Perturbation Solution Research Articles

Analytic approximation of periodic orbits with renormalization group

Abstract The renormalization group analysis has been proposed to eliminate secular terms in perturbation solutions of differential equations and thus expand the domain of their validity. Here we extend the method to treat periodic orbits or limit cycles. Interesting normal forms could be derived through a generalization of the concept 'resonance', which offers nontrivial analytic approximations. Compared with traditional techniques such as multi-scale methods, the current scheme proceeds in a very straightforward and simple way, delivering not only the period and the amplitude but also the transient path to limit cycles. The method is demonstrated with several examples including Duffing oscillator, Van der Pol equation and Lorenz equation. The obtained solutions match well with numerical results as well as those derived by traditional analytic methods.

Heat transfer analysis in hydromagnetic two‐phase Williamson fluid through tilted channel: Applications of gold and silver nanoparticles in solar thermal energy

AbstractThe heat transfer analysis in Poiseuille hydromagnetic multiphase flow of Williamson fluid suspended by gold and silver nanoparticles is discussed through the inclined channel under the impact of no‐slip boundary conditions. The present conducted research is novel as no one has explored the heat transfer analysis in the fluid‐particle suspension model of electrically conducting Williamson fluid mixed with gold and silver nanoparticles through steeply inclined channel. The highly nonlinear complex partial differential equations are transformed into simplified form by using the appropriate dimensionless transformation then obtain the analytical solution by adopting the regular perturbation method. The perturbation solution is also compared with the numerical solution. The velocity and temperature profiles are significantly affected by Hartmann number. Approximately 19% (gold suspension) and 20% (silver suspension) increment is observed in fluid phase and particle velocities respectively. The multiphase flow of non‐Newtonian (Williamson) fluid provides greater heat transfer to the system with the suspension of gold (10%) and silver (13%) nanoparticles as compared to multiphase Newtonian fluid with these suspensions. The heat transfer rate of multiphase flows is greater than single‐phase flows. The current study will help to understand the flow and heat transfer mechanism of two‐phase flows with the suspension of gold and silver nanoparticles through inclined parallel plates under the impact of the constant magnetic field. Further, the current study can help to design modern solar cell devices to store more solar energy due to the low heating release of considered nanoparticles. The present research is original and has not been conducted before.

Dual solution of thin film flow of fuzzified MHD pseudo-plastic fluid: numerical investigation in uncertain environment

The pseudoplastic fluids have wide range of applications in industrial areas including cyclone separation, bearings, paper fibre separation, heat exchangers and also in food industry. In this regard, the current manuscript investigates the impact of transverse magnetic field on thin pseudo-plastic film flow on a vertical wall in a fuzzy (uncertain) environment. The uncertainty in a model is characterized through triangular fuzzy numbers (TFNs) along with r -cut approach, which is computationally effective in capturing the uncertainties in physical phenomena. This results in the modelling of highly nonlinear fuzzified problem. For solution and analysis purposes, Runge–Kutta Fehlberg (RKF) is utilized. Also, RKF solutions are validated by comparing them to homotopy perturbation solutions in the current manuscript. The impact of r -cut, and fluid parameters including non-Newtonian parameter β, magnetic field M and Stoke's number S t on the upper and lower velocity profiles are captured and analysed numerically and graphically. Analysis reveals that velocity profile decreases with an increase in applied magnetic field at upper and lower bounds. Also, increase in S t and β increases the velocity profile at lower bound, while inverse behaviour is recorded in the case of upper bound. The results also indicate that as r goes from 0 to 1, the crisp solution always lies between upper and lower profiles, and becomes coherent at 1. Moreover, all fuzzy level set values of r ∈ [ 0 , 1 ] satisfy the fuzzy solution in the form of TFN.

