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An analytical study on the propagation characteristics of shear wave in a spherical layered structure comprising of an isotropic sandy material medium covered by a concentric spherical transversely isotropic sandy material layer of finite thickness has been carried out in the present work. The source of excitation for wave propagation is considered in the outer layer and the analytical treatment with contour integration and Fourier-Legendre Expansion theorem is utilized to accomplish closed form of dispersion relation. Further, Debye Asymptotic Expansion analysis is used to deal the complication caused by the inclusion of Bessel's function, Hankel's function and their derivatives in the solution procedure. With help of Debye Asymptotic Expansion analysis, the obtained dispersion relation converts in the form of elementary functions and found to be in well-agreement with pre-established and classical results through the particular cases. Numerical computations are done to illustrate the influence of the affecting parameters (anisotropy, sandiness and radii ratio) on dispersion curves graphically. Moreover, to expose the impact of anisotropy, numerical data of three distinct transversely isotropic materials (Magnesium, Zinc and Beryl) are taken into the account to establish the comparative study with isotropic material which serves the major highlights of the present study.

The focus of this paper is to obtain explicit solutions to initial value problems, where numerical methods cannot provide one, and to verify the accuracy orders of the explicit solutions. One of available methods to obtain an explicit solution is the asymptotic (formal) expansion method. However, we must be sure with the accuracy order of the explicit solution. In this paper, an order verification method is proposed for truncated asymptotic formal expansion solutions to initial value problems. A least-squares fit of error data is used in the existing order verification method. The method that we propose does not involve any application of least-squares fit of error data, so is simpler, yet produces accurate expected accuracy orders of solutions of explicit truncated asymptotic formal expansions. With our proposed method, we are successful in verifying the accuracy orders of solutions of truncated asymptotic formal expansions to the linear and nonlinear initial value problems accurately.

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

We study the existence and stability of standing pulse solutions to a singularly perturbed three-component reaction–diffusion system with one activator and two inhibitors. We apply the MAE (matched asymptotic expansion) method to construct solutions and the SLEP (singular limit eigenvalue problem) method to evaluate their stability. This approach is not just an alternative approach to geometric singular perturbation and the associated Evans function, it also yields two advantages: one is extendability to higher dimensional cases, and the other is that it allows us to obtain more precise information about the behaviors of critical eigenvalues. This implies the existence of codimension-two singularities of drift and Hopf bifurcations for the standing pulse solution. Moreover, it is numerically confirmed that stable standing and traveling breathers emerge around the singularity in a physically acceptable region.

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles.

In this article, we study the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise Holder continuous perturbation and investigate how the Holder constant can affect the eigenvalues. More precisely, we derive several first terms in the asymptotic expansion for the eigenvalues.

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

An algorithm for the approximate calculation of the coefficients of the Dulac series (an asymptotic series of the monodromy transformation) in the space of vector fields with a Newton diagram containing more than one edge and a monodromic singular point is proposed. The conditions for the applicability of this algorithm are obtained. The algorithm is implemented in the MAPLE package. Examples are given for the case of a Newton diagram consisting of two edges.

Asymptotic expansion based homogenization has been widely used to predict the effective macroscopic properties of periodic unit cells (PUCs). In this work, we show that the homogenization process can be done in a much more elegant manner for both continuum and discrete PUCs by taking advantage of the Cauchy–Born’s hypothesis, which is a widely used rule in the area of solid physics to relate the position of the atoms in a crystal lattice and the overall strain of the medium. It is shown that in the proposed method, the derivation process of the effective elasticity tensor is quite easy and can rely entirely on commercial CAE software (e.g., ANSYS, ABAQUS, etc.) to accomplish the homogenization task. In detail, after the discretization of the unit cell with finite elements, one only needs to apply affine boundary conditions at the exterior boundaries of the unit cell and then call the FEA solver to find the static displacement field under such affine boundary conditions. The entries of the elasticity tensor can then be expressed using the stain energy of the unit cell. After deriving the sensitivity information of the Cauchy–Born hypothesis based homogenization process, the inverse homogenization process, which attempts to find the optimal layout exhibiting pre-determined desirable material properties, can be implemented in a straightforward way as well. Some numerical examples are tested and compared with the results in the literature. It is showed that the results of both the homogenization and inverse homogenization examples obtained by our method agree very well with the ones in the literature, demonstrating the validity of the Cauchy–Born hypothesis based numerical homogenization method.

