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A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f ( α ) is transcendental or algebraic if f ( z ) is an E-function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f ( z ) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f ( 0 ) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6] .

Plasma generated by hypersonic aerocraft is one of the research subjects in computational electromagnetics. When the aerocraft flies at an ultra-high speed in the near space, plasma sheath will be generated in the surroundings and it will trigger varying changes in aerocraft radar electromagnetic scattering properties. Plasma sheath brings about tremendous impacts on the communication between the aerocraft and the outside world, and even ends with "blackout effects" to interrupt communication in severe cases. The interplay between the plasma sheath and the electromagnetic wave involves multiple disciplines covering aerodynamics, plasma sheath physics, and electromagnetic field theory. Therefore, it is challengeable to build a rigorous numerical modeling of the plasma sheath. In this paper, the latest technologies in the electromagnetic simulation of hypersonic aerocraft with plasma sheath are reviewed. Applied cases review and illustrate extensive approaches from approximate calculations to accurate calculations, including analytical methods, differential equation methods, integral equation methods and high-frequency approximation methods. The paper assesses the RCS of electromagnetic wave in analyzing the electromagnetic scattering problem in plasma. With the considerations the reality of electromagnetic simulation and future challenges, the paper talks about the efficiency, advantages and disadvantages of these approaches in simulating hypersonic aerocraft targets.

Ballistic Missile Trajectory Prediction (BMTP) is critical to air defense systems. Most Trajectory Prediction (TP) methods focus on the coast and reentry phases, in which the Ballistic Missile (BM) trajectories are modeled as ellipses or the state components are propagated by the dynamic integral equations on time scales. In contrast, the boost-phase TP is more challenging because there are many unknown forces acting on the BM in this phase. To tackle this difficult problem, a novel BMTP method by using Gaussian Processes (GPs) is proposed in this paper. In particular, the GP is employed to train the prediction error model of the boost-phase trajectory database, in which the error refers to the difference between the true BM state at the prediction moment and the integral extrapolation of the BM state. And the final BMTP is a combination of the dynamic equation based numerical integration and the GP-based prediction error. Since the trained GP aims to capture the relationship between the numerical integration and the unknown error, the modified BM state prediction is closer to the true one compared with the original TP. Furthermore, the GP is able to output the uncertainty information of the TP, which is of great significance for determining the warning range centered on the predicted BM state. Simulation results show that the proposed method effectively improves the BMTP accuracy during the boost phase and provides reliable uncertainty estimation boundaries.

An adaptive decentralized asymptotic tracking control scheme is developed in this paper for a class of large-scale nonlinear systems with unknown strong interconnections, unknown time-varying parameters, and disturbances. First, by employing the intrinsic properties of Gaussian functions for the interconnection terms for the first time, all extra signals in the framework of decentralized control are filtered out, thereby removing all additional assumptions imposed on the interconnections, such as upper bounding functions and matching conditions. Second, by introducing two integral bounded functions, asymptotic tracking control is realized. Moreover, the nonlinear filters with the compensation terms are introduced to circumvent the issue of “explosion of complexity”. It is shown that all the closed-loop signals are bounded and the tracking errors converge to zero asymptotically. In the end, a simulation example is carried out to demonstrate the effectiveness of the proposed approach.

The paper deals with an integral equation arising from a problem in mathematical biology. We propose approximating its solution by Nystrom methods based on Gaussian rules and on product integration rules according to the smoothness of the kernel function. In particular, when the latter is weakly singular we propose two Nystrom methods constructed by means of different product formulas. The first one is based on the Lagrange interpolation while the second one is based on discrete spline quasi-interpolants. The stability and the convergence of the proposed methods are proved in uniform spaces of continuous functions. Finally, some numerical tests showing the effectiveness of the methods and the sharpness of the obtained error estimates are given.

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.

A single-source surface integral equation (SS-SIE) for transverse electric (TE) scattering from cylindrical multilayered objects is proposed in this article. By incorporating the differential surface admittance operator (DSAO) and recursively applying the surface equivalence theorem from innermost to outermost boundaries, an equivalent model with only electric current density on the outermost boundary can be obtained. In addition, an integration approach is proposed, where the small argument expansion of the Hankel function is used to evaluate the singular and nearly singular integrals. Compared with other SIEs, such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation, the computational expenditure is reduced for multilayered structures because only a single source is needed on the outermost boundary. As shown in the numerical results, the proposed method generates only 19% of unknowns, uses 26% of memory, and requires 29% of the CPU time of the PMCHWT formulation.

