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In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, physics‐informed neural networks (PINNs) and the deep energy method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well‐known DEM is based on the principle of minimum potential energy, but it lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results of classical linear elasticity using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM‐Plus (DCEM‐P), adding terms that satisfy PDEs. Furthermore, we propose a deep complementary energy operator method (DCEM‐O) by combining operator learning with physical equations. Initially, we train DCEM‐O using high‐fidelity numerical results and then incorporate complementary energy. DCEM‐P and DCEM‐O further enhance the accuracy and efficiency of DCEM. [ABSTRACT FROM AUTHOR]

The application of an electric field to a semiconductor can alter its absorption properties. This electroabsorption effect can have a significant impact on the quantum efficiency of detector structures. The photocurrents in bulk InGaAs and GaAsSb p-i-n photodiodes with intrinsic absorber layer thicknesses ranging from 1 to 4.8 μm have been investigated. By using phase-sensitive photocurrent measurements as a function of wavelength, the absorption coefficients as low as 1 cm−1 were extracted for electric fields up to 200 kV/cm. Our findings show that while the absorption coefficients reduce between 1500 and 1650 nm for both materials when subject to an increasing electric field, an absorption coefficient of 100 cm−1 can be obtained at a wavelength of 2 μm, well beyond the bandgap energy when they are subject to a high electric field. The results are shown to be in good agreement with theoretical models that use Airy functions to solve the absorption coefficients in a uniform electric field. [ABSTRACT FROM AUTHOR]

Acoustic radiation from double composite laminated cylindrical shells with cyclic rectangular rib-plates is analytically investigated by using the Sommerfeld-Watson transform. High-frequency acoustic model of double composite cylindrical shells with rectangular rib-plates is established in terms of creeping waves, geometrical acoustic waves and sound pressure asymptotic integrals. One kind of bending wave boundary layers for the shear deformable rectangular rib-plate is shown. Each set of creeping wave poles for the composite cylindrical shells with rectangular rib-plates is uniformly distributed by the cyclic period. This characteristic still holds for the cyclic structures with the smallest period. Creeping wave poles of the stiffened composite cylindrical shells appear in the entire complex circumferential wavenumber plane. The asymptotic integrals with Airy function are proposed to fast calculate acoustic pressure in the shadow and penumbra regions without making use of creeping wave poles. [ABSTRACT FROM AUTHOR]

Chiral anomalies resulting from the breaking of classical symmetries at the quantum level are fundamental to quantum field theory and gaining ever-growing importance in the description of topological materials in condensed matter physics. Here we present analytical solutions of the Dirac equation for massless 3 + 1 fermions confined to an infinite stripe and placed into a background gauge field forming a triangular potential well across the width of the stripe. Such an effective 1 + 1 system hosts zero-energy modes resulting in the gauge field-dependent chiral anomaly structure. This problem has a direct relation to a half-bearded graphene nanoribbon placed into an in-plane external electric field and offers it an exact solution in terms of new special functions that are similar but not reducible to Airy functions. [ABSTRACT FROM AUTHOR]

Maxwell's equations with temporal dispersion and localized initial data are considered. Using the Maslov canonical operator, we construct a uniform asymptotic solution expressed in terms of the derivative of the Airy function of composite argument in a neighborhood of regular points of the leading wave front. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to introduce innovative orthogonal power function series aimed at obtaining accurate nonlinear analytical solutions for axisymmetric circular thin plates. The main features of this paper are as follows: The deflection is expanded by the innovative orthogonal power function series. The Airy stress function, which satisfies the geometric deformation compatibility equation, responds to the nonlinear coupling relationships between the plate deflection and the in-plane force or displacement boundary conditions. The nonlinear algebraic equations are obtained by the energy variational method. Many comparisons are made with the results of related researchers. The present accurate solutions not only allow the problems to be solved perfectly and provide the most reliable basis for engineering design but also set new benchmarks for the verification of various nonlinear numerical and approximate analytical solutions. The developed methodology represents a significant improvement, providing better accuracy and computational efficiency compared to historical approaches. Therefore, the present method is more worthy of promotion. [ABSTRACT FROM AUTHOR]

