Approximate Analytical Method Research Articles

Analytical approximate solutions of some fractional nonlinear evolution equations through AFVI method

In this article, we construct and investigate a new fractional variational iteration technique which is named as AFVI method. After that, we formulate some IVPs corresponding to the fractional nonlinear evolution equations NLTFFWE, mNLTFFWE, TFmCHE and TFmDPE. The Caputo fractional order derivative has been used to fractionalize the considered NLEEs. Then, we solve the considered IVPs by applying the formulated AFVI method. Lastly, we check the accuracy of the attained AASs by making some graphical and numerical comparisons with their corresponding exact solutions and other existing equivalent AASs. The result of this article confirm that the efficiency, appropriateness and time spent capability of the newly constructed AFVI method are better than that of the other existing analogous fractional analytical approximation methods. Here, we apply the Maple 2021 programming software to obtain the AASs of the formulated IVPs and drawing the 3D graphs of the attained AASs. Finally, in this article we validate the applicability of the Caputo fractional order derivative to form a novel analytical approximation method as well as to fractionalize some essential NLEEs of Mathematical physics.

Experimental and analytical investigations on single and repetitive impact failure responses of composite sandwich panels with orthogonal woven GFRP facesheets and PVC foam cores

Due to the unique design requirements for high load-bearing capacity of ship structures, composite materials used in the marine industry are gradually being developed with the characteristics of multi-layer sandwich structures with large thickness and cross-section, which are different from thin-walled composite materials widely applied in the aeronautical industry. However, sandwich composite materials used in marine structures have high susceptibility to impact event during in-service applications. The impact induced damage may seriously affect their mechanical performance and structural safety. Therefore, a comprehensive investigation in this paper on the failure mechanisms of composite sandwich panels with orthogonal woven GFRP facesheets and a PVC foam core layer is carried out with a series of single and repetitive low-velocity impact tests. The variations of impact force, dent depth, structural stiffness, failure modes, energy absorption and so on of composite sandwich panels against impact energy levels and impact numbers were explored. The results demonstrate that the delamination damage threshold at the first impact, the sudden drop of peak force and the slow descent process of impact force are the three typical characteristics of impact responses that corresponded to delamination initiation and fibre breakage of the upper panel, and compression damage of the core layer, respectively. The accumulated impact-induced damage has a significant negative effect on load-bearing and energy-absorption capabilities of composite sandwich panels. Moreover, an approximate theoretical analytical method is presented to solve the impact resistance of composite sandwich panels. The analytical results are compared well with experimental results. This research provides a detailed understanding of the damage mechanisms of composite sandwich panels under impact loadings and a guidance for impact resistant design of ship protective structures.

Linear–quadratic GUP and thermodynamic dimensional reduction

In this paper we investigate the statistical mechanics within the Linear–Quadratic GUP (LQGUP, i.e, GUP with linear and quadratic terms in momentum) models in the semiclassical regime. Then, some thermodynamic properties of a system of 3-dimensional harmonic oscillators are investigated by calculating the deformed partition functions. According to the equipartition theorem, we show that the number of accessible microstates decreases sharply in the very high temperatures regime. When the thermal de Broglie wavelength is of the order of the Planck length, three degrees of freedom are frozen in this setup. In other words, it is observed that there is an effective reduction of the degrees of freedom from 6 to 3 for a system of 3D harmonic oscillators in this framework. The calculations are carried out using both approximate analytical and exact numerical methods. The results of the analytical method are also presented in the form of thermal wavelengths for better understanding. Finally, the case of a 2-dimensional harmonic is treated as another example to comprehend the results, leading to a reduction of the degrees of freedom from 4 to 2.

Optimal multi-segment trajectory of solar sail with analytical approximation

This paper focuses on the optimization of two-dimensional transfer trajectories for solar sails using analytical approximation and modified analytical approximation methods. Instead of assuming circular orbits [29-31,37], the proposed approach allows for the analysis of solar sails launched from general heliocentric elliptical orbits. By employing linear perturbation theory, analytical propelled trajectories with fixed pitch angles are derived, where the perturbation coefficient serves as the reference lightness coefficient. To obtain the optimal transfer orbit between general heliocentric elliptical orbits, the analytical segments corresponding to different pitch angles are spliced together. A nonlinear optimization algorithm is then utilized to optimize the multi-segment trajectory, aiming to achieve the optimal performance index while satisfying boundary and intermediate constraints. Remarkably, this method requires the optimization of only the pitch angle for each segment, resulting in a small number of optimization variables and significantly improved efficiency in transfer orbit optimization. Additionally, the analytical approximation approach offers substantial time savings compared to numerical integration, particularly when repeated calculations are involved. Furthermore, the modified approximate trajectory exhibits high accuracy, with the optimal analytical transfer trajectory closely approximating the optimal actual trajectory acquired through the shooting method. This paper extensively validates and analyzes the precision of the proposed analytical transfer orbit through numerical verification. Additionally, when contrasted with actual transfer orbit, the suggested analytical approach can significantly decrease the computation time for multi-object challenges by a minimum of 80%. This makes the proposed method highly suitable for rapid design of optimal transfer orbits in multi-target mission scenarios for solar sail-based probes.

