String Equation Research Articles

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

AbstractThe paper is concerned with the Bari basis property of a boundary value problem associated in with the following 2 × 2 Dirac‐type equation for : with a potential matrix and subject to the strictly regular boundary conditions . If , this equation is equivalent to one‐dimensional Dirac equation. We show that the normalized system of root vectors of the operator is a Bari basis in if and only if the unperturbed operator is self‐adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let , , boundary conditions be strictly regular, and let be the sequence biorthogonal to the normalized system of root vectors of the operator . Then, These abstract results are applied to noncanonical initial‐boundary value problem for a damped string equation.

Desingularization of Black Hole Space-Times

The desingularization of a class of black hole space-times arising as solutions to the string equations is considered in connection with the consistency of the quantum theory and the description of nonperturbative states with quantum numbers from the particle spectrum. The geometry arising in an extreme limit of one of the singular solutions to the gravitational field equations is demonstrated to be a background of N = 2 string theory. The positivity of the masses in the particle spectrum is proven through quasilocal integrals near the resolved singularities in these limits of black hole space-times.

Anomalous curvature evolution and geometric regularization of energy focusing in the snapping dynamics of a flexible body

We examine the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a flexible structure. This brief but violent event appears to be a regularized finite-time singularity, with remarkably large spikes in velocity, acceleration, and tension easily induced by generic initial and boundary conditions. A numerical scheme for the inextensible string equations is validated against available experimental data for a falling chain and further employed to explore the phenomenon. We determined that the discretization of the equations, equivalent to the physically discrete problem of a chain, does not provide the regularizing length scale, which in the absence of other physical effects must then arise from the geometry of the problem. An analytical solution for a geometrically singular limit, a falling perfectly folded string, accounts surprisingly well for the scalings of several quantities in the numerics, but can only indirectly suggest a behavior for the curvature, one which seems to explain prior experimental data but does not correspond to the evolution of the curvature peak in our system, which instead displays a newly observed anomalously slow scaling. A simple model, incorporating only knowledge of the initial conditions along with the anomalous and singular-limit scalings, provides reasonable estimates for the amplifications of relevant quantities. This is a first step to predict and harness arbitrarily large energy focusing in structures, with a practical limit set only by length scales present in the discrete mechanical system or the initial conditions.

Direct and Inverse Problems of String Equation by Numerov’s Method

It is well known that free vibration of a taut string having mass per unit m(x) and frequency $$\omega$$ is governed by ordinary differential equation $$y^{\prime \prime }+\omega ^2m(x)y=0.$$ In this paper, first we discretize the differential equation by using Numerov’s method to obtain a matrix eigenvalue problem of the form $$-Au=\Lambda B M u$$ , where A and B are constant tridiagonal matrices and M is a diagonal matrix related to mass function m(x). In direct problem, for a given m(x), we approximate the first N eigenvalues of the string equation by making a new correction on the eigenvalues of matrix pair $$(-A,BM)$$ . Also we obtain the error order of corrected eigenvalues. For inverse problem, we propose an efficient algorithm for constructing unknown mass function m(x) by using given spectra by solving a nonlinear system. We solve the nonlinear system by using modified Newton’s method and a regularization technique. The convergence of Newton’s method is proved. Finally, we give some numerical examples to illustrate the efficiency of the proposed algorithm.

Blowup equations for little strings

We propose blowup equations for 6d little string theories which generalize Nakajima-Yoshioka’s blowup equations for the 4d/5d instanton partition functions on Omega background. We find that unlike the blowup equations for standard SQFTs, we need to sum over auxiliary magnetic fluxes on the blown-up ℙ1 for a non-dynamical 2-form gauge field which plays a role in canceling the mixed anomalies of the gauge symmetries. We demonstrate with explicit examples that the blowup equations, when combined with the modular properties, can be solved in order to determine the elliptic genera of little strings.

A Globally Smooth Solution to the Relativistic String Equation

A Globally Smooth Solution to the Relativistic String Equation

The Use of Fourier Transform in Heat Equation and Wave Equation

In the solve for the equation for heat transfer and wave movement, using Fourier Transform to simplify the differential equation and solve it. With the periodic property of the wave including standing wave and vibrating string and heat equation and its stable solution, it can provide many new properties of the equation. Then using the same method to solve the heat equation in space, it gives a similar equation like the plane heat transfer equation. The passage mainly discussed about using Fourier transform to solve standing waves equation heat equation in steady state and in the time dependent state. The general way of solving wave equation both standing wave and traveling wave) is finding differential equation for a half period, expending it into the R, using Fourier transform to solve the equation.

