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In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen–Cahn type in R n u ( x ) = l ⃗ + C * ∫ R n u ( y ) ( 1 − | u ( y ) | 2 ) | 1 − | u ( y ) | 2 | p − 2 | x − y | n − α d y , when |x| → ∞. Here u : R n → R k is uniformly continuous, and k ⩾ 1, n ⩾ 2, α ∈ (0, n) and }\frac{n}{n-\alpha }$?> p − 1 > n n − α . In addition, l ⃗ ∈ R k is a constant vector and C * is a real constant. If 1 − | u | 2 ∈ L s ( R n ) for some s ∈ [1, ∞), we know that |u| → 1 when |x| → ∞. Furthermore, we prove that if 1 − | u | 2 ∈ L s ( R n ) for some s ∈ [ 1 , n α ( p − 1 ) ) , then u → l ⃗ when |x| → ∞, and hence | l ⃗ | = 1 . When 1 − | u | 2 ∈ L s ( R n ) for some s ∈ [ 1 , n α ( p − 2 ) ) , then there exists some positive constant C such that |1 − |u(x)|2| ⩽ C|x| α−n for large |x|. Here the Harnack type estimate and the regularity lifting lemma come into play in those proofs.

We study the existence and uniqueness as well as the asymptotic behaviour of solutions of a certain boundary-value problem for a convolution integral equation on the whole line with monotone non-linearity. In some special cases, there are concrete applications to -adic string theory, the mathematical theory of the geographical spread of an epidemic, the kinetic theory of gases and the theory of radiation transfer. We prove the existence and uniqueness of an odd bounded continuous solution. The monotonicity and the integral asymptotics of this solution is also discussed. We finally give particular application-oriented examples of the equations considered, which illustrate the special nature of our results.

Using convexity properties of the images of completely continuous non-linear integral operators, we describe the closed convex cones lying either in the recessive cone, or in the tangent cone of the closed image of the operator being studied (depending on the nature of the integrand). These cones are determined by the principal part of the asymptotics of the integrand at infinity, independently of the variation of the subordinate part. We discuss applications to the generalized solubility of non-linear integral equations of the first kind.

The paper aims at solving a complex equation with Gamma - integral. The solution is the infected size (p) at equilibrium. The approaches are both numerical and analytical methods. As a numerical method, the higher-order composite Newton-Cotes formula is developed and implemented. The results show that the infected size (p) increases along with the shape parameter (k). But the increase has two phases: an increasing rate phase and a decreasing rate phase; both phases can be explained by the instantaneous death rate characteristics of the Gamma distribution hazard function. As an analytical method, the Extreme Value Theory consolidates the numerical solutions of the infected size (p) when k ≥ 1 and provides a solution limit ( p = 1 − 1 2 R ) as k goes to +∞.

The problem of excitation of a rectilinear ribbon vibrator near a perfectly conducting infinite screen is considered. A general equation, a two-dimensional integral equation, and a one-dimensional integral equation with respect to the current density are obtained. The results of numerical calculations are presented.

Data-driven reduced order models (ROMs) are combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.

In this work, we adopt B. spline function of two variables basis for solving linear multi-dimensional delay Volterra integral equations nonhomogeneous of the second type. We employ the two methods to approximate solution via B. spline function of two variables basis that yields linear system. Some examples are given, the results shown in tables and figures. These methods are very effective, convenient and overcome the difficulty of traditional methods. We solve this problem with the assistance of Matlab18.

The nonlinear evolution equations of fourth-order have been established for two surface gravity waves in infinite depth of water including the effect of air flowing over water. In the present paper, we have applied a general approach depending upon the integral equation due to Zakharov. Based on these equations, the stability analysis has been studied in the appearance of a uniform gravity wave packet with another wave packet of the identical group velocity. Graphs have been drawn for the instability growth rate of the uniform wave packet with shorter wave number versus the perturbed wave number for several values of the amplitude of the wave packet of larger wave-number and several values of wind velocity. We have observed from the figures that the instability growth rate of the second wave packet enhances with the enhancement of nondimensional wind velocity for the settled value of the amplitude of the first wave packet.