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular ‎Fredholm integral equations of the ‎second kind. The kernel is ‎interpolated twice concerning ‎both variables and then is transformed into the product of five ‎matrices; two of them are monomial basis ‎matrices. To isolate the singularity of the kernel, we ‎developed two techniques based on a good choice of ‎different two sets of nodes to be distributed ‎over the integration domain. Each set is specific to one of the ‎kernel arguments so that the kernel ‎values never become zero or imaginary. The significant advantage of the ‎two presented ‎techniques is the ability to gain ‎‎access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the ‎mean of the interpolant solution ‎and the maximum error norm estimation are studied. The ‎interpolate solutions of the illustrated four ‎examples are found strongly converging uniformly to the ‎exact solutions.

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

We study the oblique scattering of arbitrarily polarized plane waves from radially inhomogeneous and bianisotropic cylinders in the general case where all the elements of the four constitutive tensors are nonzero. The analysis relies on an exact reformulation of Maxwell’s equations as a second-kind linear Volterra matrix integral equation. The discretization scheme uses a high order, entire-domain, Nystrom method that employs the Gauss–Legendre–Radau quadrature. The entries of the Nystrom matrix have simple closed-form expressions. The proposed algorithms have exponential convergence and can handle both single-layer and multilayer cylinders with either continuous or sectionally continuous inhomogeneity profiles.

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

In this paper we give conditions in which the integral equation \centerline { $ x(t)=\displaystyle\int_a^c K(t,s,x(s))ds+\int_a^t H(t,s,x(s))ds+g(t),\ t\in [a,b], $ } \noindent where $a

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

The receding contact problem between a graded piezoelectric layer and a homogeneous piezoelectric substrate is considered in this paper. It is assumed that the gradation of the elastic piezoelectric graded layer is of exponential type through its thickness. Using standard Fourier transform, the contact problem is converted to a system of two singular integral equations in which the contact pressures and the electric charge displacement in addition to the contact dimensions are the unknowns. The integral equations are then solved numerically using Gauss–Jacobi integration formula. The primary objective of this paper is to investigate the effect of material gradation on the contact pressure, electric charge distribution and on the length of the receding contact. The main findings of the paper are that the inhomogeneity parameter has a strong effect on the contact pressure and the electric charge distribution at the receding contact interface. It is concluded that a softer FGPM in $$+z$$ direction results in lower contact pressure and electric displacement.

This article proposes two efficient methods to solve nonlinear stochastic Ito–Volterra integral equations. The shifted Jacobi spectral Galerkin method and shifted Jacobi operational matrix method h...

In this article, two numerical methods, known as Gegenbauer wavelet method and cosine-and-sine (CAS) wavelet method, have been introduced. Gegenbauer wavelet method based on conventional Gegenbauer polynomials has been proposed for solving two-dimensional nonlinear Volterra integral equations. CAS wavelet method based on cosine-and-sine wavelet approximations has been also utilized for comparison of obtained solutions. The different collocation points are used for these two different wavelet methods along with the Gaussian quadrature method to reduce the given integral equations to the system of algebraic equations. The convergence and error analyses for the both Gegenbauer wavelets method and CAS wavelet method have also been presented in this article. Illustrative examples are provided to describe the efficiency and validity of the proposed techniques through the comparison of errors for two proposed methods. Furthermore, some figures are plotted to demonstrate the error analysis of the proposed scheme.

A particular case of the Hermite interpolation method, namely the two-point Taylor formula, is utilized to construct a numerical technique for solving Fredholm integral equations (FIEs) of the second kind. This method can be applied to approximate the solution of both linear and nonlinear FIEs, and systems of nonlinear FIEs. The sufficient conditions to guarantee the convergence of the proposed method are provided through our analytical studies. Also, the error estimation is presented for this method. Furthermore, the efficiency of the method is confirmed by applying it to solve several illustrative examples. Numerical experiments confirm that the method is easy to implement and gives accurate approximations in acceptable computational times.