Fourier series and its property

Fourier analysis appeared to solve some partial differential equations at first. In 1747, D’ Alembert solved the vibrating string equation using the method of traveling waves. In 1753, D. Bernoulli proposed the solution which for all intents and purposes is the Fourier series, but Euler did not convince of its full generality entirely since he could not make sure whether any function could be expanded in the Fourier seriesFourier analysis is an important theorem of modern mathematics. It deeply influences the development of partial differential equations and information science. This paper will talk about some basic propositions of the Fourier series and some simple theorems of the Fourier transform. It will use an algebraic view to talk about convolution and good kernels. It will refer to some methods of Hilbert space and complex analysis.

Global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law

AbstractIn this paper, we consider the global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law. First, we will give the representation of the traveling wave solution to the recast hyperbolic system. Then, by using the weighted function method, we obtain the global stability of longitudinal wave solution for nonlinear elastic string equation.

Inverse Nodal Problem for p-Laplacian String Equation with Prüfer Substitution

We consider an inverse nodal problem for p-Laplacian string equation under some boundary conditions. Asymptotic formulas for eigenvalues and nodal parameters are constructed by modified Pr¨ufer substitution. The most important process is to apply modified Pr¨ufer substitution to get an exhaustive asymptotic estimate for eigenvalues. Moreover, a reconstruction formula for density function of p-aplacian string equation is obtained by nodal parameters. Generated outcomes are the generalization of the known string problem.

High-accuracy estimation method of typhoon storm surge disaster loss under small sample conditions by information diffusion model coupled with machine learning models

Typhoon storm surge disaster is the most severe marine disaster in China. Accurate estimation of typhoon storm surge disaster loss (TSSDL) is significant for emergency decisions and economic sustainability development. However, the TSSDL estimation is limited by small sample conditions, resulting in the low accuracy of the TSSDL estimation model based on machine learning. To solve the problems of easy overfitting and poor generalization ability of machine learning models under small sample conditions, the high-accuracy TSSDL estimation method was proposed. Firstly, this estimation method combines Gaussian Noise with the Information Diffusion Model based on the Vibrating String equation (named GN-VSIDM) to generate virtual samples to augment the original training set. Then, the augmented training set was applied to train three machine learning models. The results are as follows: the virtual sample generation method, i.e., GN-VSIDM, solves the small sample problem in the TSSDL estimation process and improves the machine models’ accuracy and robustness. Based on the GN-VSIDM and the eXtreme Gradient Boosting (named XGBoost) methods, the joint model GN-VSIDM-XGBoost is the optimal TSSDL estimation model. Compared with the original XGBoost model, the RMSE and R2 of the GN-VSIDM-XGBoost model are 0.1089 and 0.8292, which reduces 25.67% and improves 19.88%, respectively. Besides, the GN-VSIDM-XGBoost model possesses excellent robustness. The GN-VSIDM overcomes the limitation on the performance of TSSDL estimation models based on machine learning under small sample conditions. This study provides an effective case and method for solving the small sample problem in disaster loss assessment.

A Construction of Open Descendant Potentials in All Genera

Abstract We present a construction of an open analogue of total descendant and total ancestor potentials via an “open version” of Givental’s action. Our construction gives a genus expansion for an arbitrary solution to the open WDVV equations satisfying a semisimplicity condition and admitting a unit. We show that the open total descendant potentials we define satisfy the open topological recursion relations in genus $0$ and $1$, the open string and open dilaton equations. We finish the paper with a computation of the simplest nontrivial open correlator in genus $1$ using our construction.

Characteristic analysis and experimental study of rotary impact screw drilling tool

With the development of petroleum exploration and drilling, the current drilling tools are facing more difficulty situations, which lead to these problems such as low drilling speed, weak rock breaking ability or short service time. Based on this background, we propose a new drilling tool is proposed in this paper. By analyzing the force of the cam roller system, the dynamic differential equations of the horizontal drill string are established. The key parameters’ influences on the axial impact force of the new tool are studied first. Based on the new outputs, the vibration characteristics of drill string system are studied. For discovering the internal mechanism, the impact part dynamic analysis of this new tool is carried out, so as the contact force between roller and cam, the impact force between upper shaft body and pad. Furthermore, for verifying the rationality and accuracy of theoretical calculation, these parameters’ results are compared and analyzed under different drilling weights. Through laboratory and field tests, the impact frequency and impact force of this new drilling tool are analyzed, the effectiveness of theoretical models are also demonstrated. This research can provide an effective technical references to solve the current drilling difficulty in new situations. HIGHLIGHTS The rotary impact screw drilling tool has the function of impact while rotating, realizing the combination of shear and impact rock breaking, so as to improve the mechanical drilling speed. The impact load on the drill bit is produced by the combined action of hydraulic and mechanical forces. The impact force is generated in the form of mechanical structure movement, and it is a continuous and stable periodic impact force, which makes the bottomhole assembly produce benign vibration. Through experiments, the effectiveness of theoretical calculation and dynamic model is proved successfully, and the drilling tool can effectively improve the drilling rate.