The ground state of the linear harmonic oscillator is the solution to a minimization problem, and this opens up two new ways to completely solve Schrödinger’s time-independent equation. The first exploits systematic patterns that emerge from a formal relationship between two different integral representations of the energy. A remarkable invariant form of the ground-state probability-density equation under specific conditions is the leading principle of the second procedure, where a uniform notation for all mathematical object involved facilitates calculations and generates useful schemes to quickly deduce the main functional relationships of the problem. We believe these procedures may represent useful educational alternatives to standard treatments that address this fundamental problem.

The behaviour of the electromagnetic field in a three-mirror laser cavity is described using the method of integral equations. The results of numerical calculations and experimental studies for particular examples implementing the considered configuration are presented. The conditions for optimal tuning of a laser interferometer-deformograph with a three-mirror cavity are determined. The contribution of the reflected and scattered light and arising additional seismic noises to the resulting error of laser gravitational wave detectors is studied.

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory.

In many branches of pure analysis, Integral Equations are one of the most useful techniques, such as functional analysis theories and stochastic processes. It is one of the most significant branches of mathematical analysis, in many fields of mechanics and mathematical physics,. In this research, we will address the integral equations in many physical issues and their applications. They are also associated with mechanical vibration problems, analytic function theory, orthogonal systems, quadratic form theory of infinitely many variables.

We consider difficulties and algorithms of numerical solution of nonlinear equations of statistical physics of liquids. Analysis of nonlinear singlet equations of the surface phenomena in liquids is carried out. To solve Fredholm integral equations of second kind we compare the known algorithms and evaluate their application for linear singlet equation.

We study the scattering of - mesons using quadratic and Cornell potentials in our tetraquark ( ) system. The system’s wavefunction in the restricted gluonic basis, which is written by utilizing the adiabatic approximation and Hamiltonian, is used via a quark potential model. The resonating group technique is used to obtain the integral equations, which are solved to obtain the unknown inter-cluster dependence of the total wavefunction of our tetraquark system. T-Matrix elements are calculated from the solutions, and eventually, the scattering cross sections are obtained using the two potentials. We compare these cross sections and find that the magnitudes of scattering cross sections of quadratic potential are higher than the Cornell potential.

We present a fast Nyström algorithm to solving the inhomogeneous Lippmann– Schwinger equation for scattering wave field problems. This algorithm transforms the discrete matrix of the linear integral operator into a sparse matrix utilizing the compression property of the wavelet transform. The sparse linear equations are solved by the double conjugate gradient method. In the numerical part, we verified the feasibility and effectiveness of our method by modeling the wave field for a synthetic model.

While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

Extra-deep azimuthal resistivity measurements improve the depth of investigation up to 30 m from the wellbore. Interpretation of electromagnetic logging data in the neighbourhood of a well becomes an important technical problem. We present an efficient parallel method for computation of induction tool responses with multiple transmitter–receiver configurations in 2D pixel-based anisotropic model crossed by an arbitrary well trajectory. The cornerstone of the approach is volume integral equation method. We consider the conductivity distribution as a sum of background and anomalous conductivities. Background conductivity is 1D-layered. Anomalous conductivity has arbitrary 2D distribution. Electromagnetic fields are the superposition of background and anomalous fields. Background fields are calculated exactly using rigorous analytical solution for 1D-layered background model. With this approach, the 2D pixel-based model is treated as an extension of the 1D-layered model. The anomalous fields are required only in pixels with conductivity different from the conductivity of the 1D-layered model. Anomalous fields are calculated using convergent series of integral operators [1]. The approach takes into account conductivity anisotropy and allows obtaining both the exact solution and the fast approximate one. The convergence of approximate solution is investigated on some synthetic examples.