An interesting class of exact solutions to Binet’s equation, with references to Newton’s and Einstein’s theories of gravity

Abstract To adequately describe the motion of a particle in a central force field, one has to specify the apsidal angle (α). The particle's trajectory can only be truly periodic (closed), if α/π is a rational number. For example, F∼1/r^2 then α=π, and the trajectory is an ellipse. This is the simplest case when the trajectory equation (Binet's equation) is exactly solvable. In this article, bounded motion in a two-component central potential field V(r)=-a_1/r-a_2/r^2 will be examined and illustrated. This case is more complex, but the corresponding Binet's equation is still exactly solvable. Moreover, the perturbational solution to the relativistic trajectory equation is, in fact, the exact solution for a force field of the same type. It means that the relativistic force F∼1/r^4 is formally replaced with an effective F∼1/r^3 force. As will be shown, this differently interpretable solution gives a very accurate prediction to the anomalous shift of Mercury's perihelion.

Dynamic analysis of the air–water interface instability in a wind-wave flume: Initial conditions for wave generation

This paper aims to investigate the dynamic mechanism for a previously calm water surface under a background wind field, which would lose stability due to minor perturbations. When spatial distribution and growth rate of the shear wind field satisfy a certain relationship, an Orr–Sommerfeld equation can be derived from linearized Navier–Stokes equations, for the amplitude of perturbation waves at the air–water interface. After transforming its coordinates to a finite interval, this fourth-order linear ordinary differential equation with variable coefficients can turn into a linear singular differential equation. It is solved under the background flow fields of air and water using the corrected Fourier method that we previously developed, to yield two sets of four analytical solutions with physical significance. The perturbation flow at the air–water interface must satisfy the kinematic and dynamic matching conditions (continuity of velocity and smooth transition of shear stress), which results in a fourth-order homogeneous linear algebraic equation for four undetermined constants. Setting various parameters in accordance with measured data with a wind-wave flume, the corresponding perturbation waves solution is identified. Our calculation densely distributes the most unstable and readily growing waves among perturbation waves with small phase speeds, consistent with the experimental results. Our perturbation solutions over a flat air–water interface are only applicable to the short period after instability onset, which provides an initial condition for wave generation. The consequent interaction between small-amplitude waves and wind field is beyond the scope of our linear theory, which has indeed been sufficiently addressed in the literature.

One trick to treat them all: SuperEasy linear response for any hot dark matter in N-body simulations

We generalise the SuperEasy linear response method, originally developed to describe massive neutrinos in cosmological N-body simulations, to any subdominant hot dark matter (HDM) species with arbitrary momentum distributions. The method uses analytical solutions of the HDM phase space perturbations in various limits and constructs from them a modification factor to the gravitational potential that tricks the cold particles into trajectories as if HDM particles were present in the simulation box. The modification factor is algebraic in the cosmological parameters and requires no fitting. Implementing the method in a Particle-Mesh simulation code and testing it on subdominant HDM cosmologies up to the equivalent effect of ∑ m ν = 0.315 eV-mass neutrinos, we find that the generalised SuperEasy approach is able to predict the total matter and cold matter power spectra to ≲ 0.1% relative to other linear response methods and to ≲ 0.25% relative to particle HDM simulations. Applying the method to cosmologies with mixed neutrinos+thermal QCD axions and neutrinos+generic thermal bosons, we find that non-standard subdominant HDM cosmologies have no intrinsically different non-linear signature in the total matter power spectrum from standard neutrino cosmologies. However, because they predict different time dependencies even at the linear level and the differences are augmented by non-linear evolution, it remains a possibility that observations at multiple redshifts may help distinguish between them.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

One-dimensional primordial gravitational waves in pure quadratic gravity

The almost-scale-invariant spectrum for the stochastic background wave (in primordial Universe) is a firm prediction of inflationary scenarios. In this work, to study of primordial Gravitational Waves, one-dimensional toy model in generalizations of the Einstein–Hilbert frame, described by second-order curvature invariant (called pure Quadratic Gravity) is considered. Solutions to the primordial perturbations are more varied than Einstein–Hilbert frame and include simple incoming and outcoming waves. By examining the spectrum diagram of the perturbation solutions, it can be seen that the oscillating solutions are ruled out due to not having the necessary spectral power, but the solutions as damping exponential have the ability to produce a scale-invariant spectrum (confirmed by CMB data).

