Approximate Analytical Method Research Articles

A Modified Perturbation Method for Global Dynamic analysis of Generalized Mixed Rayleigh–Liénard Oscillator with Cubic and Quintic Nonlinearities

Abstract 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 Rayleigh–Liénard 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.

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.

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.

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.

Wavelet-based Bayesian approximate kernel method for high-dimensional data analysis

Wavelet-based Bayesian approximate kernel method for high-dimensional data analysis

Comparison of new computational methods for spatial modelling of malaria

BackgroundGeostatistical analysis of health data is increasingly used to model spatial variation in malaria prevalence, burden, and other metrics. Traditional inference methods for geostatistical modelling are notoriously computationally intensive, motivating the development of newer, approximate methods for geostatistical analysis or, more broadly, computational modelling of spatial processes. The appeal of faster methods is particularly great as the size of the region and number of spatial locations being modelled increases.MethodsThis work presents an applied comparison of four proposed ‘fast’ computational methods for spatial modelling and the software provided to implement them—Integrated Nested Laplace Approximation (INLA), tree boosting with Gaussian processes and mixed effect models (GPBoost), Fixed Rank Kriging (FRK) and Spatial Random Forests (SpRF). The four methods are illustrated by estimating malaria prevalence on two different spatial scales—country and continent. The performance of the four methods is compared on these data in terms of accuracy, computation time, and ease of implementation.ResultsTwo of these methods—SpRF and GPBoost—do not scale well as the data size increases, and so are likely to be infeasible for larger-scale analysis problems. The two remaining methods—INLA and FRK—do scale well computationally, however the resulting model fits are very sensitive to the user’s modelling assumptions and parameter choices. The binomial observation distribution commonly used for disease prevalence mapping with INLA fails to account for small-scale overdispersion present in the malaria prevalence data, which can lead to poor predictions. Selection of an appropriate alternative such as the Beta-binomial distribution is required to produce a reliable model fit. The small-scale random effect term in FRK overcomes this pitfall, but FRK model estimates are very reliant on providing a sufficient number and appropriate configuration of basis functions. Unfortunately the computation time for FRK increases rapidly with increasing basis resolution.ConclusionsINLA and FRK both enable scalable geostatistical modelling of malaria prevalence data. However care must be taken when using both methods to assess the fit of the model to data and plausibility of predictions, in order to select appropriate model assumptions and parameters.

Derivation of Key Characteristics of Columns in Frames with Sway

The paper paper examines columns of elastic frames that are able to displace laterally under axial load. Columns on the same level (story) of such frames may contribute to lateral support (stability) or require lateral support (supported columns). In the former case, maximum moments will occur at column ends, but may occur between column ends in the latter case. The column response, including the formation of moments and shears, is often complicated, often calling for approximate analysis methods and information that presently may not be readily available. Main attention of the paper is paid to providing results that will be helpful in providing, in a rather simple manner, improved understanding of column mechanics, and that may be helpful also as a supplement to full second-order analyses. Towards this goal, the major objective of the paper is threefold: (i) to identify characteristic points, or behavioral“landmarks” in the axial load-moment solution space, through the study of rotationally restrained elastic columns and two-column panels with sidesway; (ii) to derive simple, novel, closed-form expressions for these, thereby providing a tool simplifying the establishment of the variation of end and maximum moments versus axial load; (iii) to identify to what extent isolated (single) column analyses may adequately represent horizontally interacting columns, and potential unwinding, in frames.

Analysis of functionally graded rotating disks via analytical approximation methods

Analytical solutions for functionally graded (FG) rotating disks are limited due to the functional representation of the cross-sectional geometry, the density, and the modulus of elasticity which lead to a governing differental equation with variable coefficients. Analytical solutions for such disks are limited according to the functional form chosen for the variation of the property. Hence, analytical approximation approaches gain importance for such problems. In this study, two analytical approximation approaches are proposed for the analysis FG rotating disks. The methods used are the Improved Adomian decomposition method (IADM) and the optimal homotopy asymptotic method (OHAM). The resulting solution is an analytical solution in power series form. It is assumed that the disks considered are subjected to constant angular velocity. Material properties such as modulus of elasticity, density and yield strength are assumed to vary along the radial direction according to the power law function. Stresses are obtained for different values of the gradient parameters to investigate the effects of the parameters. Von Mises yield criterion is used to determine elastic limit angular velocities. The results from this study are in very good agreement with previous results available in the literature. In addition, it is observed that OHAM converges to results much faster than IADM.