This paper describes new versions of the least-squares collocation method for solving differential and integral equations. A p-version of the method has been proposed and implemented to solve nonlinear systems of partial differential equations. The stationary Navier-Stokes equations are used as an example. A hp-version of the method has been implemented for the numerical solution of the Fredholm integral equations of the second kind in the one-and two-dimensional cases. This paper shows that approximate solutions obtained by various versions of the least-squares collocation method converge with a high order and agree with analytical solutions of test problems with a high degree of accuracy.

We presented a modification of the fictitious absorption method (FAM), which is used to solve integral equations (IE) of mixed dynamic problems. The method was generalized for the case of the defector stamp occupying a non-convex in plane area. For regions of complex configuration, we assumed the possibility of their representation as a union of convex bounded closed regions, possibly with common boundary sets. We proposed to modify the method in terms of the selection of basis functions that are present in the solution only under the signs of integral operators. The derivatives of Dirac delta functions with supports in the boundary sets of the considered regions were chosen as such. A more complex form of the functions used provided a more convenient form of the solution. As an illustration of the modified MFA use in the basis functions we present solutions of integral equations of the first kind for an axisymmetric problem of steady-state vibrations of an elastic layer with a clamped bottom edge under the action of a surface load and a horizontally oriented internal source.

in this paper, the approximate solution of linear Fredholm integral equations of the second type on a closed interval is studied. The Galerkin method enhanced with Chebyshev polynomials was used to improve the approximate solution of our study, the related linear system was solved by the Boole’s numerical integrated rule. for testing the efficiency of the proposed method, the comparison with related updated methods was executed.

The paper focuses on the non-classical Volterra dual equation of the first kind, to which the problem of identifying non-symmetric Volterra kernels is reduced. For this equation, we obtained difference analogs based on the application of the Runge-Kutta type method. Such equations arise in the problem of modeling nonlinear dynamical systems of black-box type, which employ the Volterra integro-power series, when the input perturbation is a vector function of time.

The study of electromagnetic scattering from Gaussian rough surface is of great significance in radar reconnaissance, target tracking and ocean remote sensing. The moment method (MOM) is a commonly used method with high accuracy. However, in the past research, Monte Carlo method is used to simulate the Gaussian rough surface. In this method, a large number of random simulation operations are carried out, and then a statistical convergence result is obtained by counting the results of all operations. In this paper, the stochastic integral equation method SIEM(Stochastic Integral Equation Method) is proposed to calculate the electromagnetic scattering of Gaussian rough surface. SIEM avoids the difficulty of high difficulty precise modeling and repeated calculation many times, and only single modeling and calculation are carried out for the target problem. In this paper, the electromagnetic scattering of two-dimensional cylinder with Gaussian rough surface is solved, and the effectiveness of SIEM is proved by comparing with mom.

A single cathode linear plasma device has been designed and constructed to investigate the interactions between plasma and materials at the Sichuan University. In order to further investigate the Ohmic power of the device, the output heat load on the specimen and electric potential difference (between cathode and anode) have been tested under different discharge currents. This special power distribution in the radial direction of the plasma discharge channel has also been discussed and described by some improved integral equations in this paper; it can be further simplified as P ∝ α−2 in one-parameter. Besides, we have measured the power loss of the channel under different discharge currents by the calorimetric method, calculated the effective power of the device and evaluated the performances of the plasma device through the power efficiency analysis.

In this paper, an efficient collocation method based on two dimensional barycentric Gegenbauer interpolation is used to solve a kind of special two dimensional Fredholm-Volterra integral equations (2D-FVIEs). The explicit barycentric weights for the Gegenbauer-Gauss nodes not only reduce the complicated calculation but also preserve the numerical stability. The combination of the barycentric interpolation and the Legendre-Gauss quadrature formula transforms the 2D-FVIEs into a system of discrete algebraic equations whose solution is a set of the nodal function values. The numerical experiments provide the maximum absolute error and the convergent order to assess the accuracy and the convergence of the method.