The two-dimensional deformation of a uniform, elastically isotropic layer of a finite thickness (FT) over a rough-rigid base caused by surface loads has been solved analytically. The stresses and the displacements for an isotropic layer are obtained in the integral form by applying the Airy's stress function approach. Using the appropriate boundary conditions, the problem of surface loads is discussed in detail. The integrals cannot be evaluated analytically due to the complicated expression of denominator. The linear combination of exponential terms occurring in the denominator is expanded by a finite sum of exponential terms (FSET) using the method of least squares and then the integrals are evaluated analytically. The analytical displacements are obtained for normal strip loading, normal line loading and shear line loading. The displacements have been plotted numerically for normal strip loading to find the effect of layer thickness for a Poissonian layer and compared with the half-space. [ABSTRACT FROM AUTHOR]

Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic expansions. From these, asymptotic expansions are derived for the zeros of Bessel functions that are valid for large positive values of the order, uniformly for all the zeros. The coefficients in the expansions are explicitly given elementary functions, and similar expansions are derived for the zeros of the derivatives of Bessel functions. [ABSTRACT FROM AUTHOR]

The paper studies the regularities of wave propagation in an inhomogeneous elastic layer with gradient properties varying along the layer thickness. The laws of change in elastic parameters are established at which the waves propagating in the direction of parameter change are harmonic. The waves propagating in the layer with a linear law of change in the elastic parameters are noт-harmonic. Analytical solutions obtained for a monochromatic wave defined on the boundary of a plane layer with a rigidly fixed second boundary are represented by special functions. In the case of gradient variation of the layer density, the solution is determined by a superposition of the Airy functions, in the case of equal relative variations of the density and elastic moduli, they are expressed through the modified Bessel functions, and in the case of arbitrary linear laws of variation in the elastic parameters, they are expressed through the confluent hypergeometric functions. Based on the analytical solution obtained for a layer with inhomogeneous density, the conditions for extreme values of displacements and strains are determined, and their distributions over the layer thickness are calculated and analyzed. [ABSTRACT FROM AUTHOR]

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi 2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature. • Exact solutions for stress/displacement in gradient elasticity models for anisotropic functionally graded nano-beams. • Analysis focuses on constructing stress fields using "gradient Airy stress function" notation. • Solutions reveal significant strain gradient effects in anisotropic nano-beams. • Literature solutions for CT beams match findings by taking gradient coefficient c to zero. • Gradient stress fields vanish except in y-direction as c approaches infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the electro-mechanical buckling analysis of different kinds of smart sandwich shells with functionally graded porous core and piezoelectric sensor–actuator face sheets. Different types of shallow shells with double curvature including convex shells, cylindrical shells and concave shells are analyzed. It is assumed that effective properties of the porous core are functionally graded to vary as a special function of the thickness parameter. The equilibrium equations are established for the doubly-curved smart sandwich shells using the energy method. The stability equations governing the equilibrium position of the smart sandwich shells are obtained as a set of coupled partial differential equations. Closed-form expressions are generated to obtain the critical buckling loads of the smart sandwich shells using the airy stress function. Sandwich shells under different types of electro-mechanical loadings including axial, lateral and hydrostatic pressures are analyzed. These numerical results show the effects of feedback gain, porosity coefficient and piezoelectric layer thickness on the buckling resistance of these smart sandwich structures. [ABSTRACT FROM AUTHOR]

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot's poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures. [ABSTRACT FROM AUTHOR]

In this paper, a series of ripple waves with decay modes for the (3+1)$$ \left(3+1\right) $$‐dimensions Kadomtsev–Petviashvili equation, always viewed as nonintegrable KP equation, are investigated. The decay mode wave solutions including ripplon, lump ripples, and lump chain ripples are described by Airy function, and all solutions are constructed from the Gram determinant form. Their propagation dynamics behavior is studied, and a comprehensive analysis of the asymptotic properties of these solutions has been diligently conducted. [ABSTRACT FROM AUTHOR]

