through reasonable mesh division of wellbore and gradual iteration of the calculation process. The resultsdemonstrate that the maximum wellhead casing pressure value and the time node of peak value are both highly congruent with the measured data, indicating the method’s high reliability and highlighting its significance for the progress of oil and gas well control safety.

Analytical and numerical solution techniques for a class of time-fractional integro-partial differential equations

Analytical and numerical solution techniques for a class of time-fractional integro-partial differential equations

Porosity-dependent free vibration and transient responses of functionally graded composite plates employing higher order thickness stretching model

Tailoring the mechanical properties in hybrid materials, like functionally graded materials (FGMs), has become more pragmatic with the recent progress in material manufacturing and design. The porosities in FGMs, which develop during the manufacturing process, may become detrimental to armored plates or bulletproof designs. On the other hand, gradient-porosity distributions may have broad advantages in the design of lightweight and variable-stiffness aircraft components. The present study investigates the free vibration and transient responses of porosity-gradient FGM plates. A higher order theory, which considers the transverse shear and normal deformation, warping of the transverse cross-section, and higher order rotary inertia, is utilized for the first time to model the deformation of the porous FGM plate. The advantages of this model are (i) higher order kinematic terms for membrane and bending deformations due to the coupled membrane-bending behavior of FGMs in addition to the thickness-stretching effects, and (ii) the flexibility of applying external loads at different points along the thickness direction of the FGM plate. An analytical solution technique, popularly known as Navier’s approach, is adopted for the spatial solutions, and Newmark’s average acceleration method is utilized for the temporal solutions. The influence of the geometrical and porosity parameters on the structure’s vibration response is studied. Natural frequencies are affected significantly with uniform porosity distribution, while graded porosities can tweak vibration response in a small range but with better control. In the absence of the elasticity solutions, the present results may be considered the possible benchmark solutions for comparison of the other two-dimensional (2D) kinematic models.

Magneto-hygro-thermal vibration analysis of the viscoelastic nanobeams reinforcedwith carbon nanotubes resting on Kerr’s elastic foundation based on NSGT

This paper studies the vibration analysis of the viscoelastic nanobeams reinforced with functionally graded carbon nanotubes (FG-CNTs), considering the thickness stretching effect subjected to magneto-hygro-thermo loading based on a novel higher-order shear deformation theory. It is assumed that the nanosize structure is resting on Kerr’s elastic foundation. The effective material properties of FG-CNTs are changed through the thickness direction based on five diverse distributions of CNTs. All of the effective material properties of the nanobeam are temperature-dependent. Nonlocal strain gradient theory is applied to consider the size effect. A higher-order shear deformation beam theory is utilized to consider the shear impacts. Three kinds of relations are employed to study the effects of the hygro-thermo loading comprehensively on the frequency of the nanobeams. An analytical solution technique is utilized to solve the nonlocal governing equations achieved from Hamilton’s principle for several boundary conditions. The influences of size effect parameters, elastic foundation factors, distributions of CNTs, hygrothermal environment, and viscoelastic parameter on the frequency of nanobeams are investigated.

Nonlinear suppression using time-delayed controller to excited Van der Pol–Duffing oscillator: analytical solution techniques

To suppress the nonlinearity of an excited Van der Pol–Duffing oscillator (VdPD), time-delayed position and velocity are used throughout this study. The time delay is supplemental to prevent the nonlinear vibration of the considered system. The topic of this work is extremely current because technologies with a time delay have been the subject of several studies in the latest days. The classical homotopy perturbation method (HPM) is utilized to extract an approximate systematic explanation for the system at hand. Furthermore, a modification of the HPM reveals a more accurate approximate solution. This accuracy is tested through a comparison with the numerical solution. The practical approximate analytical methodology makes the work possible to qualitatively evaluate the results. The time histories of the obtained solutions are drawn for various values of the natural frequency and the time delay parameters. Discussion of the results is presented in light of the plotted curves. On the other hand, the multiple scale procedure examines the organized nonlinear prototypical approach. The influence of the diverse regulatory restrictions on the organization’s vibration performances is explored. Two important cases of resonance, the sub-harmonic and super-harmonic, are examined according to the cubic nonlinearity. The modulation equations achieved for these cases are examined graphically according to the impact of the used parameters.

An Improved Two-states Cyclical Dynamic Model for Plastic Waste Management

The panacea to the global challenge of plastic waste management is the transition towards plastic circular economy, which can be sustained through tailor-made management strategies. However, cutting-edge strategic solutions are constrained by inadequate data due to inadequate plastic-based predictive models. This paper presents an improved version of an existing two-state cyclical dynamic closed (CDC) model. The CDC model was formulated using a homogeneous linear system of ordinary differential equations (ODEs) and was modified by introducing a separation target which plays an essential role in determining both quantity and quality of recycled plastics. The Laplace transforms technique was the main analytic solution technique used. Values of the parameters were computed using the global plastic data applied for the existing CDC model, and with a technique termed the nth-order product derivative proximity, alternating pairs of initial values were selected each for the global annual plastic production and the global annual plastic waste generation. The validation process of the new CDC model was accomplished using the root mean squared error (RMSE) and the mean average percentage error (MAPE), which are measures of the model’s predictive power. Comparatively, RMSEs of the new CDC model were smaller than the RMSEs of the existing CDC model. MAPEs for the new CDC model were 6.5% and 7% (as against 13% and 18% in the existing model) respectively for the global annual plastic: production and waste generation, indicating that the new model predicts with 93.5% and 93% degrees of accuracy respectively for the global annual plastic: production and waste generation. Therefore, the new CDC model has outperformed the existing CDC model in terms of predictive power, and thus, establishing the new CDC model as an improved version of the existing one. The model can therefore make impactful policy decisions for sustainable plastic waste management thereby aiding to achieve the transition towards circular economy in plastic waste management.