Understanding the nonlinear behavior of Rayleigh–Taylor instability with a vertical electric field for perfect dielectric fluids

It is well known that, influenced only by gravity, the fluid interface is unstable when a light fluid supports a heavy fluid and is stable when a heavy fluid supports a light fluid. The situation becomes much more complicated when a vertical electric field is externally applied to the dielectric fluids. We present a nonlinear perturbation solution for an unstable interface between two incompressible, inviscid, immiscible, and perfectly dielectric fluids in the presence of vertical electric fields and gravity in two dimensions. Our nonlinear stability analysis shows that even when the linear theory indicates that the interface is stable, this system is actually unstable. The destabilization effects of the vertical electric field always dominate when gravity provides stabilization effects. This is true even when the applied vertical electric field is very weak. Analytical expressions for the overall amplitude and velocity of the interface are derived up to an arbitrary order in terms of the initial perturbation amplitude and are displayed explicitly up to the fourth order. A comparison study between the predictions of the nonlinear perturbation solution and the numerical results shows that the derived solutions capture the primary nonlinear behavior of the unstable fluid interface. By analyzing the electrical force at the interface, we provide theoretical explanations for the nonlinear phenomena induced by the vertical electric field.

Nonlinear Perturbations and Weak Shock Waves in Isentropic Atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low-amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane-parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham’s approximation can be used to solve for the shock’s dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate (∼few–15) Mach numbers, where the accuracy increases with the adiabatic index.

Application of the frozen-modes approximation to classical harmonic oscillator systems

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.

Energy-Preserving Perturbation Method for Thermally Induced Dynamics in Aerospace Tensegrity Structures

Tensegrity structures are emerging as pivotal components in the assembly of ultralarge modular spacecraft in orbit, primarily due to their exceptional volume-to-mass ratio and robust controllability. During on-orbit operations, given the low damping and high flexibility of tensegrity, its vibration-induced deformations affect the magnitude of the heat flux, presenting a typical force–shape coupling problem. This paper introduces an energy-preserving matrix perturbation solution designed to efficiently and accurately address the thermally induced vibration challenges in tensegrity. This method effectively mitigates the energy dissipation and numerical instability issues. Furthermore, the analysis incorporates the impact of the axial component of heat flux on structural vibration, uncovering the influence mechanism of this often-overlooked slow variable on the dynamic behavior of tensegrity. The proposed method demonstrates a deviation of less than 1% between the thermal response and the results obtained from finite element analysis, while achieving a 70% reduction in computation time. The varying axial thermal load leads to a rapid decrease in system frequency, resulting in divergent thermal vibrations. Additionally, the significant dynamic stress can potentially surpass the structural stress limits. These findings underscore the critical importance of considering such effects in the design and calculation of tensegrity structures for aerospace applications.

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Machine Learning Guided AQFEP: A Fast and Efficient Absolute Free Energy Perturbation Solution for Virtual Screening.

Structure-based methods in drug discovery have become an integral part of the modern drug discovery process. The power of virtual screening lies in its ability to rapidly and cost-effectively explore enormous chemical spaces to select promising ligands for further experimental investigation. Relative free energy perturbation (RFEP) and similar methods are the gold standard for binding affinity prediction in drug discovery hit-to-lead and lead optimization phases, but have high computational cost and the requirement of a structural analog with a known activity. Without a reference molecule requirement, absolute FEP (AFEP) has, in theory, better accuracy for hit ID, but in practice, the slow throughput is not compatible with VS, where fast docking and unreliable scoring functions are still the standard. Here, we present an integrated workflow to virtually screen large and diverse chemical libraries efficiently, combining active learning with a physics-based scoring function based on a fast absolute free energy perturbation method. We validated the performance of the approach in the ranking of structurally related ligands, virtual screening hit rate enrichment, and active learning chemical space exploration; disclosing the largest reported collection of free energy simulations to date.

Calculation of Green's functions for heat conduction in curved anisotropic media

AbstractSolving the inverse problems of heat conduction often requires performing a very large number of Green's function evaluations. This paper addresses the calculation of Green's functions in anisotropic curved media, such as composite lamina, where the principal axes of thermal conductivity follow the curved surface. We start with established exact series solutions for cylindrical convex and concave shapes. These solutions are extended to accommodate geometries of real‐world composite materials that do not form a closed surface like a cylinder. Unfortunately, the exact solutions have the form of an infinite series of sums or integrals and are computationally infeasible for the inverse problems of interest, especially for large radii of curvature. This motivates a perturbation solution that is accurate and computationally efficient at large radii of curvature. In addition, it motivates a phenomenological approximation that is extremely computationally efficient over a broad range of curvatures, but with some sacrifice in accuracy. These solutions are compared with exact solutions, with each other, and with numerical finite difference calculation. We identify regions in parameter space where the different approaches are preferable and where they lead to the same numerical result.