Relativistic electromagnetic plasma waves are described by a dynamical equation that can be solved not only in terms of plane waves, but also for several different accelerating wavepacket solutions. Depending on the spatial and temporal dependence of the plasma frequency, different kinds of accelerating solutions can be obtained, for example, in terms of Airy or Weber functions. Also, we show that an arbitrary accelerated wavepacket solution is possible, for example, for a system with a luminal plasma slab. [ABSTRACT FROM AUTHOR]

Accurate calculation of the tunneling currents in a scanning tunneling microscope (STM) is needed for developing image processing algorithms that convert raw data of the STM into surface topographic images. In this paper, an accurate calculation of the tunneling current for several tip–sample distances, bias voltages, and tips of a hyperboloidal shape with several radii of curvature is carried out. The main features of this calculation are the following. Non-WKB exact solutions to the trapezoidal (linear) potential in the barrier region are used to calculate the tunneling probabilities. Pauli blocking effects on both forward and reverse current densities are introduced. Finite temperature (viz. 300 K) calculation in which electrons belonging to a narrow band of energy about the Fermi level contribute to tunneling is carried out. Integration over a field line method is used to obtain tunneling currents for the nonplanar hyperboloidal shaped tips, using the expressions obtained in the paper, for planar model current densities. An estimate of the lateral resolution is introduced. Earlier works do not consider all these aspects together in a single calculation. Tunneling currents are found to increase rapidly with increasing bias voltage and decrease exponentially with increasing tip–sample distances. Airy function determined currents are a more accurate function of a tip–sample distance than the WKB determined currents. Pauli effects are found to not always reduce currents from their non-Pauli values. The lateral resolution is found to be degraded for blunter tips, larger bias voltages, and larger tip–sample distances. [ABSTRACT FROM AUTHOR]

The Goos–Hänchen (GH) shift and the spin polarization are theoretically investigated under the anti-parallel double δ -magnetic-barrier nanostructure model by applying a bias using Airy's function formalism and transfer-matrix technique, which is an improvement on previous calculation. The results show the magnitude and sign of GH shift and spin polarization can be easily changed by modulating the applied bias. The closer the applied bias to the critical value is, the more intensive GH shift and spin polarization will be. Furthermore, two critical values come through modulating the strength of magnetic fields, and an intensive GH shift and a spin polarization also exist near each critical value. These findings can provide a practical approach to designing and fabricating spin beam filters or splitters modulated by applied bias. [ABSTRACT FROM AUTHOR]

In the present paper, we have studied the two-dimensional deformation of an elastically orthotropic layer of finite thickness overlying a rough-rigid base due to surface loads. For a two-dimensional approximation, the governing equations are split into two cases: plane strain and anti-plane strain case. We consider plane strain case only. The Airy stress function approach is used to obtain the stresses and displacements in the integral form at any point of the layer. To solve the integrals analytically, the linear combination of exponential terms occurring in the denominator is expanded by binomial expression and then approximated by a finite sum of exponential terms using the least square method. Then, the integrals are evaluated using standard integral tables. The analytical expressions for the displacements are obtained for the normal line loading. Two orthotropic materials, such as Topaz and Barytes, have been considered and graphs showing the displacement field for Topaz, Barytes and the isotropic case have been plotted. [ABSTRACT FROM AUTHOR]