Introduction. The paper deals with the calculation of wooden arches taking into account the nonlinear relationship between stresses and instantaneous deformations, as well as creep and geometric nonlinearity, are considered. The analysis is based on the integral equation of the viscoelastoplastic hereditary aging model, originally proposed by A.G. Tamrazyan [1] to describe the nonlinear creep of concrete. Materials and Methods. The creep measure is taken in accordance with the work of I.E. Prokopovich and V.A. Zedgenidze [2] as a sum of exponential functions. The transition from the integral form of the creep law to the differential form is shown. The relationship between stresses and instantaneous deformations for wood under compression is determined from the Gerstner formula, and elastic work is assumed under tension. The solution is carried out using the finite element method in combination with the Newton-Raphson method and the Euler method according to the scheme that involves a stepwise increase in the load with correction of the stiffness matrix taking into account the change in the coordinates of the nodes with the sequential calculation of additional displacements of the nodes, which are due to the residual forces. The proposed approach for increasing the accuracy of determination of creep deformations at each step provides using the fourth-order Runge-Kutta method instead of the Euler method. Results. Based on the Lagrange variational principle, expressions are obtained for the stiffness matrix and the vector of additional dummy loads due to creep. The method developed by the authors is implemented in the form of a program in the MATLAB environment. Calculation examples are given for parabolic arches simply supported at the ends without an intermediate hinge and with an intermediate hinge in the middle of the span under the action of a uniformly distributed load. The results obtained are compared in the viscoelastic and viscoelastic formulation. The reliability of the results is validated through the calculation in the elastic formulation in the ANSYS software package. Discussion and Conclusions. For the arches considered, it is found that even with a load close to the instant critical, the growth of time travel is limited. Thus, the nature of their work under creep conditions differs drastically from the nature of the deformation of compressed rods.

In this paper, first, Bernstein multi-scaling polynomials (BMSPs) and their properties are introduced. These polynomials are obtained by compressing Bernstein polynomials (BPs) under sub-intervals. Then, by using these polynomials, stochastic operational matrices of integration are generated. Moreover, by transforming the stochastic integral equation to a system of algebraic equations and solving this system using Newton’s method, the approximate solution of the stochastic Ito-Volterra integral equation is obtained. To illustrate the efficiency and accuracy of the proposed method, some examples are presented and the results are compared with other methods.

A technique is proposed for solving a 3D problem of elasticity theory for a transversely isotropic half-space. The isotropy plane is parallel to the half-space boundary where arbitrary local force loadings are given. By the direct integration method, the original equations of the problem are reduced to the governing integral equations of second kind for individual stress tensor components. Explicit solutions of the equations are obtained in the space of Fourier double-integral transform by the resolvent method. The solution constructed exactly satisfies the original equations and boundary conditions of the problem and ensures its decaying, in accordance with the Saint-Venant principle, away from the loaded segment of the boundary for all possible relationships between the elastic moduli of transversely isotropic media.

By employing the kernel functions with the form of piecewise polynomials in the Sobolev reproducing kernel Hilbert spaces (RKHSs), a globally superconvergent numerical technique is proposed to solve the second kind linear integral equations of Fredholm type. This method has an order of global convergence O ( h 4 ) and O ( h 6 ) based on the kernel functions in the Sobolev RKHSs H 1 and H 2 , respectively. Three linear Fredholm integral equations, one Volterra-Fredholm integral equation and one nonlinear Fredholm integral equation are numerically solved by the present approach to verify the superconvergence and effectiveness.

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

Sufficient conditions in this paper are presented for the existence of periodic solutions of a second order totally nonlinear iterative differential equation. The equivalent integral equation of the given equation defines a fixed point mapping written as a sum of a large contraction and a compact map. The main tool used here is Krasnoselskii-Burton’s fixed point technique.

In this study, the Volterra integral equations are defined in the non-Newtonian calculus with the aid of $$*$$ -integral. The focus of this study is obtaining the solution of linear non-Newtonian integral equations. The $$*$$ -resolvent kernel method, successive approximations method and Adomian decomposition method are applied for solving non-Newtonian integral equations in the non-Newtonian sense. Some numerical examples are presented to explain the procedure of these methods. Finally, a model that describes a type of the exponential growth model is proposed in the sense of non-Newtonian calculus as an example.

A reduced spectral problem is considered for an elliptic type operator, in particular, for the Cauchy - Riemann operator with regular boundary value conditions to a quasi-singular integral equation with continuous kernel. Structure of the kernel of the quasi-singular integral equation is studied in explicit form. Index is calculated and condition for Noetherity of the investigated quasi-singular integral equation is established by the topological method on the complex plane. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. The given problem is a nonlocal nature, and similar problems for the Cauchy-Riemann operation are described by M. Otelbaev in 1982. The solution of some singular integral equations with kernels depending on the difference and the sum of the arguments were studied in the works of I.I. Kalmushevsky. This interest is explained both by the theoretical significance of the results obtained and by the possibilities of important applications. Similar boundary value problems arise in the mathematical modeling of gas dynamics problems, soil moisture prediction, radiation transfer, population genetics, etc. The simplest examples of these boundary conditions were formulated by V.A. Steklov 1922. In this paper, the fundamental difference from the above work is the topological approach to calculating the index and establishing the Noether condition on the complexs plane.