Parameter Regression for Porous Electrodes Employed in Lithium-Ion Batteries and Application to Ni0.89Co0.05Mn0.05Al0.01O2

We develop a parameter regression scheme that can be used with battery models of interest to the battery-analysis community. We show that the recent reduced order model (ROM1, 2022 J. Electrochem. 169 070520, DOI: 10.1149/1945–7111/ac7c93), which is based on a perturbation solution, can be used in place of the full system of nonlinear partial differential equations with minimal loss of accuracy for the conditions of this work, which are relevant for electric vehicle applications. The use of the computationally efficient ROM1, cast in the Python programming language, along with a routine native to Python for the nonlinear regression of model parameters through the minimization of the squared differences between experimental results and model calculations, provides a fast method for the overall endeavor. We apply the procedure to examine Ni0.89Co0.05Mn0.05Al0.01O2, a high-capacity material that is of increasing interest with respect to electric vehicles and other products that rely on batteries of high energy density. Difficulties encountered in this work include the large number of parameters governing the battery model, parameter sensitivity in the regression analyses, and the potential for multiple solutions. We close this publication with a discussion of these challenges and open questions with respect to parameter identification.

Revisiting flat rotation curves in Chern-Simons modified gravity

We revisit slow rotating black hole (BH) solutions in Chern-Simons modified gravity (CSMG) by considering perturbative solution about Schwarzschild BH. In particular, the case when nondynamical CSMG with noncanonical CS scalar is considered. We provide a new solution different from the previously obtained one [1] which we refer to as KMT model. The present solution accounts for frame dragging effect which includes not only radial dependence as in the KMT. Nevertheless, it reduces to KMT as a particular case. We show that the tidal gravitational force (Kretschmann invariant) associated to the present solution contains a term of order 1/r3 additional to Schwarzschild but absent from directional divergence, unlike KMT model which diverges along the axis of symmetry. We derive the corresponding circular velocity of a massive test particle in which the KMT velocity is recovered in addition to an extra term ∝r. We investigate possible constraints on KMT and the present solutions from the observed rotation curve of UGC11455 galaxy as an example. We show that perturbation solutions cannot physically explain the flattening of galactic rotation curves.

Application of Perturbation Method to Analytical H2 Optimization for a Damped Structure with Inerter-Based Control

Inherent damping exists in all real engineering systems and it affects the dynamic behavior of the structures in vibration reduction. However, it is generally neglected in existing analytical H2 optimization for inerter-based vibration control. Hence, an analytical H2 optimization of a tuned inerter damper (TID) based on the perturbation method is presented to control the vibration of a damped structure subjected to stochastic acceleration ground excitations. Since relative displacement and velocity are crucial to structural damage and absolute acceleration is related to vibration comfort, three sets of closed-form solutions for the optimal stiffness ratio and damping ratio are derived to suppress the corresponding responses. The expressions of these optimal parameters are functions of the inherent damping and inerter-mass ratio. The accuracy of these analytical solutions is demonstrated by comparison with the results provided by the numerical optimization procedure. The relevant parameters affecting the vibration control performance and the robustness of the TID in a damped structure are analyzed in frequency domain. Finally, time–history response and energy dissipation of TID are also explored by numerical examples under real and artificial seismic waves. The results highlight that more than 10% reduction ratio of the mean square value can be further achieved for a structure with a low inherent damping ratio ([Formula: see text]) if the present multi-order perturbation solutions are used for optimization, instead of the existing method which neglects the inherent damping. This superiority of the present perturbation method will become more prominent with the increase of the inherent damping ratio. Therefore, the present perturbation method has a great significance for designing a TID to achieve the best control performance.

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem—too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, Kα(x) .