An attempt was made to map the distribution of stress in fibrous composites with imperfect bonding. Two analytical micro-mechanics models were developed. In the first model, the composite was subjected to axial tensile loading, parallel to the fiber direction, and the assumption of iso-strain was employed to derive the control equations. In the second model, the composite was loaded in the direction transverse to the fiber. An iso-stress condition was employed, and Airy stress function was utilized to articulate the stress and displacement equations. An assumption of how the stress is transferred between the matrix and the fiber was introduced in both models. To investigate and validate the models, specimens were fabricated using a carbon plain weave fabric and a geopolymer matrix. Single fiber pullout and three-point bending tests were carried out. The maximum average tensile stress obtained from the three-point bending tests, as well as the mechanical properties of the fiber and geopolymer, served as input for the models. Results indicate that the effect of the level of bonding is very high in the transverse direction while almost negligible in the axial direction. The difference in the maximum value of the axial tensile stress at the fiber-matrix interface was used to calculate the numerical value of the interfacial shear strength, and the numerical result matched the data obtained from the single fiber pullout test. [ABSTRACT FROM AUTHOR]

The stress state of a soft interlayer under contact strengthening, when tensile stresses are greater than the yield strength of the interlayer metal and less than the stresses causing a general yield, is considered. The analytical expressions under plain strain and axisymmetric tension are obtained. In the first case, the stresses are determined using the Airy stress function as a corresponding polynomial, in the second case – based on the stress function of the fifth degree, built on the corresponding Legendre polynomial. The stresses satisfy the differential equations of equilibrium and boundary conditions. [ABSTRACT FROM AUTHOR]

A new set of Hadamard series expansions for the Airy functions, Ai(z) and Bi(z), is presented. Previous Hadamard expansions were defined in terms of an infinite number of integration path subdivisions. Unlike the earlier expansions, the expansions in the present work originate in the splitting of the steepest descent into a number of segments that is not only finite but very small, and these segments are defined on the basis of the location of the branch points. One of the segments reaches to infinity, and this gives rise to the presence of upper incomplete gamma functions. This is one of the most important differences from the Hadamard series as defined in the work of R.B. Paris, where all the incomplete gamma functions are of the lower type. The interest of the new series expansion is twofold. First, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path in a process that can be named "exactification." Second, the inverse of the phase function that is part of the Laplace-type integral is Taylor-expanded around branch points to produce Puiseux series when necessary. Regarding their computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision. [ABSTRACT FROM AUTHOR]

In this paper, we model, classify and investigate the solutions of (normalized) second order ode's with non-constant continuous coefficients. We introduce a generalized frequency function as the solution of a nonlinear integro-differential equation , show its existence and then derive a representation formula for (all) solutions of (normalized) second order ode's with non-constant continuous coefficients. Because this formula specifies the interplay between the coefficients of the ode, the relaxation function ("strongly" decreasing positive function) and the frequency function of the oscillation, it can be applied to design models of dissipative oscillations. As an application, we present and discuss some oscillation models that stop within a finite time period. Via two examples about the Airy function and the Bessel functions, we show that the formula might be useful in analysing certain special functions. Moreover, we demonstrate that a large class of oscillations can be used to design and analyze dissipative waves. In particular, it is easy to model dissipative waves that cause in each point of space an oscillation that stops after a finite time period. Finally, we derive an alternate form of the dissipative wave equation (for a large class of dissipative waves), which is uniquely defined by the sound speed c 0 and the parameter functions in the oscillation equation. [ABSTRACT FROM AUTHOR]

We identify Whittaker vectors for W k (g) -modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P 2 for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N = 2 four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods. [ABSTRACT FROM AUTHOR]

In this paper, a new type of structured electron beam is introduced, which is the electron Airy tornado wave (eATW) with the characteristics of abrupt auto‐focusing and rotation property in free space. It is also found that the eATWs propagate periodically in a magnetic field, and their patterns of intensity distributions show the time‐like axial symmetry about a half‐period point in a period of time. Furthermore, the influence of the magnetic field's magnitude and direction on eATWs with different parameters has also been discussed. [ABSTRACT FROM AUTHOR]

A family of 2D light fields consisting of the product of three Airy functions with linear arguments has been studied theoretically and experimentally. These fields, called three-Airy beams, feature a parameter shift and have a cubic phase and a super-Gaussian circular intensity in the far zone. Transformations of three-Airy beams in the Fresnel zone have been studied using theoretical, numerical, and experimental means. It has been shown that the autofocusing plane of a three-Airy beam is similar to the square root of the shift parameter. We also introduce generalized three-Airy beams containing nine free parameters, and obtain their Fourier transform in a closed form. [ABSTRACT FROM AUTHOR]