Variable storage behavior controlled by rainfall intensity and profile structure upon saturation excess overland flow generation

The effects of rainfall intensity and terrain structure on hillslope storage behavior upon saturation excess overland flow (SOF) generation are not yet fully understood. Analytical solution techniques that provide direct insights into the hillslope storage behavior were adopted in this study. The Threshold of StOrage (TSO) at critical rainfall intensity N0 (the minimum rainfall intensity to generate SOF) and the storage upon SOF generation (SO) at rainfall intensity N (>N0) were derived using analytical solutions of the kinematic wave equation over unit-width shallow sloping aquifers. The derived analytical solutions match the results of the numerical models of the Boussinesq equation for 25 idealized hillslopes under various rainfall intensities. It was found that as N = N0, TSO is determined by hillslope structures, i.e., TSO on the convex hillslopes is always greater than that on concave hillslopes with the same gradient, and TSO on convex and concave hillslopes gradually approaches as the gradient increases. As N > N0, hillslope storage dynamics is characterised by variable storage behavior as rainfall intensity increasing, i.e., SO increases with N. An inflection point of rainfall intensity N1, below which the curvature of the bedrock profile has a limited effect on SO, was found. Although N1 was greater on steeper hillslope, the corresponding values of SO1 on hillslopes under different gradients are close, which means that the effects of rainfall intensity and profile structure on SO is balanced at N = N1. This study helps to deepen our understanding of the quantitative role of rainfall intensity and profile structure in the process of runoff generation, and has the potential to improve the free storage capacity parameter in conceptual models.

Analytical Investigation of Some Dynamical Systems by ZZ Transform with Mittag–Leffler Kernel

In this work, ZZ transformation is combined with the Adomian decomposition method to solve the dynamical system of fractional order. The derivative of fractional order is represented in the Atangana–Baleanu derivative. The numerical examples are combined for their approximate‐analytical solution. It is explored using graphs that indicate that the actual and approximation results are close to each other, demonstrating the method’s usefulness. Fractional‐order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series form solution that is quickly convergent to exact solutions. The acquired results suggest that the novel analytical solution technique is simple to use and very successful at assessing complicated problems that arise in related fields of research and technology.

Optical Modeling and Characterization of Demyelinated Nerve Using Graphene-Based Photonic Structure

The myelin sheath, as an insulation layer in nerve cells, plays an essential role in neural communication and signal conduction. Loss of myelin, referred to as demyelination, is associated with many mental disorders. Detecting the tiny demyelinated parts in nerve fibers can provide early diagnosis of some mental disorders and create effective treatment plans. This paper establishes a new engineering approach for differentiation between demyelinated and myelinated axons by analyzing spectral responses resulting from the optical simulation framework. We propose computational modeling on the photonic communication of nerve fibers and develop a graphene-based neurophotonic device that can be used to detect the regions demyelinated on the nerve fiber. We first model a nanoscale thin-film configuration of the multilayered myelinated axon to evaluate photons transmission in the nerve fibers under geometric defects as demyelination. Then, the nerve’s optical characteristics are achieved by focusing on the reflectance of light incidence on the nerve model with the change of the demyelination size to distinguish demyelinated—from myelinated nerves by the spectral contrast. Undertaking the different levels of demyelination progression, we theoretically explore the variations of effective refractive index using an analytical solution technique. Ultimately, we design a nanostructure configured with silicon dioxide, graphene, and gold nanoparticles to function as a biochip recognizing myelinated axon damage under the surface plasmon effect. This device can promote a practical procedure to distinguish nanoscale demyelinated and myelinated axons, which can be utilized for neural sensing of tiny brain tissues as a neurophotonic needle.

Stress Impact of the Annealing Procedure of Cu-Filled TSV Packaging on the Performance of Nano-Scaled MOSFETs Evaluated by an Analytical Solution and FEA-Based Submodeling Technique.