Sustainable development is now engaged in many leading countries. The notion is connected with the activity of countries, their economies, social and political institutions. One of the criteria of sustainability is the economic sphere, the study of which will allow you to control and predict its activity. It is known that the description of continuous processes that arise in economic problems, using definite integrals. In the case when the studied continuous development is carried out in small jumps, when it is the description of the use of fractional integral and differential equations. However, their exact solution is possible only in special cases. Therefore, the construction of approximate methods for solving fractional integral and differential equations is of great importance. In this paper, we give an approximate solution of fractional integral equations of the first kind, where the fractional integral is constructed using the Weyl fractional integral. It turns out that the constructed operator of fractional integration is symmetrical and positively definite. These properties are built by the operator enables to introduce the scalar product on the basis of the specified operator and the corresponding energy space. Next, using the same methodology that is used in the finite element method, approximate solution is constructed, constructed a rough scheme of the method and rationale of the proposed method to construct the energy space. The integral equation of the standard form is given as a specific example, as well as an example solution to illustrate the effectiveness of the proposed method.

We solve several applied superconductivity problems using the series in Chebyshev polynomials. ‎Although this method may be less general than the usually employed finite element methods, ‎accuracy of the obtained solutions is often much higher due to the very fast convergence of the ‎Chebyshev expansions. First, as an introduction, we consider the thin strip magnetization and ‎transport current problems for which the analytical solutions are known. Then we apply this method ‎to two-dimensional problems for stacks and pancake coils made of coated conductors assuming the ‎Bean critical-state model. Finally, we extend this approach to evolutionary stack problems with an ‎arbitrary current-voltage relation. Simulation of coils with a large number of turns is also discussed.‎

The use of HTS materials for the construction of magnetic bearings has shown their high efficiency. Currently, more and more research is focused on the use of tape superconductors for magnetic bearings. Application of HTS tapes allows simplifying the manufacturing technology of these devices and improving the specific power characteristics. However, when modelling devices with HTS tapes, computational difficulties arise due to the small thickness of the active layer. The paper considers the possibility of using models for magnetization to describe the properties of HTS tapes in the calculation method based on the integral equations of an electromagnetic field. The features of the proposed models are illustrated by the results of solving simple test tasks and calculations of real designs of magnetic bearings. To confirm the correctness of the simulation, the calculation data are compared with the data of experimental studies.

A mathematical model of the current density distribution in an axially symmetric system of superconductors in an external magnetic field is formulated within the framework of the integral form of London equations. The discretization of the model is carried out and the stability and convergence of the computational scheme are investigated. The distribution of magnetic induction, the density of currents in the ring and the sphere and the attractive force between them are calculated. The possibility of stable levitation of the sphere near the center of the ring is shown.

Paraxial transmission characteristic of a second-order frequency chirped finite Olver Gaussian beam (CFOGB) in the uniaxial crystal orthogonal to the optical axial is investigated based on the generalized Huygens-Fresnel integral equation. The exact expression for the CFOGB passing through the uniaxial crystal is derived. The contour graph of the CFOGB intensity distribution on some transversal cross sections, the side view of this beam propagating in three different kinds of uniaxial crystal models are discussed, the influence of the frequency chirp parameter on the beam evolution properties is explored as well. We make sure that the formulae and the conclusions obtained can provide an effective and quick method to adjust and control this CFOGB evolution path and profile through choosing the proper uniaxial crystal structure and frequency chirp value to meet the practical usage.

The contact with adhesion between the FGM coating and axisymmetric indenter is probed into in the present paper. Coating is made of FGM with linear varying modulus. Poisson’s ratios of coating and substrate are chosen to 1/3. The Maugis-type adhesive model is used to describe the adhesive interaction in the adhesion zone. According to the basic equations for axisymmetric problem, the controlling singular integral equations of the contact problem with adhesion are gained by making use of Hankel integral transformation. These integral equations are solved by applying a numerical collocation technique. The results show that the contact damage and deformation on the coating surface can be reduced by adjusting the inhomogeneity of the coating.