Method for determining eigen numbers in heat conduction problems for a cylinder

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

Analysis of the Axial Vibration of Non-Uniform and Functionally Graded Rods via an Analytical-Based Numerical Approach

In this study, an analytical-based numerical approach was proposed for the analysis of the free axial vibration of homogeneous and functionally graded rods with varying cross-sectional areas. The proposed approach is based on analytical approximation techniques, such as the Adomian decomposition method, variational iteration method, and homotopy perturbation method. However, the governing equations of the problems solved in this study were variable coefficient differential equations. These equations provide analytical solutions for strictly limited cases. Analytical approximation methods easily handle problems with uniform material properties and constant cross-sections, whereas with varying cross-sectional areas, the analytical integration process becomes a difficult task for the software. If the rod’s material is functionally graded with varying cross-sectional areas, the analytical integration process becomes a cumbersome task. The proposed approach eliminates all difficulties and requires computation within several seconds. The application of this method is straightforward, and the results obtained in this study are in excellent agreement with the solutions provided in the literature.

Towards Electricity from the Combustion of Agricultural Waste in Boilers with Low CO2 Emissions

To meet the growing energy needs of the world’s population, international governments have turned to biomass energy from agricultural waste as a form of renewable energy. To this end, this study is based on the experimental and practical determination of certain physicochemical properties of three agricultural residues (cocoa cortex, oil palm fibres and sugar cane bagasse) using approximate and structural analysis methods. Then, using a METERK electronic detector, the small quantities of CO2 emitted during combustion in 0.5 g boilers of each of these waste products used to produce energy (heat and/or electricity) were determined. The gross and net calorific values of these biomasses were also determined. Here are some of the results obtained for these three agricultural residues: variable moisture content (from 3.6% to 43.12%), variable organic matter content (from 76.12% to 90.31%), cellulose content (from 32.74% to 55.91%), lignin content (from 8.26% to 27.05%), variable net calorific values (from 6.581 kWh/kg to 10.461 kWh/kg) and low CO2 gas emissions (from 439 ppm to 447 ppm). This proves that these three agricultural residues are very good potential sources of energy that can be used as fuel in boilers to produce large quantities of heat that can be converted into electricity by turbines connected to alternators using electromagnetic induction phenomena. Finally, this work provides credible information for public and private decision-makers in the field of electricity production from the combustion of these agricultural residues in boilers.

Efficient Approximate Analytical Methods to Solve Some Partial Differential Equations

The goal of this research is to solve several one-dimensional partial differential equations in linear and nonlinear forms using a powerful approximate analytical approach. Many of these equations are difficult to find the exact solutions due to their governing equations. Therefore, examining and analyzing efficient approximate analytical approaches to treat these problems are required. In this work, the homotopy analysis method (HAM) is proposed. We use convergence control parameters to optimize the approximate solution. This method relay on choosing with complete freedom an auxiliary function linear operator and initial guess to generate the series solution. Moreover, the method gives a convenient way to guarantee the convergence of series solutions via the control parameter curve graphical method to rate the convergence and obtain the best solution. Decoding and analyzing potential Korteweg-de-Vries, Benjamin, and Airy equations, followed by convergence analysis to demonstrate the applicability of the method. By using the programs Mapel and Mathematica, the obtained results are very effective, accurate, and convergent to the exact solution after a few iterations, as shown in the tables and figures of this work.

A predictive method for pipe in pipe, cylindrical modules and spherical packed bed latent thermal energy storage systems

Due to the transient operation of latent thermal energy storage (LTES) heat exchangers, methods from classical heat exchanger theory such as the effectiveness number of transfer unit method are not applicable. As a result, the study of these systems lacks a general theoretical framework. Recently, an approximate analytical solution method was developed which integrates phase fraction under a constant boundary condition to an LTES heat exchanger with a varying heat transfer fluid temperature. In this paper, the method is applied to three geometries: pipe in pipe, cylindrical modules and spherical packed bed geometries. The resulting model predicts the phase change fraction and effectiveness as a function of time, based on one-dimensional solutions for the movement of the phase change front. The predictions of the analytical model are compared to results of a simplified numerical method of an LTES heat exchanger. The average deviation of the predicted phase change fraction is 3.3 10−2 %, 4.6 10−1 % and 3.7 10−1 % for the pipe in pipe, cylindrical and spherical module respectively. The difference in model performance stems from the description of the fundamental heat transfer behavior. In a second validation step, the outlet temperature predicted by the model is compared to an experimental dataset for a LTES shell and tube heat exchanger. The predictions of the model can be improved further by including sensible heat, heat losses and allowing time-varying inlet conditions.