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

Using the so-called direct scheme method, an asymptotic expansion of n-th order to the solution of a class of singularly perturbed linear-quadratic optimal problem with integral constraint on control is constructed. The expansion contains three type functions. Two of them are boundary layer functions in the vicinities of two fixed end-points, and the remain is regular function. A numerical example is represented to illustrate the obtained results.

In this paper, we construct the utility-based optimal hedging strategy for a European-type option in the Almgren-Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first order terms in the associated HJB equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.

We are interested in problems related to forward starting options for Hull–White stochastic volatility model. Our objective is to obtain analytical, semi-analytical, or approximated expressions of its price for simulation. To obtain an analytical representation of the price, we use Yor’s formula. However, the analytical formula is difficult to implement. Next we consider semi-analytical expressions for the price. In order to have them, we use the tower property for conditional expectations with a certain filtration and explicitly calculate it. Then, we consider an expansion expression for the price using the semi-analytical expression to have a simple expression. The semi-analytical expressions and the expansion expression can reduce computational costs and standard errors when the Monte Carlo method is used. Finally, some numerical results are given to show their accuracy and efficiency.

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

The main results of this paper are related to studying the dynamics of the Van-der-Pol oscillator, which is under the influence of a driving force and a hysteretic effect, formalized by the Bouc–Wen phenomenological model. The regularizing role of the hysteretic link in the reduction of chaotic regimes is established. The main results are obtained as part of the asymptotic expansion method (small parameter method). A system of two coupled Van-der-Pol oscillators is considered under the conditions of a driving force acting on one of them: the relationship between the amplitudes and frequencies of the natural oscillations of the oscillators is established in an analytical form. The paper also describes a system of cross-related Van-der-Pol oscillators, in which the driving force acting on one of them is determined by the output of a hysteretic transducer, to the input of which the mismatch between the oscillators’ velocities is applied. It is found that the coupling parameter between the oscillators and the parameter that determines the influence of the hysteretic link have a significant effect on the synchronization in the system.

We consider scalar real Riccati equations with coefficients expanding in convergent power series in a neighborhood of infinity. Continued solutions of such equations are studied. Power geometry methods are used to obtain conditions for expanding these solutions in asymptotic series.

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

The authors provide a complete asymptotic expansion for a class of functions in terms of the complete Bell polynomials. In particular, they obtain known asymptotic expansions of some Keller type sequences.

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .

The asymptotic behavior of weak time-periodic solutions to the Navier–Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.

Two types of existing iterative methods for solving the nonlinear balance equation (NBE) are revisited. In the first type, the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution. In the second type, the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively. The two methods are rederived by expanding the solution asymptotically upon a small Rossby number, and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution. For each rederived method, two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration. Upon testing with analytically formulated wavering jet flows on the synoptic, sub-synoptic and meso-α scales, the iterative procedure designed for the first method with the Poisson solver, named M1a, is found to be the most accurate and efficient. For the synoptic wavering jet flow in which the NBE is entirely elliptic, M1a is extremely accurate. For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic, M1a is sufficiently accurate. For the meso-α wavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed, M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow, but not for the anti-cyclonically curved part.

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 ( u 3 ) x x , u ( x , 0 ) = u 0 ( x ) ∈ H ( R ) , where H ( R ) = W 3 , 1 ( R ) ∩ H 2 , 2 ( R ) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale ( y , t ) . In any fixed space-time cone C ( y 1 , y 2 , v 1 , v 2 ) = { ( y , t ) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , we compute the long time asymptotic expansion of the solution u ( x , t ) , which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N ( I ) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second | t | − 1 / 2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O ( | t | − 1 ) from a ∂ ‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable.