Numerical methods for the computation of the parabolic cylinder function U (a , z) for real a and complex z are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5 × 10 − 13 relative accuracy in double precision floating point arithmetic. [ABSTRACT FROM AUTHOR]

In this article, we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process is identical in distribution to a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent ν > 1 and skewness parameter β, with 0 < | β | < 1. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator. [ABSTRACT FROM AUTHOR]

The Mellin transforms of some products of two shifted Airy functions are evaluated in a closed form. The relation between the Mellin transforms of a single Airy function and the square of the function is obtained. The shifted Airy function moments of integer and half-integer order are expressed in terms of Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

It is known that random directed graphs D(n,p)$$ D\left(n,p\right) $$ undergo a phase transition around the point p=1/n$$ p=1/n $$. Earlier, Łuczak and Seierstad have established that as n→∞$$ n\to \infty $$ when p=(1+μn−1/3)/n$$ p=\left(1+\mu {n}^{-1/3}\right)/n $$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as μ$$ \mu $$ goes from −∞$$ -\infty $$ to ∞$$ \infty $$. By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of μ$$ \mu $$ and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter p$$ p $$, we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study. [ABSTRACT FROM AUTHOR]

In information transfer, the dissipation of a signal is of crucial importance. The feasibility of reconstructing the distorted signal depends on the related permanent loss. Therefore, understanding the quantized dissipative transversal mechanical waves might result in deep insights. In particular, it may be valid on the nanoscale in the case of signal distortion, loss, or even restoration. Based on the description of the damped quantum oscillator, we generalize the canonical quantization procedure for the case of the transversal waves. Then, we deduce the related damped wave equation and the state function. We point out the two possible solutions of the propagating-damping wave equation. One involves the well-known Gaussian spreading solution superposed with the damping oscillation, in which the loss of information is complete. The other is the Airy function solution, which is non-spreading–propagating, so the information loss is only due to oscillation damping. However, the structure of the wave shape remains unchanged for the latter. Consequently, this fact may allow signal reconstruction, resulting in the capability of restoring the lost information. [ABSTRACT FROM AUTHOR]

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫ 0 b x α f (x) Ai (− ω x) d x over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω ≫ 1 . The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [ 0 , 1 ] and [ 1 , b ] . For integrals over the interval [ 0 , 1 ] , we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [ 1 , b ] , we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [ 0 , + ∞) , which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

The goal is to take advantage of the Earth-ionosphere wave guide’s fundamentally two-dimensional wave propagation, utilizing the normalized Airy functions (NAFs) in a complex domain. It is demonstrated that the typical working formula of VLF radio-mode theory may be obtained simply from orthogonality reflections, devoid of the requirement of sophisticated argumentation in the open unit disk. The combination of the expressions is given by considering the symmetry-convex illustration of the NAFs.

In the current research, we delve into the intricate realm of bifurcation analysis for Föppl–von Kármán plates, employing a precise numerical tool. This innovative numerical approach melds the power of spectral discretization with the prowess of a high-order continuation method-based Taylor series development (HODC). It is worth noting that combining the high-order continuation method with such discretization techniques offers an efficient path-following approach, complete with adaptive step lengths, capable of tackling a wide array of nonlinear problems. Despite the extensive applications of nonlinear elasticity, the spectral method remains relatively uncharted territory within this context. However, our deep-rooted understanding and expertise in the field drive us to embrace this method alongside high-order development continuation for bifurcation analysis of Föppl–von Kármán plates. The governing equations governing thin elastic plates experiencing significant elastic deflections manifest as a pair of coupled nonlinear differential equations, famously known as the von Kármán (vK) equations, presented in a strong form with two principal unknowns: deflection (w) and the Airy stress function (F). Leveraging Chebyshev decomposition matrices, we approximate these fourth-order elliptic nonlinear partial differential equations. Subsequently, we harness high-order development continuation techniques to morph these nonlinear systems into linear ones. Our rigorous evaluation and validation of this numerical approach's precision and performance come to fruition through a comprehensive buckling analysis encompassing multiple illustrative examples. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

