Asymptotic Expansion Research Articles

Asymptotic expansion of the Lebesgue constant for even-order Hermite-Fejér interpolation on Chebyshev nodes

An integral representation is obtained for the Lebesgue constant associated with even-order Hermite-Fejér interpolation on Chebyshev nodes, and this result is extended to give a complete asymptotic expansion with a remainder term.

Assessment of effective elastic constants of U-10Mo fuel: A multiscale modeling and homogenization study

The significant microstructural changes that U-Mo fuel undergoes during operation degrades its mechanical properties and structural integrity. Microstructural evolution entails the formation, evolution, and redistribution of porosity in conjunction with grain refinement. In the present paper, we employ numerical approaches to assess the impact of the various microstructural features—grains, nanoscale intragranular fission gas bubbles, and mesoscale intergranular voids—on the degradation of elastic constants. Phase-field microstructure models are combined with the asymptotic expansion homogenization technique in order to derive the effective elastic constants as a function of porosity and fission density. The results are verified and compared against theoretical bounds. Using this approach, elastic degradation in operating nuclear fuels can be quantified when the distributions of microstructural features are known.

Asymptotic expansions for variants of the gamma and Post–Widder operators preserving 1 and xj

Recently, the authors constructed operators acting on a space of functions defined on and preserving 1 and for a given . To this end, they considered suitable modifications of the Post–Widder and gamma operators. They proved convergence results and established the associated Voronovskaja type formulas. In this paper, we give asymptotic expansions for the sequences of these operators.

Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

Design of a Switching Strategy for Output Voltage Tracking Control in a DC-DC Buck Power Converter

This work proposes the design of a commutation function to solve the output voltage trajectory tracking problem in the DC-DC Buck power electronic converter. Through a Lyapunov-type analysis, sufficient conditions are established, taking into account the discontinuous model, to ensure asymptotic convergence to the desired trajectories. Based on this analysis, a state-dependent switching function was designed to guarantee the closed-loop stability of the tracking error. To validate the control performance, circuit numerical simulations were carried out under abrupt disturbances in the source and load of the converter. The results demonstrate that the voltage tracking at the output of the converter is satisfactorily achieved.

Synergistic graph fusion via encoder embedding

In this paper, we introduce a method called graph fusion embedding, designed for multi-graph embedding with shared vertex sets. Under the framework of supervised learning, our method exhibits a remarkable and highly desirable synergistic effect: for sufficiently large vertex size, the accuracy of vertex classification consistently benefits from the incorporation of additional graphs. We establish the mathematical foundation for the method, including the asymptotic convergence of the embedding, a sufficient condition for asymptotic optimal classification, and the proof of the synergistic effect for vertex classification. Our comprehensive simulations and real data experiments provide compelling evidence supporting the effectiveness of our proposed method, showcasing the pronounced synergistic effect for multiple graphs from disparate sources.

A model ODE for the exponential asymptotics of nonlinear parasitic capillary ripples

Abstract In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.

DECAY RATES AND INITIAL VALUES FOR TIME-FRACTIONAL DIFFUSION-WAVE EQUATIONS

We consider a solution u(·,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α ∈ (0,2)\ {1} where t is a time variable. We first prove that a suitable norm of u(·,t) is bounded by the rate t −α for 0 α 1 and t 1−α for 1 α 2 for all large t > 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α = 1, the decay rate can keep some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.

Shape and location recovery of laser excitation sources in photoacoustic imaging using topological gradient optimization

This study focuses on source term inversion in fractional partial differential equations, specifically applied to photoacoustic imaging. This work contributes to advancing imaging techniques and provides practical insights for medical diagnostics and materials characterization. Our aim in this paper is to develop an accurate method for recovering the location and the shape of a laser excitation source from partial boundary data. To achieve this, we reformulate our inverse problem as an optimization challenge. We utilize topological sensitivity analysis to establish an asymptotic expansion of a relevant shape function. These theoretical findings form the basis for a rapid and precise detection algorithm. Additionally, we present several numerical experiments that demonstrate the effectiveness and accuracy of our proposed approach.

Effective boundary conditions for second-order homogenization

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter ɛ≪1, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

Comparative study on excess distribution estimation in iid settings

. This study considers excess distribution estimation in iid settings. There are two ways for the estimation; the fitting to the generalized Pareto distribution and the fully non parametric estimation. The fitting estimator is justified by the approximation proven in the extreme value theory; however, the accuracy depends on how extremely large the target is. The non parametric estimator does not need an approximation and has the advantage of wide applicability. This study conducts both theoretical and numerical comparative study on excess distribution estimation. Asymptotic convergence rates of two estimators are obtained, and the mean integrated squared errors are numerically surveyed by simulation study. An illustrative example of Abisko rainfall amount is presented.