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

The article investigates the Robin boundary-value problem for a linear inhomogeneous ordinary differential equation of the second order on a segment with a small parameter at the highest derivative of the unknown function. The aim of the study is to construct a uniform asymptotic expansion of the solution of the Roben problem with an arbitrary degree of accuracy on the considered segment, when a small parameter tends to zero. The singularities of the problem under study are the presence of a small parameter at the highest derivative of the unknown function (i.e., a singular perturbation) and the corresponding unperturbed differential equation of the first order has a regularly singular point at the left end of the segment. The Roben problem under consideration is included in the class of bisingular problems in the terminology of A. M. I’lin. The complete asymptotic solution of the Robin problem is constructed by the generalized method of boundary functions, i.e. the modified method of the boundary functions of Vishik–Lyusternik–Vasilyeva–Imanalieva. The asymptotic solution is justified using the maximum principle.

For the two-dimensional elasticity problems, we give a uniqueness and existence analysis via the single-layer potential approach leading to a system of integral equation that contains a weakly singular operator. For its numerical solutions we describe a O ( h 3 ) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O ( h 3 ) is got and the accuracy of numerical approximations can be improved to the order of O ( h 5 ) by the Richardson extrapolation algorithm (EA) once. The efficiency of the method is illustrated by two numerical examples.

In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain Ω : = R + × R 2 . More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the suitable initial approximation function such that the linearized equations around it admitted a partial dissipative structure when we carry out the weighted energy estimate. Meanwhile, the asymptotic expansion of Sobolev regularity solutions is given.

In this paper, testing the equality of two mean vectors is considered when each dataset has a monotone pattern of missing observations. A simplified Hotelling’s T 2-type test statistic in a two-sam...

In this paper, we consider the Rasch model and suggest novel point estimators and confidence intervals for the ability parameter. They are based on a proposed confidence distribution (CD) whose construction has required to overcome some difficulties essentially due to the discrete nature of the model. When the number of items is large, the computations due to the combinatorics involved become heavy, and thus, we provide first- and second-order approximations of the CD. Simulation studies show the good behavior of our estimators and intervals when compared with those obtained through other standard frequentist and weakly informative Bayesian procedures. Finally, using the expansion of the expected length of the suggested interval, we are able to identify reasonable values of the sample size which lead to a desired length of the interval.

In this paper we study both the fast computation and the related error analysis of many different finite and infinite oscillatory Bessel transforms. By transforming the Bessel function into Confluent hypergeometric function and using its asymptotic expansion, we develop an efficient approach by the contours in the complex plane, corresponding to the paths of steepest descent. Furthermore, we conduct their error analyses in inverse powers of the frequency ω. The related error analyses show that the precision can be improved by either adding more Gauss-Laguerre nodes or increasing the frequency ω. In particular, the method exhibits the high error order. Meanwhile, we apply the proposed method to some other kinds of oscillatory integrals through some simple changes of variables. Numerical experiments can verify our theoretical analysis. It is also shown that the new algorithms are more efficient than the existing Filon-type method and can achieve the same level of accuracy with the method (the XMX method) proposed in Xu, Milovanovic and Xiang [46] by some numerical experiments under the given conditions. A combination of the n 1 -point Gauss-Laguerre quadrature rule and the n 2 -point generalized Gauss-Laguerre quadrature rule are used in the XMX method. However, we only use the Gauss-Laguerre quadrature rule for one time in the established method, which makes the theoretical and error analysis simpler than the XMX method.

It is long known that the distribution of a sum Sn of independent non-negative integer-valued random variables can often be approximated by a Poisson law: Sn≈πλ, where . The problem of evaluating the accuracy of such approximation has attracted a lot of attention in the past six decades. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, it provides insights into Kolmogorov’s problem. Among popular metrics considered in the literature is the Gini–Kantorovich distance dG. The task of establishing an estimate of dG(Sn;πλ) with correct (the best possible) constant at the leading term remained open for a long while. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We show that the accuracy of approximation can be considerably better if the random variables obey an extra condition involving the first two moments. A sharp estimate of the accuracy of shifted (translated) Poisson approximation is established as well.