A Mathematical Study on a Steady MHD Flow in Double Stratification Medium

The detailed analysis on the flow of MHD fluid in double stratification medium across a stretching sheet with exponential permeability is examined in this research. The approximate analytical solution for the governing equations in steady state is found by using a new approximate analytical method called Ananthaswamy-Sivasankari Method (ASM) and Modified Homotopy Analysis Method (MHAM). The approximate analytical expressions for the dimensionless velocity, dimensionless temperature and dimensionless concentration are derived using these methods. The analytical and numerical results (previous work) are compared and there is a good agreement between our analytical results and numerical works. The impacts of several parameters including porosity, magnetic, suction and heat source parameters are shown in graphical representation. The error table for the physical parameter namely Nusselt number for various values of Prandtl number has been provided. Both ASM and MHAM are very useful to solve some other non-linear boundary value problems especially in MHD fluid flow

Extended melt‐conveying models for single‐screw extruders: Integrating domain knowledge into symbolic regression

AbstractThe literature provides several analytical approximation methods for predicting the flow of non‐Newtonian fluids in single‐screw extruders. While these are based on various flow conditions, they were developed mostly for extruder screws with standard geometries. We present novel analytical melt‐conveying models for predicting the flow and dissipation rates of fully developed flows of power‐law fluids within three‐dimensional screw channels. To accommodate a broad range of industrial screw designs, including both standard and high‐performance screws, the main intention of this work was to significantly extend the scope of existing theories. The flow equations were first rewritten in a dimensionless form to reduce the mathematical problem to its dimensionless influencing parameters. These were varied within wide ranges to create a set of physically independent modeling setups, the flow and dissipation rates of which were evaluated by means of a finite‐volume solver. The numerical results were then approximated analytically using symbolic regression based on genetic programming. To support the regression analysis in finding accurate solutions, we integrated domain‐specific process knowledge in the preprocessing of the dataset. We obtained three regression models for predicting the flow and dissipation rates in melt‐conveying zones and tested their accuracy successfully against an independent set of numerical solutions.Highlights Flow of power‐law fluids in three‐dimensional screw channels Identification of independent influencing parameters by dimensional analysis Numerical parametric design study for a broad range of industrial applications Integration of domain knowledge in symbolic regression Surrogate models derived from numerical simulation results

Stationary response analysis for a linear system under non-Gaussian random excitation by the equivalent non-Gaussian excitation method and the Hermite moment model

An approximate analytical method is developed to obtain the stationary response probability density and mean upcrossing rate for a linear system subjected to non-Gaussian random excitation. The random excitation is a stationary stochastic process prescribed by an arbitrary first-order probability density and a class of power spectral density with a bandwidth parameter, and is described by an Itô stochastic differential equation. In the proposed method, the equivalent non-Gaussian excitation method is first utilized to derive a closed set of moment equations for the system response. The exact solutions of response moments up to the fourth order are calculated from the moment equations. Next, using the response moments, the Hermite moment model is built for the stationary displacement response, and its probability density function is obtained through the model. Finally, the mean upcrossing rate of the displacement is determined from the response probability distribution by making two approximations that the displacement and velocity of the system are independent and the velocity is Gaussian. The proposed method can be used for any excitation probability density and bandwidth parameter value as long as the skewness and kurtosis of the response are within the applicable range of the Hermite moment model. In illustrative examples, the proposed method is applied to linear systems under random excitations with two-types of non-Gaussian probability densities. The effectiveness of the method is demonstrated by comparing the analytical results with the pertinent Monte Carlo simulation results while widely varying the bandwidth ratio between the excitation spectral density and the frequency response function of the system. Moreover, the effect of excitation non-Gaussianity on the mean upcrossing rate is discussed. It is shown that for small bandwidth ratio, the crossing rate differs considerably depending on the excitation probability density. The difference due to the excitation distribution decreases monotonically as the bandwidth ratio increases, and the crossing rate becomes almost the same as that in the case of Gaussian excitation when the bandwidth ratio is large.