Approximate Analytical Method Research Articles

Symmetries, Reductions and Exact Solutions of Nonstationary Monge–Ampère Type Equations

A family of strongly nonlinear nonstationary equations of mathematical physics with three independent variables is investigated, which contain an arbitrary degree of the first derivative with respect to time and a quadratic combination of second derivatives with respect to spatial variables of the Monge–Ampère type. Individual PDEs of this family are encountered, for example, in electron magnetohydrodynamics and differential geometry. The symmetries of the considered parabolic Monge–Ampère equations are investigated by group analysis methods. Formulas are obtained that make it possible to construct multiparameter families of solutions based on simpler solutions. Two-dimensional and one-dimensional symmetry and non-symmetry reductions are considered, which lead to the original equation to simpler partial differential equations with two independent variables or ordinary differential equations or systems of such equations. Self-similar and other invariant solutions are described. A number of new exact solutions are constructed by methods of generalized and functional separation of variables, many of which are expressed in elementary functions or in quadratures. To obtain exact solutions, the principle of the structural analogy of solutions was also used, as well as various combinations of all the above-mentioned methods. In addition, some solutions are constructed by auxiliary intermediate-point or contact transformations. The obtained exact solutions can be used as test problems intended to check the adequacy and assess the accuracy of numerical and approximate analytical methods for solving problems described by highly nonlinear equations of mathematical physics.

Random P-bifurcation in a Duffing-van der Pol vibro-impact system with a Bingham model.

The present work investigates stochastic P-bifurcation phenomena in a Duffing-van der Pol vibro-impact oscillator containing a Bingham model under Gaussian white noise excitation. By employing non-smooth transformations and stochastic averaging techniques, an approximate analytical method is proposed to analyze the stochastic response and bifurcation behavior of nonlinear systems with friction and vibro-impact effects. Using a non-smooth transformation, the stochastically excited vibro-impact oscillator is converted into an approximately equivalent system without velocity discontinuities. Subsequently, the friction term is handled, and stochastic averaging is applied to derive the averaged stochastic Itô equation. The corresponding Fokker-Planck-Kolmogorov equation is then solved to obtain the probability density function of the system's steady-state response. Numerical simulations are conducted to verify the reliability of the proposed method. Based on these results, the critical parameter conditions for stochastic P-bifurcation are derived using singularity theory, considering both the amplitude probability density and the joint probability density of system displacement and velocity. Bifurcation diagrams, extreme value plots, amplitude probability density plots, velocity probability density plots, and joint probability density plots of system displacement and velocity are constructed for different parameter spaces. The findings demonstrate that changes in the viscous damping coefficient of the magnetorheological damper, Coulomb damping force, noise intensity, vibro-impact coefficient, and nonlinear damping coefficient can all induce stochastic P-bifurcations.

Piecewise Analytical Approximation Methods for Initial-Value Problems of Nonlinear Ordinary Differential Equations

Piecewise analytical solutions to scalar, nonlinear, first-order, ordinary differential equations based on the second-order Taylor series expansion of their right-hand sides that result in Riccati’s equations are presented. Closed-form solutions are obtained if the dependence of the right-hand side on the independent variable is not considered; otherwise, the solution is given by convergent series. Discrete solutions also based on the second-order Taylor series expansion of the right-hand side and the discretization of the independent variable that result in algebraic quadratic equations are also reported. Both the piecewise analytical and discrete methods are applied to two singularly perturbed initial-value problems and the results are compared with the exact solution and those of linearization procedures, and implicit and explicit Taylor’s methods. It is shown that the accuracy of piecewise analytical techniques depends on the number of terms kept in the series expansion of the solution, whereas that of the discrete methods depends on the location where the coefficients are evaluated. For Riccati equations with constant coefficients, the piecewise analytical method presented here provides the exact solution; it also provides the exact solution for linear, first-order ordinary differential equations with constant coefficients.

Selecting analytical method of leakage inductance calculation for optimizing high-voltage test transformer design

AN ACTUAL PROBLEM in transformer design is to optimize its design, since it allows you to create a competitive transformer. One of the objective functions in optimization algorithms for high-voltage test transformers is the ratio of the leakage inductive reactance of the transformer, reduced to the secondary side, to the load capacitance. Since the objective functions in the optimization procedure must be found many times, it is advisable to use approximate analytical calculation methods to calculate the leakage inductance. THE PURPOSE of the article is to analyze the error of the traditional analytical method for calculating the leakage inductance of power transformers with a rectangular axial cross-sections of the windings, and to develop a refined analytical method that takes into account the main design feature of high-voltage test transformers, which consists in the trapezoidal shape of the secondary winding cross-section. The refined method takes into account the change in the number of turns in the radial direction while maintaining the basic assumptions of the traditional method. METHODS. The error is studied by comparing the calculation results using these methods with the results of a numerical calculation of leakage inductance based on a 3D magnetostatic field, which are accepted as accurate. RESULTS. It is shown that the calculation error using the method proposed in the article is significantly reduced compared to the error of the method developed for power transformers with rectangular winding cross-sections. The ranges of changes in the relative geometric parameters of the base transformer have been determined, in which the error of this calculation method does not exceed 10%. The TGI 50/100 transformer, whose rated power is 5 kVA, was chosen as the base transformer. The developed analytical method is recommended for high-voltage test transformers optimization algorithms.

