Discretization Of Integral Equations Research Articles

B\"{a}cklund transformations as integrable discretization. The geometric approach

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that B\"{a}cklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

High order multiscale analysis of discrete integrable equations

In this article we present the results obtained applying the multiple scale expansion up to the order $\varepsilon^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which seem to be new, depending at most on 2 parameters.

Quasi-periodic solutions of a discrete integrable equation with a finite-dimensional integrable symplectic structure

Through the paper, we research the quasi-periodic solutions to a semi-discrete hierarchy which has Hamiltonian structure and integrable symplectic map. Firstly, a semi-discrete hierarchy is derived by use of the discrete zero-curvature representation and then its integrability is proved under the Liouville condition. Using the binary nonlinearization approach, the integrable symplectic map and finite-dimensional Hamiltonian system of the hierarchy are obtained. Moreover, the trigonal curve Kq−1 is denoted through the characteristic polynomials for the Lax pair, as well as the related Baker–Akhiezer function and meromorphic function are introduced, from which the asymptotic properties and divisors of the two functions mentioned above are analyzed. Finally, we introduce the three kinds of Abel differentials and straighten out of corresponding continuous and discrete flows, the quasi-periodic solutions of the equations are received via the Riemann theta function.

A GPU parallel randomized CUR compression method for the Method of Moments

In this work, we propose a GPU parallel implementation of the randomized CUR (or Pseudo Skeleton) Approximation to compress the H-matrices of linear systems that arise in the discretization of integral equations modeling electromagnetic scattering problems. This compression method is highly parallelizable, in contrast with other similar methods such as the Adaptive Cross Approximation. It involves dense linear algebra computations that can be efficiently implemented on a GPU device. Besides, a stochastic convergence criterion is introduced to minimize the communication between the host and the device. Testing the code with standard cases shows the efficiency and accuracy of the method.

Characteristic analysis of scattering field in two-layer media by Green’s function

The problem of three-dimensional (3D) acoustic scattering in a complex medium has aroused considerable interest of researchers for many years. An ultrasonic scattered field calculating technique is proposed to study the scattering echo from strongly scattered materials in a two-layer medium in this work. Firstly, with the high frequency stationary phase method, the Green's function of two-layer fluid media is derived. And then based on the idea of integral equation discretization, the Green’s function method is extended to two-layer fluid media to derive the scattering field expression of defects in a complex medium. With this method, the scattering field of 3D defect in a two-layer medium is calculated and the characteristics of received echoes are studied. The results show that this method is able to solve the scattering P wave field of 3D defect with arbitrary shape at any scattering intensity in two-layer media. Considering the circumstance of water-immersion ultrasonic non-destructive test (NDT), the scattering sound field characteristics of different types of defects are analyzed by simulation, which will help to optimize the detection scheme and corresponding imaging method in practice so as to improve the detection quality.

Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Abstract In order to explore the fractional differential equations in accounting informatization financial software, the author proposes a system for fractional diffusion wave equations and fractional differential equations, two numerical algorithms with higher precision are given, and the amount of computation is reduced at the same time. First, based on the equivalent integral form of the time fractional diffusion wave equation, using the fractional echelon method and the Crank-Nicolson method, for the time fractional diffusion wave equation, a finite difference scheme is designed, this format has second-order accuracy in both the temporal and spatial directions and is computationally stable. Numerical examples verify the accuracy and effectiveness of this format. Then when dealing with the initial value problem of fractional differential equations with Caputo derivative operator, convert it to the equivalent Voltera integral equation system, an initial approximate solution is obtained by a low-order method, derive the residual and error equations, the idea of applying the stepwise correction of spectral delay correction improves the numerical accuracy of the solution, at the same time, the Richard Askey integral equation is used to reduce the amount of calculation. At last, the high precision and effectiveness of the new method are verified by numerical experiments. Experiments show that: Starting from the equivalent integral form of the fractional diffusion wave equation, a second-order finite-difference scheme of the fractional-order diffusive wave equation is constructed, through numerical experiments, it is verified that the scheme has good accuracy and efficiency. In numerical solution, discrete integral equations have better numerical stability than differential equations, therefore, the format also has better stability. When taking different fractional derivative indices a=1.5 and a=1.8, it can be seen that the difference format constructed by the author, in the time direction, has second-order precision, as expected.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3×3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3×3 Lax pair of this equation, the discrete generalized (m,3N−m)-fold Darboux transformation was constructed for the first time and extended from the 2×2 Lax pair to the 3×3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants.