Позиционирование автономного необитаемого подводного аппарата с одновременной обработкой текущих и сохраненных измерений дальностей от менее чем трех гидроакустических маяков

The paper presents a recursive algorithm for determining the coordinates of an autonomous underwater vehicle (AUV) using time-different ranges to the acoustic beacons with log and heading indicator aiding with unknown a priori coordinates of the underwater vehicle. Any number of beacons can be used simultaneously, but at least at one moment the measurements from minimum three beacons are available. At this moment the initial linearization point is obtained using an approximate analytical method. At all the previous moments the measurements are saved for further processing. The algorithm uses two filters parallely processing the saved and current measurements. The solutions of the forward-time filter are updated by the backward-time filter data using the dummy measurements. The results of modeling and test this real data are presented, confirming the effectiveness of the developed algorithm.

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

Viscous dissipation effects on time-dependent MHD Casson nanofluid over stretching surface: A hybrid nanofluid study

The study of MHD Couple stress Casson flow of a hybrid nanofluid is paramount due to its potential applications in advanced engineering systems, particularly in enhancing thermal management and efficiency in various industrial processes. In this report, the time-dependent MHD Couple stress Casson flow of a hybrid nanofluid, considering the impact of viscous dissipation over a stretching surface is presented. The main objective of this work is to enhance the transformation heat ratio to meet the requirements of the engineering and manufacturing industries. After conducting a continuity check, the issue is analyzed by applying the principles of momentum and energy conservation. This modeling process generates nonlinear partial differential equations (PDEs), which are subsequently converted into nonlinear ordinary differential equations (ODEs) using similarity transformation and thermophysical characteristics. To solve these ODEs, the Approximate Analytical Method HAM is employed, which is specifically designed for nonlinear problems. Velocity decreases with increasing values and higher nanoparticle volume fraction further slows the velocity. Increasing the couple-stress parameter reduces velocity; increasing the Casson parameter slows the velocity profile. Increasing magnetic parameter enhances temperature profile; profile escalates with Eckert number magnitude. These findings contribute significantly to understanding the structure, interactions, and dynamics of such liquids, aligning with the focus on exploring the fundamental properties of simple, molecular, ionic, and complex liquids.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

New Lie Symmetries and Exact Solutions of a Mathematical Model Describing Solute Transport in Poroelastic Materials

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

Efficient slope reliability and sensitivity analysis using quantile-based first-order second-moment method

This paper introduces a novel approach for parameter sensitivity evaluation and efficient slope reliability analysis based on quantile-based first-order second-moment method (QFOSM). The core principles of the QFOSM are elucidated geometrically from the perspective of expanding ellipsoids. Based on this geometric interpretation, the QFOSM is further extended to estimate sensitivity indices and assess the significance of various uncertain parameters involved in the slope system. The proposed method has the advantage of computational simplicity, akin to the conventional first-order second-moment method (FOSM), while providing estimation accuracy close to that of the first-order reliability method (FORM). Its performance is demonstrated with a numerical example and three slope examples. The results show that the proposed method can efficiently estimate the slope reliability and simultaneously evaluate the sensitivity of the uncertain parameters. The proposed method does not involve complex optimization or iteration required by the FORM. It can provide a valuable complement to the existing approximate reliability analysis methods, offering rapid sensitivity evaluation and slope reliability analysis.

Exploring Advanced Analysis Technique for Shallow Water Flow Models with Diverse Applications

The present article focuses on the analytical approach using fractional orders and its applicationin the dynamics of physical processes. Fractional order models align better with experimental data compared tonon-fractional ones. This study primarily focuses on employing the new approximate analytical method to solveshallow water models with fractional orders. Numerical examples within the Caputo fractional derivative showcasethe method’s application. Results for both integer and fractional orders are graphically depicted, demonstrating thefractional solutions’ closeness to actual data. Analysis of 3D and 2D fractional order graphs reveals convergencetoward integer order graphs as fractional derivatives approach non-fractional ones. This method shows promise fordirect application in solving targeted problems and can be easily adapted for other fractional nature problems.