In this article, a new inversion model for 2-D microwave imaging is introduced by means of a convenient rewriting of the usual Lippmann–Schwinger integral scattering equation. Such a model is derived by decomposing Green’s function and the corresponding internal radiation operator in two different contributions, one of them easily computed from the collected scattered data. In the case of lossless backgrounds, the resulting model turns out to be more convenient than the traditional one, as it exhibits a lower degree of nonlinearity with respect to parameters embedding the unknown dielectric characteristics. This interesting property suggests its exploitation in the solution of the inverse scattering problem. The achievable performance is tested by comparing the proposed model with the one based on the usual Lippman–Schwinger equation in both cases of linearly approximated and full nonlinear frameworks. Both numerical and experimental data are considered.

A high-order solution of electromagnetic (EM) scattering from electrically large targets with uncertain shapes is proposed in this letter. At first, the uncertain geometry can be described with several independent variables by using the nonuniform rational B-spline surfaces. Then, the second-order Taylor series expansion is used to present the geometrical uncertainty in terms of the combined field integral equation. As a result, higher accuracy can be obtained by using this high-order solver. Finally, the mean value and variance of the radar cross section (RCS) are derived and can be compared with the traditional Monte Carlo (MC) method. Numerical results show the proposed method can provide higher accuracy than the traditional one-order solver and higher efficiency than the traditional MC method.

Using the technique associated with the classical Schauder fixed point theorem we demonstrate the existence of solutions of a class of quadratic integral equation of Fredholm type in Holder spaces....

In this paper, we discuss the numerical solution of Adomian decomposition method and Taylor’s expansion method in Volterra linear integral equation. And we apply modified Adomian decomposition method and Newton Raphson method in Volterra nonlinear integral equation with the help of example and estimated an error in MATLAB 13 versions.

This study proposes a novel estimation-based approach to solving asset pricing models for both stationary and time-varying observations. Our method is robust to misspecification errors while inheriting a closed-form solution. By representing the Euler equation into a well-posed integral equation of the second kind, we propose a penalized two-stage nonparametric estimation method and establish its optimal convergence under mild conditions. With the merit of penalized splines, our estimate is less sensitive to the spline setting and we also design a fast data-driven algorithm to effectively tune the key smoother, i.e. the penalty amount. Our approach exhibits excellent finite sample performance. Using the US data from 1947 to 2017, we reinvestigate the return predictability and find that the estimated implied dividend yield significantly predicts lower future cash flows and higher interest rates at short horizons.

I develop inverse scattering methods for velocity reconstruction in the subsurface, based on multiple scattering theory, Gaussian beams and nonlinear Born approximation. This method is based on a new version of the distorted Born iterative inverse scattering method. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method. I propose three direct scattering methods for forward modeling, namely, Gaussian beam integral equation propagators, nonlinear Born approximation, and multiple scattering. First, I apply a sensitivity kernel incorporating the Gaussian beam summation-based integral equation method and compute the background medium Green’s function in an inhomogeneous medium. Then I extend an approximate solution of the Lippmann–Schwinger equation, referred to as nonlinear Born approximation to inverse scattering problem. I apply the nonlinear Born approximation to forward modeling at each iteration. Furthermore, I combine the Gaussian beams with the multiple scattering theory for nonlinear scattering problems. I obtain the inversion results using the proposed approaches, which shows the capability for inverse scattering imaging. I have carried out several numerical experiments that involve reconstructing the resampled Marmousi and Society of Exploration Geophysicists (SEG)/European Association of Geoscientists and Engineers (EAGE) salt models from its smoothed version.

Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called “analytical” preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples., Numerische Mathematik, 148 (2), ISSN:0029-599X, ISSN:0945-3245

Incompressible turbulent boundary layer problems are solved based on certain assumed boundary conditions, by integrating Prandtl integral equation. Representing the velocity profile parameter as a function of dimensionless similarity number of turbulent boundary layers, a separation criteria can be achieved. This helps in solving the attached boundary layers with max mum deceleration. The obtained theoretical results are compared with previous available experimental data obtained for radial diffuser. Moreover, this investigation contains relation between the turbulent velocity profile and shear stress profile in a parametric form for equilibrium boundary layers and agree well with that obtained from Mellor-Gibson theory.

We consider the Dirichlet problem for the inhomogeneous Lame system in the plane with constant leading coefficients in a (finite or infinite) domain, bounded by a Lyapunov contour. For domains of finite diameter we use the weighted Holder class of functions with power behavior at infinity. We propose an equivalent reduction of the problem to a system of Fredholm integral equations in the domain and on the boundary contour.