The Landau damping which reveals the characteristic of relaxation dynamics for an equilibrium state is a universal concept in the area of complex system. In this paper, we study the Landau damping in the phase oscillator system by considering two types of coupling heterogeneity in the Kuramoto model. We show that the critical coupling strength for phase transition, which can be obtained analytically through the balanced integral equation, has the same formula for both cases. The Landau damping effects are further explained in the framework of Laplace transform, where the order parameters decay to zero in the long time limit.

Hypersingular integral equation associated with the modified complex potential is formulated to solve the three inclined cracks problems in an elastic half-plane with free traction boundary condition. The modified complex potential possesses two parts; the principal and the complementary parts. The principal part is derived from the original complex potential of the crack problem in an infinite plate. The complementary part eliminates the traction along boundary of half-plane caused by the principal part. The crack opening displacements (COD) is the unknown function and the traction is the right hand terms. The appropriate quadrature formula is adapted to solve the integral equation numerically and the stress intensity factor (SIF) is computed. The behaviour of SIF at crack tips is analysed. Numerical examples show that the SIF increases as the angle of inclined cracks and the distance of cracks from the boundary of half-plane increase. Our results are agreeable with the previous works.

The three-dimensional diffraction problem of stationary acoustic waves on a homogeneous inclusion is considered. It is reduced to the weakly singular boundary Fredholm integral equations of the first kind with one unknown function, each of which is conditionally equivalent to the original problem. By using the original method of averaging the integral operators kernels, these equations are approximated by systems of linear algebraic equations. The resulting systems are solved numerically by the generalized minimal residual method (GMRES). Then the solution of the initial problem is calculated. To find the solution on the spectrum of integral operators, where the condition of equivalence of integral equations to the original problem is violated, the interpolation solution method is proposed. It does not require knowledge of the spectrum and allows us to find the approximated solutions with high accuracy. The proposed algorithms have been implemented in the computing cluster of Computing Center FEB RAS. The results of the calculations that allow us to assess the possibilities of this approach are presented.

In this study, we propose Alternating Group Explicit (AGE) iterative method to solve the problem of linear fuzzy Fredholm integral equations of the second kind (FFIE-2) based on Trapezoidal quadrature rule. First, we generate a system of linear equations by using approximation equations. Next, we use AGE iterative method to solve the generated linear system of FFIE-2. Then, we illustrate the applicability of AGE iterative method on some numerical examples. Finally, we do the comparison to test the efficiency of AGE with Jacobi and Gauss-Seidel (GS) iterative method based on three parameters: number of iterations, execution time and Hausdorff distance. From our findings, we found that AGE is a better iterative method compared with Jacobi and GS to solve FFIE-2.

In this paper, a piecewise collocation discretization scheme based on the piecewise constant approximation with the concept of quarter-sweep Jacobi (QSJ) iteration is discussed in solving the linear Fredholm integral equations of second kind. By considering the piecewise approximation equations, the generated linear system has been constructed with its large scale coefficient matrix. The purpose of this quarter-sweep iteration concept is to reduce the computational complexity of the linear system. For the purpose of comparison, the formulation and implementation of full-sweep Jacobi (FSJ), half-sweep Jacobi (HSJ) and QSJ iterative methods are also included. The results of these three proposed methods showed that the QSJ method is better than others Jacobi iteration family.

The article presents the derivation of the integral equation for a cylindrical perfectly conducting symmetric vibrator located vertically at a certain height from the interface of the two media. An example of numerical calculation is considered.

In the article the output of integral equations symmetric cylindrical dipole antenna of superconductive material. The antenna is located parallel to the interface between the two media. An example of calculating the current distribution over the antenna depending on the parameters of the medium is given.