Stress-induced performance change in electron packaging architecture is a major concern when the keep-out zone (KOZ) and corresponding integration density of interconnect systems and transistor devices are considered. In this study, a finite element analysis (FEA)-based submodeling approach is demonstrated to analyze the stress-affected zone of through-silicon via (TSV) and its influences on a planar metal oxide semiconductor field transistor (MOSFET) device. The feasibility of the widely adopted analytical solution for TSV stress-affected zone estimation, Lamé radial stress solution, is investigated and compared with the FEA-based submodeling approach. Analytic results reveal that the Lamé stress solution overestimates the TSV-induced stress in the concerned device by over 50%, and the difference in the estimated results of device performance between Lamé stress solution and FEA simulation can reach 22%. Moreover, a silicon–germanium-based lattice mismatch stressor is designed in a silicon p-type MOSFET, and its effects are analyzed and compared with those of TSV residual stress. The S/D stressor dominates the stress status of the device channel. The demonstrated FEA-based submodeling approach is effective in analyzing the stress impact from packaging and device-level components and estimating the KOZ issue in advanced electronic packaging.

A Unified Simulation Framework for Fluid–Structure–Control Interaction Problems with Rigid and Flexible Structures

Vortex-induced vibrations are often unwanted as they can lead to catastrophic failure of the associated structures, warranting countermeasures to mitigate or suppress these vibrations. Due to the nature of nonlinearities in fluid–structure interaction (FSI) problems, analytical solution techniques for FSI problems have limited applicability at best. On the other hand, the numerical solution of vortex-induced vibrations is quite challenging, especially for problems undergoing large structural deformations, let alone the simulation of effective control techniques for suppressing unwanted vibrations. In this work, a robust and computationally efficient simulation framework is proposed to simulate fluid–structure–control interactions problems involving rigid or flexible structures. The effectiveness of the proposed framework in designing effective controller mechanisms for controlling the onset as well as the amplitude of vortex-induced vibrations is illustrated by studying the examples of lock-in of a circular cylinder, galloping of a square body and vortex-induced vibrations of a flexible, lightweight beam under different parameter settings for the controllers.

Multi-step fractional differential transform method for the solution of fractional order stiff systems☆

In this study, the multi-step fractional differential transform method (MSFDTM) is employed to obtain approximate analytical solutions of stiff systems of fractional order. The fractional derivative is described in the Caputo sense. Several numerical experiments are carried out to illustrate the feasibility of MSFDTM as an approximate analytical solution technique for stiff systems of fractional order.

Effects of fractional order time derivative on the solitary wave dynamics of the generalized ZK–Burgers equation

The generalized ZK-Burgers equation has nonlinear characteristics of the governing equation and helps to understand nonlinear waves in many fluid problems of plasma mediums. This work aims to study the effects of fractional order of derivative on the wave solutions of this interesting equation via exact and analytical approximate solution techniques. The analytical approximate wave solutions are retrieved using homotopy analysis method taking the fractional derivative in Caputo’s sense, while exact solutions are reaped with the help of first integral method taking the fractional derivative in Jumarie’s modified Riemann–Liouville sense. The obtained solutions will be helpful to understand the dynamical framework of many nonlinear physical phenomena.

A new multistage technique for approximate analytical solution of nonlinear differential equations

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h<R, the domain of the problem is split into subintervals. By stepping out in a multistage manner, corresponding subproblems are defined which are then subsequently solved with conventional Parker-Sochacki method to get a piecewise continuous solution with very high accuracy. The method is applied to SIR epidemic model, stiff differential equation modelling combustion, Lorenz chaotic problem, and the Troesch's boundary value problem. The results obtained showed a remarkable accuracy when compared with Runge-Kutta Method of order 4. The article showcased the proposed method as a simple, yet accurate approximate analytical technique for nonlinear differential equations.

Verification of the iterative procedure for solving the elastoplastic Kirsch problem on the Lame problem

An iterative procedure for solving the elastoplastic Lame problem was proposed to further develop an iterative solution of the elastoplastic Kirsch problem. A ring under an external pressure is considered at which a certain plastic zone arises in the ring. The proposed iterative procedure for solving the elastoplastic Lame problem consists in an explicit analytical representation for stresses in the plastic region and the iterative technique for an elastic analytical solution. Furthermore, the plastic zone radius is beforehand unknown. The convergence of the iterative procedure for the elastoplastic Lame problem is shown in comparison with well-known analytical solution to this problem in elastoplastic formulation. A numerical-analytical iterative solution of the Lame problem in elastoplastic formulation is given for various external pressure and internal radius of the ring. Shows the convergence of numerical-analytical iterative solutions the problem in comparison with numerical solutions in elastoplastic formulation.

Analysis of the electrical field in viscosity sensors with torsionally vibrating quartz cylinders

The potential and electric fields in viscosity sensors with torsionally vibrating quartz cylinders are calculated for various sensor geometries by means of analytical solution techniques and a numerical integration method for the Laplace equation. Sensor geometries with electrodes on the surface of the cylinder and offset at a distance from the cylinder are investigated for different angular extensions of the electrodes and dielectric constants of the fluid. It is found that the electric field is more favorable for the excitation of the torsional vibration when the electrodes are located at a distance from the cylinder surface rather than on the cylinder. When the electrodes are on the surface of the cylinder, the direction of the piezoelectrically active component of the electric field is reversed in certain regions near the edges of the cylinder, which hinders the excitation of the torsional vibration. As a result of this analysis, a novel viscosity sensor with line conductor electrodes was developed and validated in our laboratory. Furthermore, this article provides a brief review of the available literature on torsionally vibrating quartz cylinder viscometers.