An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where $$\gamma \ge 0$$ , and i, j, k are nonnegative integers. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from cartesian to spherical. The resulting expression for $$h\left( {s,{\mathbf{Q}}} \right)$$ includes a number of cases of elementary integrals, the most difficult of which is $$II\left( {n,\mu } \right) = \int_{0}^{\pi } {\cos^{n} \theta e^{ - \mu \cos \theta } } d\theta$$ , with a nonnegative integer n and positive μ. This integral can be formed by linearly combining modified Bessel functions of the first kind $$B(n,\mu ) = \frac{1}{\pi }\int\limits_{0}^{\pi } {e^{\mu \cos \theta } \cos \left( {n\theta } \right)d\theta }$$ , with a nonnegative integer n and negative μ. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for this expression for n = 0,1,2. The relative error of asymptotic function is in the order of 10−16 with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function of $$B\left( {n,\mu } \right)$$ based on the expression for $$e^{ - \mu } II\left( {n,\mu } \right)$$ when n is an integer and μ is real and large in absolute value.

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where \begin{document}$ {\delta} = {\delta}(x) $\end{document} is the delta function supported at the origin. In the \begin{document}$ L^2 $\end{document} supercritical setting \begin{document}$ p>3 $\end{document} , we construct self-similar blow-up solutions belonging to the energy space \begin{document}$ L_x^\infty \cap \dot H_x^1 $\end{document} . This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \begin{document}$ x = 0 $\end{document} imposed by the \begin{document}$ \delta $\end{document} term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \begin{document}$ 0 using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

An accurate evaluation of three-dimensional (3D) Time-Domain Green Function (TDGF) in infinite water depth is essential for ship’s hydrodynamic analysis. Various numerical algorithms based on the TDGF properties are considered, including the ascending series expansion at small time parameter, the asymptotic expansion at large time parameter and the Taylor series expansion combines with ordinary differential equation for the time domain analysis. An efficient method (referred as “Present Method”) for a better accuracy evaluation of TDGF has been proposed. The numerical results generated from precise integration method and analytical solution of Shan et al. (2019) revealed that the “Present method” provides a better solution in the computational domain. The comparison of the heave hydrodynamic coefficients in solving the radiation problem of a hemisphere at zero speed between the “Present method” and the analytical solutions proposed by Hulme (1982) showed that the difference of result is small, less than 3%.

In this study, we consider a bias reduction of the conditional maximum likelihood estimators for the unknown parameters of a Gaussian second-order moving average (MA(2)) model. In many cases, we use the maximum likelihood estimator because the estimator is consistent. However, when the sample size n is small, the error is large because it has a bias of $O({n^{-1}})$. Furthermore, the exact form of the maximum likelihood estimator for moving average models is slightly complicated even for Gaussian models. We sometimes rely on simpler maximum likelihood estimation methods. As one of the methods, we focus on the conditional maximum likelihood estimator and examine the bias of the conditional maximum likelihood estimator for a Gaussian MA(2) model. Moreover, we propose new estimators for the unknown parameters of the Gaussian MA(2) model based on the bias of the conditional maximum likelihood estimators. By performing simulations, we investigate properties of this bias, as well as the asymptotic variance of the conditional maximum likelihood estimators for the unknown parameters. Finally, we confirm the validity of the new estimators through this simulation study.

This paper is about solutions to PDEs in which the ratio between the largest and smallest values of the coefficient is very large. Such problems are referred to as high-contrast ones. A full asympt...

The article considers multidimensional improper integrals of functions that are the product of generalized polynomials in some degrees. Such integrals are found in many branches of mathematics and theoretical physics. In particular, they include Feynman integrals arising in the study of various objects of quantum field theory. The exact calculation of these integrals is a difficult and not always possible task; therefore, determining the conditions for their convergence and obtaining their asymptotic expansion in one of the parameters is of considerable practical interest. The convergence conditions for the integrals considered in the article can still be used, for example, in the study of multiple series representing the sum of the values of a rational function at the nodes of an integer lattice. The article considers the problem when the integration area is R+^n, and the generalized polynomials included in the integrand are either positive everywhere except zero or have positive coefficients. The convergence set of these integrals is described and the equivalence of the convergence condition to the condition on the Newton polytopes of polynomials in integrands is proved. The convergence criterion proved in the paper coincides in formulation with the corresponding result of the work of A. K. Tsikh and T. O. Ermolaeva, but it was obtained by other methods and for a slightly wider set of integrands. The proofs of the statements in the paper are based on the simplest properties of convex polytopes and basic facts from the theory of improper multiple integrals.