We consider the Stark operator T = -d²/dx² + x + q (x) on the semiaxis 0 ≤ x < ∞ with the Dirichlet boundary condition at the origin. The asymptotic behavior of the eigenvalues of this operator is studied. By the method of transformation operators, we study the spectral problem. We give a rigorous derivation of the main integral equation for the inverse problem. [ABSTRACT FROM AUTHOR]

We present a variational theory for lattice defects of rotational and translational type. We focus on finite systems of planar wedge disclinations, disclination dipoles, and edge dislocations, which we model as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Operating under the assumption of planar linearized kinematics, we formulate the mechanical equilibrium problem in terms of the Airy stress function, for which we introduce a rigorous analytical formulation in the context of incompatible elasticity. Our main result entails the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. By adopting the regularization approach via core radius, we show that, as the core radius vanishes, the asymptotic energy expansion for disclination dipoles coincides with the energy of finite systems of edge dislocations. This proves that Eshelby's kinematic characterization of an edge dislocation in terms of a disclination dipole is exact also from the energetic standpoint. [ABSTRACT FROM AUTHOR]

The Stark operator with a step-like potential is considered. An explicit form of special solutions to the corresponding Stark equation is found. The scattering problem for the operator L is studied. A formula for expansion in terms of eigenfunctions of a continuous spectrum is obtained. [ABSTRACT FROM AUTHOR]

The magnetotelluric (MT) responses are explicitly solved for a stratiform earth containing multiple transitional layers in which the conductivity varies linearly with depth. In the model under consideration, any one homogeneous layer with constant conductivity or transitional one may be absent in the geometry. The traditional one-dimensional (1D) models with sharp boundaries will be obtained if all the transitional layers are absent in the geometry, while a special 1D model consisting of a sequence of contiguous transitional layers may be obtained if all the homogeneous layers (except the basement layer) are removed from the geometry. The tangential electric and magnetic fields as well as the surface impedance are analytically expressed by Airy functions. The analytical formula is validated in three theoretical examples by comparing with the results from available codes. The apparent resistivity and impedance phase on the surface of three different transitional models are illustrated to analysis the influence of the transitional layers on MT responses. The new formula provides an alternative way to obtain the analytic MT responses for the special layered earth. [ABSTRACT FROM AUTHOR]

The problem of constructing asymptotics of the internal gravity waves far fields arising from an pulsed localized source of perturbations in a stratified fluid of finite depth rotating as a whole is solved. In the approximation of constancy of the buoyancy frequency, uniform and nonuniform asymptotics of solutions are constructed to describe far wave fields, which are expressed in terms of the Airy function and its derivative. The exact and asymptotic results are compared, and it is shown that at times longer than several buoyancy periods and at distances on the order of the liquid-layer thickness, the obtained asymptotics allow one to describe the amplitude–phase structure of far wave fields. [ABSTRACT FROM AUTHOR]

In this work, the analytical properties of dispersion relations of the equation of internal gravity waves with benchmark and arbitrary distributions of buoyancy frequency are investigated. To solve the problem analytically, the benchmark distribution of the buoyancy frequency is used, which is known from applied oceanological calculations in the presence of seasonal thermocline. Implicit forms of dispersion dependences are obtained; they are expressed through the Bessel function of real index. For nonzero wave numbers, an asymptotic method of studying the dispersion relation is proposed based on constructing uniform asymptotics of the Bessel functions for large values of real index and argument, expressed through the Airy functions. For an arbitrary distribution of buoyancy frequency, the asymptotic representations of dispersions relationships at small wavenumbers are obtained by means of the perturbation method and the WKB method. The solutions constructed in this work allow further computing the amplitude-phase characteristics for the fields of internal gravity waves with benchmark and arbitrary distributions of the buoyancy frequency. [ABSTRACT FROM AUTHOR]