A number of real-life applications involve the determination of scattered field of an unchanged large object and several varying small scatterers in its vicinity. Integral equation methods based on free-space Green's function (FSGF) are usually applied to solve this problem. In this letter, we propose an artificial neural network (ANN) accelerated numerical Green's function (NGF) to reduce the computational cost of conventional methods. For explicit demonstration, 2-D scattering from infinite cylinders is considered. Since the unchanged large object is considered as part of the background, its scattering is included in NGF. In our approach, its scattering is extracted via an FSGF scheme in order to exploit the regression ability of ANN efficiently. Then a dataset in possession of the scattering feature is constructed. By feeding ANN with the dataset, ANN can be trained as a good representation of NGF, where only a small number of ANN parameters are computed and stored. Since the dataset only contains part of all the possible field-source pairs in the computation region, the calculation of NGF can be accelerated. Once the ANN accelerated NGF is obtained, the scattering analysis can be easily conducted with a small numerical system where the unknowns are only associated with small scatters. The combination of ANN and NGF assists us to construct a framework that can save the CPU time and memory usage in both online scattering solver and offline NGF solver.

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

An accurate method is proposed to calculate volume integrals arising from integral equation (IE) analysis using the Schaubert–Wilton–Glisson (SWG) basis function (BF). The volume integral is transformed into a summation of some of line integrals. The dimension reduction is achieved using the divergence theorem twice. Finally, each line integral is evaluated using an accuracy-controllable quadrature rule. We compare our results with those generated by Mathematica. Good agreement is found, which verifies the effectiveness and correctness of our method.

We introduce a new approach to find the Tomita-Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo-Martin-Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann-Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered., v2: 45 pp, 2 figures, discussion of thermal states revised

We investigate a class of finite dimensional iteratively regularized Gauss–Newton methods for solving nonlinear irregular operator equations in a Hilbert space. The developed technique allows to in...

This scientific study is concerned with an integral equation, especially with Volterra integral equation and used new variational formulation which corresponds on the linear operator Lu=f. The vibrating spring equation has been solved using this approach.

We pose the direct and inverse problems of finding the electromagnetic field and the diagonal memory matrix for the reduced canonical system of integro-differential Maxwell’s equations. The problems are replaced by a closed system of Volterra-type integral equations of the second kind with respect to the Fourier transform in the space variables of the solution to the direct problem and the unknowns of the inverse problem. To this system, we then apply the contraction mapping method in the space of continuous functions with a weighted norm. Thus, we prove the global existence and uniqueness theorems for solutions to the problems.

Systematization of numerical randomized functional algorithms for approximation of solutions to second-kind Fredholm integral equation is performed in this paper. Three types of algorithms are emphasized: projection algorithms, grid algorithms, and projection-grid algorithms. Disadvantages of grid algorithms that require calculation of values of the kernel of integral equations at fixed points are revealed (practically, the kernels of equation have integrable singularities and this calculation is impossible). Thus, for applied problems related to the solution of second-kind Fredholm integral equations, projection or projection-grid randomized algorithms are more convenient.

Context. We consider the Kolmogorov-Wiener filter for forecasting of telecommunication traffic in the framework of a continuous fractional Gaussian noise model. Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding WienerHopf integral equation. Also the aim of the work is to show the convergence of the proposed method of solution of the corresponding equation. Method. The Wiener-Hopf integral equation for the filter weight function is a Fredholm integral equation of the first kind. We use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding equation. A set of Chebyshev polynomials of the first kind is used. Results. We obtained approximate solutions for the Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise. The solutions are obtained in the approximations of different number of polynomials; the results are obtained up to the nineteen-polynomial approximation. It is shown that the proposed method is convergent for the problem under consideration, i.e. the accuracy of the coincidence of the left-hand and right-hand sides of the integral equation increases with the number of polynomials. Such convergence takes place due to the fact that the correlation function of continuous fractional Gaussian noise, which is the kernel of the corresponding integral equation, is a positively-defined function. Conclusions. The Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise is obtained as an approximate solution of the corresponding Fredholm integral equation of the first kind. The proposed truncated polynomial expansion method is convergent for the problem under consideration. As is known, one of the simplest telecommunication traffic models is the model of continuous fractional Gaussian noise, so the results of the paper may be useful for telecommunication traffic forecast.

In this paper, we use definition of new transformation. (Al-Zughair Transform) to solve Volterra integral equation which can solve it by using Laplace Transform.