Simulating antennas around a conducting object is a challenge task in computational electromagnetism, which is concerned with the behaviour of electromagnetic fields. To analyze this model efficiently, an improved integral equation-fast Fourier transform (IE-FFT) algorithm is presented in this paper. The proposed scheme employs two Cartesian grids with different size and location to enclose the antenna and the other object, respectively. On the one hand, IE-FFT technique is used to store matrix in a sparse form and accelerate the matrix-vector multiplication for each sub-domain independently. On the other hand, the mutual interaction between sub-domains is taken as the additional exciting voltage in each matrix equation. By updating integral equations several times, the whole electromagnetic system can achieve a stable status. Finally, the validity of the presented method is verified through the analysis of typical antennas in the presence of a conducting object.

We study certain classes of non-linear Hammerstein integral equations on the semi-axis and the whole line. These classes of equations arise in the theory of radiative transfer in nuclear reactors, in the kinetic theory of gases, and for travelling waves in non-linear Richer competition systems. By combining special iteration methods with the methods of construction of invariant cone segments for the appropriate non-linear operator, we are able to prove constructive existence theorems for positive solutions in various function spaces. We give illustrative examples of equations satisfying all the hypotheses of our theorems.

Conditions are obtained for the image of a given set under a general completely continuous nonlinear integral operator to have convex closure. These results are used to establish the uniqueness of quasi-solutions of nonlinear integral equations of the first kind and to prove the solvability of equations of the first kind on a dense subset of the right-hand sides. Bibliography: 11 titles.

The possibility of a numerical solution of construction problems leading to Fredholm integral equations of the first and second kinds is considered. The following problems of building sciences, for example, lead to such equations: in the theory of elasticity – elastoplastic torsion of bodies, stress concentration in a body during its bending; in the theory of vibrations, in the mechanics of structural failure – in the formation of cracks; when studying non-stationary phenomena in solids – non-stationary heat transfer, quasistatic viscoelasticity, propagation of longitudinal and transverse waves and other processes. According to the formulas of numerical integration, the problem can be reduced to a system of linear algebraic equations with a matrix of coefficients, which can be ill-conditioned, and the task is classically incorrect. This circumstance complicates its decision by traditional methods. To solve ill-posed problems similar to the problem in question, there are classical regularization methods. However, the numerical implementation of regularizing algorithms is not always easy to implement. In this regard, the goal of our study was to develop a numerical method for solving the Fredholm integral equations of the first and second kinds, which allow solving such problems without using regularization methods. The authors propose to solve problems of this type on the basis of the Bogolyubov – Krylov formula for representing certain integrals, in the form of finite-dimensional sums. At the same time, it becomes possible to use a non-uniform grid of nodes for splitting the integration interval. To efficiently define the set of nodes for splitting the integration interval, it is proposed to use a priori information about the properties of the solution of the problem. For example, the number of split points of the integration interval increases where the intended solution undergoes the most rapid changes, and decreases in areas where the intended solution is close to linear. The optimal arrangement of the splitting nodes of the integration interval makes it possible to increase the conditionality of the system of linear equations corresponding to the difference analogue of the problem and, thereby, to prevent the divergence of the iterative process. The article provides an example of calculating the optimal arrangement of the nodes splitting the integration interval when solving a test problem.

In this paper, an inclined and a semi circular arc cracks lying in the upper half of bonded dissimilar materials are considered.The modified complex variable function method is used to formulate the problem into the hypersingular integral equations (HSIEs) with the help of the continuity conditions of the traction and displacement.The traction along the crack serves as the right hand side of the integral equation. The HSIEs is solved numerically for the unknown crack opening displacement function using the appropriate quadrature formulas. Numerical results presented the behaviour of the nondimensional stress intensity factorsat the crack tips which are depends on the elastic constants ratio, the position of the crack and the distance between the crack and the boundary.