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.

Analytical analysis of the generation of a rotating driving magnetic field on the outer surface of a magnetic pinch load with a helical return current post

A recently emerging approach adopts a directionally time-varying (rotating) magnetic field to drive a pinch load, aiming to mitigate the inherent magneto-Rayleigh-Taylor instability in dynamic Z pinches. A helical return current post (RCP) serves as a functional structural element capable of generating the requisite driving magnetic field for this purpose in the load region. This paper first calculates the current azimuthally induced on the outer surface of a magnetically pinched load within this type of RCP using a zero-dimensional lumped-parameter circuit model. The results show that the induced current deviates significantly from the presumed “perfect” induced current (100% amplitude) as reported in the literature [S. A. Sorokin, ); P. F. Schmit , ); G. A. Shipley , ); and P. C. Campbell , ], with an effective coefficient of current induction considerably less than 1. However, even when the load is fully compressed to the axis, the effective coefficient does not approach zero but rather converges to a finite value that solely depends on the aspect ratio of the RCP. This is quite favorable for the suppression of magneto-Rayleigh-Taylor instability in the Z pinch. As for the pointlike X pinch, the axial magnetic field does not tend to zero but a finite value, though the effective coefficient tends to zero, and this result may be used to suppress the instability in X pinch and improve the time stability and spatiotemporal unity of hot spots. In addition, the anode and cathode plates have the potential to enhance the current induced in the load. This paper then analyzes the axial distribution and time behavior of the induced current adopting an approximate analytical method and numerical integration and finds an approximate invariance that can be well characterized by δt, the product of the normalized skin depth and time. Similar values of δt indicate similar axial distribution characteristics. When δt is lower than, at, or higher than the critical region (∼0.1–0.3), the axial distribution appears dumbbell shaped, nearly flat, and arched, respectively. These distributively induced currents can be exploited to achieve quasispherical, near flat, and dumbbell-shaped implosions, respectively. Published by the American Physical Society 2024

On-Demand Centralized Resource Allocation for IoT Applications: AI-Enabled Benchmark

The development of emerging information technologies, such as the Internet of Things (IoT), edge computing, and blockchain, has triggered a significant increase in IoT application services and data volume. Ensuring satisfactory service quality for diverse IoT application services based on limited network resources has become an urgent issue. Generalized processor sharing (GPS), functioning as a central resource scheduling mechanism guiding differentiated services, stands as a key technology for implementing on-demand resource allocation. The performance prediction of GPS is a crucial step that aims to capture the actual allocated resources using various queue metrics. Some methods (mainly analytical methods) have attempted to establish upper and lower bounds or approximate solutions. Recently, artificial intelligence (AI) methods, such as deep learning, have been designed to assess performance under self-similar traffic. However, the proposed methods in the literature have been developed for specific traffic scenarios with predefined constraints, thus limiting their real-world applicability. Furthermore, the absence of a benchmark in the literature leads to an unfair performance prediction comparison. To address the drawbacks in the literature, an AI-enabled performance benchmark with comprehensive traffic-oriented experiments showcasing the performance of existing methods is presented. Specifically, three types of methods are employed: traditional approximate analytical methods, traditional machine learning-based methods, and deep learning-based methods. Following that, various traffic flows with different settings are collected, and intricate experimental analyses at both the feature and method levels under different traffic conditions are conducted. Finally, insights from the experimental analysis that may be beneficial for the future performance prediction of GPS are derived.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Approximate Analytical Solution of the Influences of Magnetic Field and Chemical Reaction on Unsteady Convective Heat and Mass Transfer of Air, Water, and Electrolyte Fluids Subject to Newtonian Heating in a Porous Medium

This paper addresses the unsteady hydrodynamic convective heat and mass transfer of three fluids namely air, water, and electrolyte solution past an impulsively started vertical surface with Newtonian heating in a porous medium under the influences of magnetic field and chemical reaction. Suitable dimensionless parameters are used to transform the flow equations and the approximate analytic method employed to solve the flow problem. The results are illustrated graphically for the velocity, temperature, and concentration profiles. Though, low Prandtl numbers produce high-thermal boundary layer thickness, however, as a novelty, the presence of the magnetic field delayed the convection motion hence, the thermal boundary layer thickness is greater for water with high Pr = 7.0 as compared to air with low Pr = 0.71 and electrolyte solution with low Pr = 1.0. Practically, water with a high-Prandtl number can effectively absorb and release heat. This makes water useful in applications such as geothermal heat pumps and solar thermal collectors, industrial processes such as chemical reactions, distillation, and drying, and in oceanography in predicting the movement and behavior of ocean currents, which in turn can impact weather patterns and climate. Another major observation from the study is that the rate of cooling associated with air, water, or electrolyte impacts differently on the product being cooled.

Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions.

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm-Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods.

Understanding fast adsorption in single-solute breakthrough curves

To design efficient adsorption separations and select high-performing materials, understanding the relationship between adsorption equilibria and the movement of solute waves along the fluid is critical. While approximate-analytical methods are useful to combat nonlinearities in the differential equations and efficiently investigate the myriad of materials, their utility is limited by the lack of a clear physical interpretation and mathematical relationship to well-established models. In this work, we show that the two popular methods can be understood with regular perturbation theory and boundary layer theory in small ɛ, where a dilute solute travels from fluid to solid much faster than along the column. We consider conditions associated with a popular experimental technique used to characterize solute wave movement, where the column is initially at steady state and the fluid concentrations exiting, or breaking through, are time-monitored in response to a step change in inlet concentration. Our transformations and asymptotic analyses illustrate the time scales associated with popular techniques. We generalize their applicability to equilibria possessing inflection points while considering both adsorption and desorption modes. We use numerical simulations to estimate convergence orders of asymptotic approximations of breakthrough concentrations. For the leading-order approximation of regular perturbation theory, rarefaction waves possess positive convergence orders, depending on the extent of nonlinearity, while shock waves do not. Boundary-layer theory affords a more accurate approximation of the breakthrough curve than a shock wave, without guaranteeing a positive convergence order. This lack of positive convergence order is likely due to the evolution of the boundary layer, as illustrated with numerical simulations. An improved understanding of fast adsorption sets the stage for more realistic selection of high-performing materials and design configurations from thousands of available candidates.

Modeling of the heterogeneous hypergolic ignition with particle packing model

The ignition process of a hypergolic hybrid rocket fuel presents a significant challenge due to its complex and interconnected nature. However, understanding its underlying physics is crucial for accurately estimating the delay time and ignition performance. In this study, we employed an approximate analytical method to directly address the transient conduction problem with nonlinear surface heat generation. By using the Arrhenius equation and considering dual heat flux directions in both the liquid and solid domains, we propose a model for determining the delay time of heterogeneous hypergolic ignition. Our solution is straightforward and computationally efficient. Notably, when subjected to identical initial conditions and property settings, our results exhibit similarity with the solutions derived from other models. Moreover, we advance our approach by developing a particle packing model through the utilization of a well-established algorithm. By simulating the particle arrangement within the pellet, the surface layer properties and the thermal time constant of certain fuel designs can be modeled. This simulation serves the purpose of incorporating the particle size effect of the fuel pellet into our ignition model. The results generated by our heterogeneous hypergolic ignition model, in conjunction with the particle packing model, demonstrated impressive agreement with the experimental data derived from droplet tests. Novelty and significance statementThis study utilized a novel solving method to address a transient conduction problem with non-linear heat generation on the surface during a heterogeneous hypergolic ignition process. The study successfully derived an analytical solution, providing a simpler method for calculating ignition delay times. A scenario involving different initial temperatures was also examined. Additionally, the research integrated particle packing model data and thermal time constants to assess the effect of particle size. The model results were in agreement with the experimental data, confirming its accuracy and offering a reliable method for estimating the heterogeneous hypergolic ignition delay time.

Approximate global mode method for flutter analysis of folding wings

The aircraft with folding wings benefits from the ability to transform its folding wings to adapt to various flight missions but suffers from the change of the wings’ aeroelastic characteristics as a result. To clarify the change, it is necessary to study the dynamical modeling of the folding wing and also its aeroelastic response. This paper proposes a novel approximate global mode method (AGMM) for dealing with complex boundary conditions and models a folding wing consisting of separate rectangular plates for nonlinear flutter analysis. A high-precision low-dimensional dynamical model of the folding wing is developed, which is first validated by comparing the modal results obtained from the AGMM model with those obtained from an equivalent finite element model and is also experimentally validated by a comparison of forced vibration responses. On this basis, an AGMM model of the folding wing in supersonic aerodynamic loads based on the piston theory is further developed. The critical flutter velocity at different folding angles and the aeroelastic response are comprehensively studied by simulations based on the model. The result shows that the first 7th-order modes are sufficient for obtaining a converged critical flutter velocity, which varies with geometric parameters and potentially increases by 714 %. In addition, the low-order torsional modes of the wing contribute more to the vibration than the other modes during flutter. This work provides a new way of developing advanced dynamical modeling methods of folding and combinational structures for their dynamics design, optimization, and system control.