On the reconstruction of the attenuation function of a return-stroke current from the Fourier Transform of finite-duration measurements

A correct representation of the lightning current is crucial when the electromagnetic field radiated to a point of interest has to be computed. Based on the engineering models of Transmission Line type, such representation involves the knowledge of the return-stroke speed, the channel-base current, the channel height and the attenuation function. Whereas the first three quantities can be measured in different ways, no measurement technique can directly provide reliable information on the attenuation function. In the past decades, researchers have applied various strategies to address this problem. These strategies are all based on a common feature, i.e., the unknown function is postulated a priori and then validated through the comparison of the computed electromagnetic fields at one or more observation points with the corresponding measured waveforms. In this paper, we propose an alternative approach for the identification of the attenuation function: starting from appropriate measurements of the return-stroke speed, the channel-base current, the channel height and the radiated electromagnetic field, we first formulate an algebraic inverse and ill-posed problem, obtained from the discretization of integral equations relating the source to the radiated field in the frequency domain, and then we solve it by means of a Tikhonov regularization technique. The proposed framework is preceded by a detailed theoretical analysis, with special emphasis on the description and filtering of measurement noise and on the minimum duration of the measurement time-windows ensuring reliable results.

Discrete integrable equations on face-centred cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centred quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang–Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centred-cube (CAFCC), which requires consistency of an overdetermined system of 14 five-point equations on the face-centred cubic unit cell. In this paper, a new formulation of CAFCC is introduced where so-called type C equations are centred at faces of the face-centred cubic unit cell, whereas previously they were only centred at corners. This allows type C equations to be regarded as independent multidimensionally consistent integrable systems on higher dimensional lattices and is used to establish their Lax pairs.

Volterra-Type Discrete Integral Equations and Spectra of Non-self-adjoint Jacobi Operators

Volterra-Type Discrete Integral Equations and Spectra of Non-self-adjoint Jacobi Operators

Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations that satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube integrability condition applicable to five-point equations in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which, namely, involves satisfying an overdetermined system of 14 five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star–triangle relations associated with the Adler–Bobenko–Suris list, 15 sets of equations are obtained, which satisfy CAFCC.

Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure

In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by using discrete non-point transformations. We use both non-invertible linearizable transformations and non-point transformations invertible on solutions of the discrete equation. As a result, we get several series of new examples of discrete equations along with their generalized symmetries and master symmetries. The generalized symmetries constructed give new integrable examples of five- and seven-point differential-difference equations together with their master symmetries. In the case of discrete equations, the method of constructing non-invertible linearizable transformations by using conservation laws is considered, apparently, for the first time.

Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure

В недавней работе авторов была построена серия интегрируемых дискретных автономных уравнений на квадратной решетке с нестандартной структурой высших симметрий. Здесь построены модифицированные серии с использованием дискретных неточечных преобразований. Используются как необратимые линеаризуемые преобразования, так и неточечные преобразования, обратимые на решениях дискретного уравнения. В результате получены серии новых примеров дискретных уравнений вместе с их высшими симметриями и мастер-симметриями. Построенные высшие симметрии дают новые интегрируемые примеры пяти- и семиточечных дифференциально-разностных уравнений вместе с их мастер-симметриями. В случае дискретных уравнений метод построения необратимых линеаризуемых преобразований при помощи законов сохранения рассматривается, по-видимому, впервые.

Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools

Surface currents provide a general way to model magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of the arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Lastly, we share examples related to magnetic shielding and the surface-coil design using the presented tools.

Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

AbstractWe study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

Discrete nonlocal nonlinear Schrödinger systems: Integrability, inverse scattering and solitons

A number of integrable nonlocal discrete nonlinear Schrödinger (NLS) type systems have been recently proposed. They arise from integrable symmetry reductions of the well-known Ablowitz–Ladik scattering problem. The equations include: the classical integrable discrete NLS equation, integrable nonlocal: PT symmetric, reverse space time (RST), and the reverse time (RT) discrete NLS equations. Their mathematical structure is particularly rich. The inverse scattering transforms (IST) for the nonlocal discrete PT symmetric NLS corresponding to decaying boundary conditions was outlined earlier. In this paper, a detailed study of the IST applied to the PT symmetric, RST and RT integrable discrete NLS equations is carried out for rapidly decaying boundary conditions. This includes the direct and inverse scattering problem, symmetries of the eigenfunctions and scattering data. The general linearization method is based on a discrete nonlocal Riemann–Hilbert approach. For each discrete nonlocal NLS equation, an explicit one soliton solution is provided. Interestingly, certain one soliton solutions of the discrete PT symmetric NLS equation satisfy nonlocal discrete analogs of discrete elliptic function/Painlevé-type equations.

A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration

The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has already been developed to discretize this well-posed equation. However, the integral computation and solution of the large multiscale collocation integral equation are two time-consuming processes. To overcome these difficulties, we propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integral, which efficiently converts the integral operation to the matrix operation and reduces costs. In addition, we also propose a multilevel iteration method (MIM) to solve the fully discrete integral equation obtained from the integral approximation strategy. Herein, the stopping criterion and the computation complexity that correspond to the MIM are shown. Furthermore, a posteriori parameter choice strategy is developed for this method, and the final convergence order is evaluated. We present three numerical experiments to display the performance and computation efficiency of our proposed methods.

Tetrahedron maps and symmetries of three dimensional integrable discrete equations

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z3 is investigated. Our approach is a generalization of a method developed in the context of Yang–Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case–by–case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.

Algebraic entropy computations for lattice equations: why initial value problems do matter

In this letter we show that the results of degree growth (algebraic entropy) calculations for lattice equations strongly depend on the initial value problem that one chooses. We consider two problematic types of initial value configurations, one with problems in the past light-cone, the other one causing interference in the future light-cone, and apply them to Hirota’s discrete KdV equation and to the discrete Liouville equation. Both of these initial value problems lead to exponential degree growth for Hirota’s dKdV, the quintessential integrable lattice equation. For the discrete Liouville equation, though it is linearizable, one of the initial value problems yields exponential degree growth whereas the other is shown to yield non-polynomial (though still sub-exponential) growth. These results are in contrast to the common belief that discrete integrable equations must have polynomial growth and that linearizable equations necessarily have linear degree growth, regardless of the initial value problem one imposes. Finally, as a possible remedy for one of the observed anomalies, we also propose basing integrability tests that use growth criteria on the degree growth of a single initial value instead of all the initial values.

Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

By considering the normalizers of reflection subgroups of types A (1) 1 and A (1) 3 in W ~ ( D 5 ( 1 ) ) , two subgroups: W ~ ( A 3 × A 1 ) ( 1 ) ⋉ W ( A 1 ( 1 ) ) and W ~ ( A 1 × A 1 ) ( 1 ) ⋉ W ( A 3 ( 1 ) ) can be constructed from a ( A 3 × A 1 × A 1 ) (1) type subroot system. These two symmetries arose in the studies of discrete Painlevé equations (Kajiwara K, Noumi M, Yamada Y. 2002 q -Painlevé systems arising from q -KP hierarchy. Lett. Math. Phys. 62 , 259–268; Takenawa T. 2003 Weyl group symmetry of type D (1) 5 in the q -Painlevé V equation. Funkcial. Ekvac. 46 , 173–186; Okubo N, Suzuki T. 2018 Generalized q -Painlevé VI systems of type ( A 2 n +1 + A 1 + A 1 ) (1) arising from cluster algebra. ( http://arxiv.org/abs/quant-ph/1810.03252 )), where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups (Howlett RB. 1980 Normalizers of parabolic subgroups of reflection groups. J. London Math. Soc. (2) 21 , 62–80; Brink B, Howlett RB. 1999 Normalizers of parabolic subgroups in Coxeter groups. Invent. Math. 136 , 323–351). This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.