Domains with elliptic boundaries are regularly pursued in Elasticity to elucidate the role of circular asymmetry and are associated with many classical closed form solutions. The work under consideration presents a unified semi-analytical approach to solve for stress and displacement field while being subjected to all kinds of plausible boundary conditions on any variant of elliptic geometry i.e. elliptic cylinder to confocal elliptic annulus to elliptic hole in an infinite plane. The generalized representation of Airy stress function in elliptical co-ordinates truncated to finite terms is considered and the associated coefficients are deduced to ensure boundary conditions using collocation technique. The correctness and effectiveness of the method is demonstrated through solution to a variety of problems and its validation via an independent finite element simulation. [ABSTRACT FROM AUTHOR]

Analytic modeling of gravitational waves (GWs) from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation (SPA). In this work, we study this breakdown within the context of catastrophe theory. We find that the SPA holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies f m i n < f < f m a x , where f m i n / m a x are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: (1) the development and improvement of effective fly-by waveforms for binary black holes, (2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric GW bursts and Bose–Einstein condensates, and (3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of GWs from eccentric binaries within the context of post-Newtonian theory. [ABSTRACT FROM AUTHOR]

The current study investigates the nonlinear vibration characteristic of rotating axially moving conical shell made of shape memory alloy (SMA). For this purpose, the material behavior of SMA is simulated via Boyd-Lagoudas and Brinson models, and three nonlinear governing equations are derived by employing Hamilton principle, Donnell's nonlinear theory assumptions, and SMA constitutive relations. By applying a suitable parametric airy stress function, three nonlinear equations of motion are reduced to one in radial direction, which must be solved with the help of the compatibility equation. By the aid of Jordan conical form and applying the Galerkin method on the equilibrium equation in the radial direction, seven nonlinear nonhomogeneous ODEs are resulted. Then, the set of nonlinear equations is solved using the fourth-order Runge–Kutta method and pseudo-arc length continuation. Furthermore, the bifurcation analysis based on the different parameters especially frequency responses along with the curves of the time histories and phase portraits mention the influence of different phases of the material, axial motion and spinning on the conical shells made of SMA. The results of the present work are validated with available approved data, which shows good agreements. [ABSTRACT FROM AUTHOR]

Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, as displayed in the well-known von Kármán equations set. The present study aims at providing explicit expressions for the membrane stresses, the deflections, and the Airy stress function for a general square plate area vertically uniformly loaded to reach large deflection state. This was obtained by using the results of a high-fidelity finite element analysis applied on a lateral loaded simply supported thin square plate, which are then casted to yield approximate Fourier series expressions for the membrane stresses, deflections, and the Airy stress function. The stress map figures provide a good understanding of the critical points on the plate, while the explicit mathematical expressions enabled the calculation of deflections and stresses for the entire plate area. Among other interesting findings, the presence of relatively high tensile and compressive membrane stresses existing near the plate edges was revealed, which might lead to potential failure hazards. The derivatives of the deflections and the Airy stress function enabled the validation of the large deflections von Kármán equations set for the investigated case, and it turned out that the generated expressions for the stresses and the lateral deflection based on a high-fidelity finite element model satisfy the second equation with a good accuracy, while the first one remains to further be investigated. Moreover, using the generated explicit equations, the load influence on the deflections and stresses was also analyzed to yield general novel expressions for the medium and very large deflections states. To generalize the investigated case, the stresses and the deflections were non-dimensionalized so they can be used for any material and plate dimensions. [ABSTRACT FROM AUTHOR]