In this study, we develop model bias estimators based on an asymptotic expansion of the model dynamics for small time scales and small perturbations in a model parameter, and then use the estimators to improve the performance of a data assimilation system. We employ the well-known Lorenz (1963) model so that we can study all aspects of the dynamical system and model bias estimators in a detailed way that would not be possible with a full physics numerical weather prediction model. In particular, we first work out the asymptotics of the Lorenz model for small changes in one of its parameters and then use statistics from cycled data assimilation experiments to demonstrate that the asymptotics accurately represent the behavior of the model and that the coefficients of the nonlinear asymptotical expansion can be reasonably estimated by solving a least squares minimization problem.\ud \ud In data assimilation, the background error covariance matrix usually estimates the uncertainty of the model background, which is then used along with the observation error covariance matrix to produce an updated analysis. If the uncertainty of the model background is strongly influenced by time-dependent model biases, then the development of nonlinear bias estimators that also vary with time could improve the performance of the assimilation system and the accuracy of the updated analysis. We demonstrate this improvement through the combination of a constant background error covariance matrix with a dynamically-varying matrix computed using the model bias estimators. Numerical tests using the Lorenz (1963) model illustrate the feasibility of the approach and show that it leads to clear improvements in the analysis and forecast accuracy.

Electrovortex flows are of great interest both from the viewpoints of theoretical magnetohydrodynamics and applications. They arise when the electric current of variable density passes through a conducting medium (such as a liquid metal). The interaction between the current and self magnetic field of the current induces a nonpotential electromagnetic force that causes a vortex flow of the liquid. As a model problem, we consider a stationary flow between two parallel planes. The flow is described by the stream function for which we obtain a nonlinear fourth-order partial differential equation. For moderate values of the electrovortex flow parameter, we investigate the problem by the asymptotic series expansion in the powers of this parameter. We describe the process of successive approximations of various degrees and present the flow pattern that is given by two first terms. This asymptotic solution is shown to be rather close to the numerical solution of the problem.

Under investigation in this work is the general three-component nonlinear Schrodinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in$$L^p$$Lp-setting,$$1 \le p \le +\infty $$1≤p≤+∞, have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order$$r \in \mathbb {N}^+$$r∈N+. Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the$$L^p$$Lpmodulus of smoothness of orderrhave been established. As a consequence, the qualitative order of approximation is also derived assumingfin suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in$$L^p(\mathbb {R})$$Lp(R),$$1 \le p \le +\infty $$1≤p≤+∞, for the sampling Kantorovich operators.

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison & Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem is briefly mentioned but left open for future research., 12 pages, 6 figures

Asymptotic solutions of the eigenvalue problem for an Euler operator in a neighborhood of a regular singular point are considered. We find a condition under which the asymptotic expansion is free of logarithms. Eigenvalues expressed in terms of elementary functions in the form of a finite sum of quasi-polynomials are obtained for third-order Euler operators and also for commuting Euler operators of sixth and ninth orders. The problem on common eigenfunctions for commuting Euler operators is examined. In the case of operators of rank 2 and 3, it can be reduced to second- and third-order Bessel equations by differential substitutions.

For the second-order nonlinear ordinary differential equation $$ {u}_{xx}^{\hbox{'}\hbox{'}}={u}^5-{tu}^3-x, $$ we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5 − tu3 − x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2 →∞.

In this paper, we consider the sine-Gordon equation with a high-frequency parametric pumping and a weak dissipative force. We examine the class of π-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable 2π-kinks, these solutions are unstable. We prove that the time of decaying of π-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametric pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a π-kink. Numerical simulations of solutions confirm the good qualitative agreement with asymptotic expansions.

We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using $$ R\left(N,p\right)\equiv \sum \limits_{n=N}^{\infty }1/{n}^p $$ as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.