Modern Nonlinear Optimization Algorithms by Using Approximate Methods for Combinatorial Numerical Analysis

Large-scale optimization has become an important research theme in the evolutionary computing field. To take advantage of this progress, sophisticated algorithms have been merged in commercially inexpensive software, like python, to increase the aptitude of large data handling. In this paper, we propose an advanced algorithm, to compare Augmented and Penalty methods for resolving large-scale constrained problems of optimization. We highlight the advantages of an Augmented method in terms of faster convergence, better numerical stability and more robust performance while noting the behavior of Penalty Methods like parameter sensitivity issues, degraded performance near optimal and numerical instabilities. Using numerical experiments, our algorithm is quickly convergent and is more reliable.

Modeling and Dynamic Analysis of Fish Robot with Soft Fluidic Actuation

A biomimetic robotic fish tailored for environmental monitoring and hydrographic exploration is examined in this study. Traditional fish robots, driven by tail oscillations, often grapple with constrained maneuverability in a single plane. This article focuses on mitigating these limitations through an exploration of the dynamic behavior of a bioinspired soft robotic fish. Unlike prior designs that employed similar actuators for the robot's tail, which either lacked the ability for out-of-plane motion or relied on other propulsion systems, the single propulsion design proposed in our model enables movement along three-dimensional trajectories, leading to improved efficiency and maneuverability due to tail oscillation dynamics. The proposed design integrates strategically positioned nozzles for out-of-plane movements, alongside parallel fluid channels on the tail's neutral plate. Actuation is achieved by manipulating the internal fluid pressure within these channels. To precisely model tail deflection, we introduce a novel method utilizing Euler–Bernoulli beam theory considering nonlinear characteristics arising from internal fluid stress. For instance, following the proposed approximate analytical method, we optimize the fluidic actuator, considering that the soft tail deformation increases by 65% as the channel shape transitions from a semicircular to a square cross-section. The comprehensive comparison with analytical nonlinear method, finite element method, and experimental-driven analytical method extends our approach as an effective tool in terms of accuracy and computation time, demonstrating the effectiveness of validation processes. This study unveils a simplified and robust fish robot design, outperforming traditional mechanisms in efficacy. Despite its simplicity, the proposed design delivers comparable performance, presenting an effective alternative for achieving requisite functionality in fish robots.

ON THE ABILITY OF DIFFERENTIAL TRANSFORM METHOD AND ITS IMPROVEMENTS TO SOLVE PROPERLY THE SHIP’S ROLL EQUATION

A major concern regarding the ship safety is related to severe rolling oscillations which, in certain circumstances, can even lead to its capsizing. Assuming the rolling motion decoupled from its other five possible motions, one results a mathematical model associated to a second order differential equation where the main parameters (inertia, damping, stiffness and excitation) reveal a significant nonlinearity. In the absence of exact analytical solutions, the amplitude and period characteristics of the ship rolling can be evaluated by approximate numerical or analytical methods. In this work, we checked if the performant differential transform method (DTM) and its improvement with Pade approximants is able to provide approximate analytical solutions for nonlinear roll equation, valid for both the transitory and stationary states. The obtained results were verified against those produced by MatLab numerical generated simulations. We noticed that for the linearized roll equation, which describes quite correctly a significant part of the situations of practical interest, the DTM doubled by the Pade approximation [4/4] offers the exact solution. If the nonlinear terms from damping and restoring moments are included in the study but the sea is considered waveless, the investigated technique proves a good accuracy in describing the attenuation over time of the initial excitation. DTM and Pade [4/4] cease to offer reasonable solutions as soon as the exciting moment of the waves is included in the procedure. We gave clear explanations for this impasse and showed that the use of higher-order Pade approximations can solve (even if with additional efforts) totally or at least partially this problem.

ANALYTIC METHOD FOR SOLVING ONE CLASS OF NONLINEAR EQUATIONS

The paper proposes an analytical approximate method for calculating multidimensional integrals of analytic integrand functions, using the approximation of the latter by a power series. This approach lets transforming the original system of nonlinear equations with integral components into a system of equations with a polynomial left-hand side. To solve equations of this class we developed an analytical method. A sequential recurrent procedure has been developed for the analytical solution of this class of nonlinear equations.

Closed-form solutions of the nonlinear Schrödinger equation with arbitrary dispersion and potential

For the first time, the general nonlinear Schrödinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class of related nonlinear partial differential equations that are often used in various areas of theoretical physics, including nonlinear optics, superconductivity and plasma physics. To construct exact solutions, a combination of the method of functional constraints and methods of generalized separation of variables is used. New exact closed-form solutions of the general nonlinear Schrödinger equation, which are expressed in quadratures or elementary functions, are found. One-dimensional non-symmetry reductions are described, which lead the considered nonlinear partial differential equation to a simpler ordinary differential equation or a system of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for integrating nonlinear equations of mathematical physics.

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.

Analytical study of MHD stagnation point flow with the impact of thermal radiation and viscous dissipation over stretching surface

This study examines the analytical study of magnetic hydrodynamic stagnation point flow with the impact variable viscosity on a movable surface along with the impact of thermal radiation. The problem is modeled with the help of momentum and energy conservation laws in the form of NLPDEs. The novelty of this study is the combined impact of variable viscosity and thermal radiation with the analytical method. Aluminum oxide nanoparticles and water are used as base fluids in this research work. The authors applied appropriate transformations to convert a collection of dimension forms of NLPDEs to dimensionless forms of NODEs. The transformed NODEs are solved with the help of an approximate analytical method known as the HAM. The effects of different parameters, including electric field, magnetic field, stagnation point flow, thermal radiation PN, and EN on energy and momentum profiles intended, and the results are planned with the help of graphs.