We consider a connection problem of the first Painlev\'e equation (PI), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable t tends to the negative infinity for real PI functions. We get a classification of the real PI functions in terms of (p, H) so that they behave differently at the negative infinity, where p is the location of a pole and H is the free parameter in the Laurent series. Some limiting-form connection formulas of PI functions are obtained for large H. Specifically, for the real tritronqu\'ee solution, the large-n asymptotic formulas of pn and Hn are obtained, where pn is the nth pole on the real line in the ascending order and Hn is the associated free parameter. Our approach is based on the complex WKB method (also known as the method of uniform asymptotics) introduced by Bassom et al. in their study on the connection problem of the second Painlev\'e transcendent [Arch. Ration. Mech. Anal., 143 (1998), pp. 241--271]. Several numerical simulations are carried out to verify our main results. Meanwhile, we obtain the phase diagram of P\I solutions in the (p, H) plane, which somewhat resembles the Brillouin zones in solid-state physics. The asymptotic and numerical results obtained in this paper partially answer Clarkson's open question on the connection problem of the first Painlev\'e transcendent. [ABSTRACT FROM AUTHOR]

A new power-exponent-phase vortex beam with nonlinear phase winding has shown flexible control freedom compared with conventional vortex beams. In order to further enrich the modulation freedom and expand the ability of self-healing to meet current application requirements, we conducted a detailed study on the characteristics of the Airy transform of the new power-exponent-phase vortex beam. The influences of the Airy function, the power exponent, and the topological charge on normalized intensity and phase distributions are investigated theoretically and experimentally. More importantly, the self-healing properties of the new power-exponent-phase vortex beam with and without the Airy transform are compared. This shows that the new power-exponent-phase vortex beam with the Airy transform exhibits better self-healing ability when obstructed by obstacles. This study has potential applications in optical trapping and free-space optical communication. [ABSTRACT FROM AUTHOR]

This paper develops a coupled chemo-mechanical model for stress-assisted diffusion in the framework of micropolar elasticity. The two-way interaction of mechanical and chemical driving forces as well as internal microstructure of the polar media is taken into account. The fundamental governing equations of chemo-mechanics with kinetics driven by diffusion and stress are developed, and mathematical expressions for stress and concentration fields are derived. Using Airy stress functions, a simplified formulation for two-dimensional chemo-elasticity problems under chemical equilibrium is presented. Using a perturbation approach to solve the presented system of equations, closed-form expressions for different field parameters can be obtained. As an illustrative case study, expressions for the stress and solute concentration in an infinite plate with circular hole embedded in a chemical medium have been written. The derived equation and proposed solution approach can be applied to various plane micropolar chemo-elasticity problems for studying the interaction between mechanical and chemical driving forces as well as material length-scale parameters. [ABSTRACT FROM AUTHOR]

Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of spider web-inspired web structures. This work generalizes previous results on the existence of cable webs that are able to support assigned sets of nodal forces under tension. We extend such a problem to the limit analysis of compression-only "strut nets" subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces ("underlying masonry structures"). It is assumed that fixed nodal forces are combined with variable forces growing proportionally to a scalar multiplier (load multiplier), and that the supporting strut net is subjected to kinematic constraints at given nodal positions. [ABSTRACT FROM AUTHOR]

The nonlinear vibration characteristics of the rotating axially moving circular cylindrical shells in subharmonic regions are investigated in the present paper. The motion equations are carried out based on the Hamilton principle in cylindrical coordinates utilizing Donnell's nonlinear shell theory. By introducing the suitable airy stress function, three equilibrium equations in the cylindrical coordinates are simplified into two nonlinear coupled nonhomogeneous PDEs, including a compatibility equation and the transverse motion equation. The compatibility equation solution is obtained employing the seven degrees of freedom for the flexural mode shape of the system. By implementation of the Galerkin method, the motion equation would be projected into seven nonlinear coupled nonhomogeneous ODEs. This set of equations is solved using a direct normal form method validated by the numerical method and available data. The effect of angular velocity and axial speed is investigated employing frequency and force response curves, bifurcation diagrams, time history, and the system's phase portraits. Always axially moving and rotation speed of the system intensifies the nonlinear behavior of the frequency responses. [ABSTRACT FROM AUTHOR]