Asymptotic Behavior of Singular Solution to the k-Hessian Equation with a Matukuma-Type Source

This paper is concerned with radially positive solutions of the [see formula in PDF]-Hessian equation involving a Matukuma-type source [see formula in PDF], where [see formula in PDF] is the [see formula in PDF]-Hessian operator, [see formula in PDF], and [see formula in PDF] is a suitable bounded domain in [see formula in PDF]. It turns out that there are two different types of radially positive solutions for [see formula in PDF], i.e., M-solution (singular at [see formula in PDF]) and E-solution (regular at [see formula in PDF]), which is distinct from the case when [see formula in PDF]. For [see formula in PDF], we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired extend. In contrast to the case [see formula in PDF], we require a more precise range of parameters due to repeated application of Taylor expansions, which also makes asymptotic expansions need more delicate investigation.

Asymptotic Expansions for the Stationary Moments of a Modified Renewal-Reward Process with Dependent Components

Asymptotic Expansions for the Stationary Moments of a Modified Renewal-Reward Process with Dependent Components

Near-Field to Far-Field Approximation Formulas Based on the Asymptotic Expansion Method for Electromagnetic Wave Propagation Across Air–Sea Boundary Due to a Vertical Magnetic Dipole

Near-Field to Far-Field Approximation Formulas Based on the Asymptotic Expansion Method for Electromagnetic Wave Propagation Across Air–Sea Boundary Due to a Vertical Magnetic Dipole

On the stability in variation of non-autonomous differential equations with perturbations

In this paper, we investigate the problem of stability in variation of solutions for nonautonomous differential equations. Some new sufficient conditions for the asymptotic or exponential stability for some classes of nonlinear time-varying differential equations are presented by using Lyapunov functions that are not necessarily smooth. The proposed approach for stability analysis is based on the determination of the bounds that characterize the asymptotic convergence of the solutions to a certain closed set containing the origin. Furthermore, some illustrative examples are given to prove the validity of the main results.

Adaptive Robust Control of Uncertain Euler-Lagrange Systems Using Gaussian Processes.

This article proposes a novel adaptive robust control approach based on Gaussian processes (GPs) for the high-precision tracking problem of uncertain Euler-Lagrange (EL) systems with time-varying external disturbances. Given a prior dynamic model, the GP regression (GPR) technique is employed to obtain a nonparametric data-based uncertainty model, including its probabilistic confidence intervals. Based on the adaptive sliding mode control (ASMC) framework, the posterior means of GPs are utilized for dynamic compensation, whereas the posterior variances are applied to adjust the feedback gains. This proposed control strategy is robust against significant system uncertainty with low feedback gains. A novel adaptive law for updating hyperparameters based on tracking error feedback is presented, thereby improving the performance of both tracking control and GP modeling simultaneously. Compared to existing likelihood-based optimization methods, this hyperparameter adaptive law enables data-efficient and fast uncertainty learning for control applications. The proposed control strategy guarantees the semiglobal asymptotic convergence to zero tracking error with a specified probability. Simulations using an underwater robot model demonstrate that the utilization of GPs and hyperparameter adaptive law significantly improves the performance of tracking control and uncertainty learning.

Transient resonance of sloshing liquid with time-varying mass

This study examines the sloshing of liquid with time-varying mass in a tank. A set of innovative experiments is carried out involving a shaking table supporting a water tank equipped with a drain pipe. Physical evidence of transient resonance is observed for the first time. Transient resonance occurs under specific excitation conditions when the instantaneous average water level (AWL) approaches a critical depth. During transient resonance, the oscillatory amplitude of the free-surface elevation increases sharply and then decreases in an envelope pattern. A bifurcation of the frequency band is first found in the Morlet-wavelet time–frequency spectrum, coinciding with the appearance of the maximum oscillatory amplitude. How the excitation conditions, drainage rate, and initial water depth affect transient resonance is recognized. Two mathematical models—one based on linear modal theory and the other based on nonlinear asymptotic theory and the Bateman–Luke variational principle—are derived to replicate the physical observations, by which application scopes of both models have been greatly broadened. The linear solution fails to predict the key feature of transient resonance, namely, the asymmetric envelopes of the oscillatory component about the AWL. By contrast, the nonlinear asymptotic solution captures this asymmetric feature accurately, and predicts both the steady and maximum oscillatory amplitudes well. The nonlinear solution is decomposed into terms of order 1/3, 2/3, and 1 using an asymptotic series for component analyses. A special nonlinear jump behavior is observed. The effects of draining and filling on transient resonance are compared.

Mean-variance portfolio with wealth and volatility dependent risk aversion

Risk aversion rate plays a significant role in mean-variance portfolio selection. Most existing literature assumes it to be constant or wealth dependent, which is unrealistic. In this study, I contend that it is both wealth and volatility dependent because it varies across economic status: either steady or fluctuated. In addition, I decompose the risk aversion rate into a wealth prudence rate and a volatility prudence rate and investigate their mutual effect on portfolio selection under a continuous-time mean-variance framework. Using Game theoretic approach and asymptotic expansion, I derive the optimal trading rule analytically.

Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows

We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on Zd -covers, thus proving a strong form of Krickeberg mixing. We also prove that the spectral measures around 0 of the Casimir operators on any increasing sequence of finite Abelian covers converge weakly to an absolutely continuous measure.