On Poncelet problem, Abel equation and a Goursat-type boundary value problem

This paper explains a connection between the famous Poncelet problem, the Abel equation, a new moment problem, problems of solution existence and uniqueness of boundary value problems for the string equation and some others in short presentation. We prove a connection of above problem with one problem of Goursat-type for a hyperbolic-like equation of fourth order. We consider a fourth-order differential equation with two multiple characteristics and parameters in the lower terms and a Goursat-type homogeneous boundary value problem with two real boundary conditions in some space of real entire functions. It is proven that this problem has a nontrivial solution in some space if and only if a parameter belongs denumerable dense set of real number.

Sharp stability of a string with local degenerate Kelvin–Voigt damping

AbstractThis paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping where on , and on for . It is known that the optimal decay rate of solution is in the limit case and exponential for . When , the damping coefficient is continuous, but its derivative has a singularity at the interface . In this case, the best known decay rate is , which fails to match the optimal one at . In this paper, we obtain a sharper polynomial decay rate . More significantly, it is consistent with the optimal polynomial decay rate at and uniform boundedness of the resolvent operator on the imaginary axis at (consequently, the exponential decay rate at as ). This is a big step toward the goal of obtaining eventually the optimal decay rate.

Weighted string equation where the weight is a noncompact multiplier: continuous spectrum and eigenvalues

The oscillation equation for a singular string with discrete weight generated by a self-similar n n -link multiplier in the Sobolev space with a negative smoothness index is considered. It is shown that in the case of a noncompact multiplier, the string problem is equivalent to the spectral problem for an ( n − 1 ) (n-1) -periodic Jacobi matrix. In the case of n = 3 n=3 , a complete description of the spectrum of the problem is given, and a criterion for emergence of an eigenvalue in a gap of the continuous spectrum is obtained.

Irreducible Metric Maps and Weil–Petersson Volumes

We consider maps on a surface of genus g with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus g and fixed number n of faces with circumferences alpha _1,ldots ,alpha _n and a beta -irreducibility constraint, which roughly requires that all contractible cycles have length at least beta . Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume V_{g,n}^{(beta )}(alpha _1,ldots ,alpha _n) of this family of maps that arises naturally from the Lebesgue measure on the edge lengths. It is shown to be a homogeneous polynomial in beta , alpha _1,ldots , alpha _n of degree 6g-6+2n and to satisfy string and dilaton equations. Surprisingly, for g=0,1 and beta =2pi the volume V_{g,n}^{(2pi )} is identical, up to powers of two, to the Weil–Petersson volume V_{g,n}^{mathrm {WP}} of hyperbolic surfaces of genus g and n geodesic boundary components of length L_i = sqrt{alpha _i^2 - 4pi ^2}, i=1,ldots ,n. For genus gge 2 the identity between the volumes fails, but we provide explicit generating functions for both types of volumes, demonstrating that they are closely related. Finally we discuss the possibility of bijective interpretations via hyperbolic polyhedra.

On Polynomials Counting Essentially Irreducible Maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.

W algebra and eigenvalues of the low-order string constraint operators for the discrete KP hierarchy

Firstly, for any p ∈ N , the string constraint operators without the redundant variables are calculated, which are imposed by the string equation on the τ -function of the p -reduced discrete Kadomtsev–Petviashvili (dKP) hierarchy. Furthermore, the commutators of the string constraint operators without the redundant variables are calculated. It is found that although the string constraint operators without the redundant variables are different from the original string constraint operators with the redundant variables, they also form a Virasoro algebra and a W algebra after a transformation. Moreover, using the algebraic structure, eigenvalues of the low-order string equation constraint operators are calculated. Based on the obtained eigenvalues, it is also proved that the τ -function of the p -reduced dKP hierarchy constrained by the string equation is a vacuum vector for a Witt algebra. When p = 2 , it is coincident with the classical fact that the τ -function of the Korteweg–de Vries hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra.