Solution Of Linear Integral Equations Research Articles

Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach

Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

Algebraic grids and their application to the numerical solution of linear integral equations

Algebraic grids and their application to the numerical solution of linear integral equations

An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind

In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels.

Exponential Stability and Asymptotic Periodic Solutions of Linear Integral Equations with Two Delays

Our concern is to consider linear integral equations with two delays. A new result on necessary and sufficient conditions for the exponential stability is presented by careful study of the characteristic equation. Explicit expressions of limits of solutions in the critical case where integral equations lose exponential stability are also given by use of an asymptotic formula for solutions.

An optimal maintenance strategy for multi-state deterioration systems based on a semi-Markov decision process coupled with simulation technique

Maintenance optimization of multi-state systems is a research topic of practical significance for various manufacturing systems. Nevertheless, design of satisfactory maintenance strategies remains a challenging problem due to the complex industrial field and large strategy space for maintenance optimization of such systems. In this study, a model of the deterioration process of multi-state systems with the multi-phase property was constructed by utilizing semi-Markov decision theory and linear systems of integral equation. An efficient maintenance strategy was developed with the purpose of maximizing the net revenue per unit period based on the simulation technique. In the constructed model, a system of linear integral equations with a recursive structure was constructed to obtain all possible transition probabilities between the states. Due to this recursive structure, conventional solutions of linear integral equations are not capable of solving this equation. For this reason, an algorithm was developed that was instrumental in resolving this issue. In the designed maintenance strategy, to avoid the potential curse of dimensionality resulting from the scales of the systems and the strategy space, a semi-Markov decision process coupled with the simulation technique was adopted in each decision period to reduce the search space by eliminating unreachable states. Finally, numerical examples of two systems with different scales based on real data of the inspection record database at Dongbei Special Steel Group Co., Ltd. are given to illustrate the effectiveness of the proposed model and the performance of the designed strategy. The numerical results demonstrate the high efficiency of the optimal strategy, especially for large-scale systems.

A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains

In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.

R\xe9nyi entanglement entropies for the compactified massless boson with open boundary conditions

We investigate the Rényi entanglement entropies for the one-dimensional massless free boson compactified on a circle, which describes the low energy sector of several interacting many-body 1d systems (Luttinger Liquid). We focus on systems on a finite segment with open boundary conditions and possible inhomogeneities in the couplings. We provide expressions for the Rényi entropies of integer indices in terms of Fredholm determinant-like expressions. Within the homogeneous case, we reduce the problem to the solution of linear integral equations and the computation of Riemann Theta functions. We mainly focus on a single interval in the middle of the system, but results for generic bipartitions are given as well.

Generalization of Kravchenko—Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions

New generalization of Kravchenko–Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions is discussed. These functions are solutions of linear integral equations of special form. The spectrum of is a multiple infinite product of the spectra of atomic functions. dilated by the argument. Constructed generalized series has fast convergence. This property is confirmed by the presented truncation error bound formula and the results of a numerical experiment.

A Generalization of the Kravchenko–Kotelnikov Theorem by Spectra of Compactly Supported Infinitely Differentiable Functions $$h_{a}^{{(m)}}(x)$$

A new generalization of the Kravchenko–Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions $$h_{{\mathbf{a}}}^{{(m)}}(x)$$ is considered. These functions are solutions of linear integral equations of a special form. The spectrum of $$h_{{\mathbf{a}}}^{{(m)}}(x)$$ is a multiple infinite product of the spectra of the atomic functions $${{h}_{a}}(x)$$ dilated with respect to the argument. The resulting generalized series is characterized by fast convergence, which is confirmed by the truncation error bound presented in the study and by the results of a numerical experiment.

Discrete Spectrum Reconstruction Using Integral Approximation Algorithm.

An inverse problem in spectroscopy is considered. The objective is to restore the discrete spectrum from observed spectrum data, taking into account the spectrometer's line spread function. The problem is reduced to solution of a system of linear-nonlinear equations (SLNE) with respect to intensities and frequencies of the discrete spectral lines. The SLNE is linear with respect to lines' intensities and nonlinear with respect to the lines' frequencies. The integral approximation algorithm is proposed for the solution of this SLNE. The algorithm combines solution of linear integral equations with solution of a system of linear algebraic equations and avoids nonlinear equations. Numerical examples of the application of the technique, both to synthetic and experimental spectra, demonstrate the efficacy of the proposed approach in enabling an effective enhancement of the spectrometer's resolution.

A new iterative method for solving linear Fredholm integral equations using the least squares method

In this paper, a new iterative method is proposed for solving linear integral equations. This method is based on the LSQR method, an algorithm for sparse linear equations and sparse least squares, reducing the solution of linear integral equations to the solution of a bidiagonal linear system of algebraic equations. A simple recurrence formula is presented for generating the sequence of approximate solutions. Some theoretical properties and error analysis of the new method are discussed. Although the new method can be used for solving the ill-posed first kind integral equations independently, combining of the new method with the method of regularization is presented to solve this kind of integral equations. Also the perturbing effect of the first kind integral equations is analyzed. Some properties and convergence theorem are proposed. Finally, some numerical examples are presented to show the efficiency of the new method.

Transport coefficients of a granular gas of inelastic rough hard spheres.

The Boltzmann equation for inelastic and rough hard spheres is considered as a model of a dilute granular gas. In this model, the collisions are characterized by constant coefficients of normal and tangential restitution, and hence the translational and rotational degrees of freedom are coupled. A normal solution to the Boltzmann equation is obtained by means of the Chapman-Enskog method for states near the homogeneous cooling state. The analysis is carried out to first order in the spatial gradients of the number density, the flow velocity, and the granular temperature. The constitutive equations for the momentum and heat fluxes and for the cooling rate are derived, and the associated transport coefficients are expressed in terms of the solutions of linear integral equations. For practical purposes, a first Sonine approximation is used to obtain explicit expressions of the transport coefficients as nonlinear functions of both coefficients of restitution and the moment of inertia. Known results for purely smooth inelastic spheres and perfectly elastic and rough spheres are recovered in the appropriate limits.

Semidefinite Optimization Providing Guaranteed Bounds on Linear Functionals of Solutions of Linear Integral Equations with Smooth Kernels

Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.

PQWs in complex plane: Application to Fredholm integral equations

In this paper, we use a numerical procedure for solving Fredholm integral equations of the second kind in complex plane. The periodic quasi-wavelets (PQWs) constructed on [0,2π] are utilized as a basis in collocation method to reduce the solution of linear integral equations to a system of algebraic equations. Convergence analysis is derived and we used some numerical examples to illustrate the accuracy and the implementation of the method.

Nonlinear collective excitations in helical magnetic structures

Based on the sine-Gordon model for a quasi-one-dimensional ferromagnet without an inversion center, it is shown that the total set of nonlinear modes of a helical structure contains particle-like solitons and dispersing spin waves. New types of solitons in the helical structure are described. It is found that collisions of solitons with each other are accompanied by variations in the coordinates of their centers and in their phases, as well as shifts of the structure. The possibilities of the diagnostics of solitons in the presence of external magnetic field perpendicular to the axis of the magnetic helix are established. The theoretical description of the nonlinear dynamics of spin waves is reduced to the solution of linear integral equations.

The Moore–Penrose Pseudoinverse: A Tutorial Review of the Theory

In the last decades, the Moore–Penrose pseudoinverse has found a wide range of applications in many areas of science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore–Penrose pseudoinverse and in related subjects, such as the singular value decomposition theorem for matrices. In this note, we present a tutorial review of the theory of the Moore–Penrose pseudoinverse. We present the first definitions and some motivations, and after obtaining some basic results, we center our discussion on the spectral theorem and present an algorithmically simple expression for the computation of the Moore–Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented, and for those who already have some working knowledge of the subject.

Support vector regression for the solution of linear integral equations

The aim of this paper is to establish a support vector regression method for semi-discrete ill-posed problems. We consider the equation Af = g, where a linear integral operator A is known; discrete measurements of the right-hand side are observed and a solution f* is sought. For the reconstruction, instead of a standard square loss function, Vapnik's ϵ-intensive function is used to measure the distance between Af and g. This avoids an overfitting to disturbed data and guarantees additional stability given that the cut-off parameter ϵ is chosen appropriately. The resulting solution procedure can be formulated as a quadratic program. Besides the method, a Sobolev error analysis and a parameter strategy for the regularization parameters are provided. The results are substantiated with a numerical example.

Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets

In this paper, we suggest an efficient method for solving Fredholm integral equations of the first kind. The continuous Legendre multi-wavelets constructed on [0, 1] are utilized as a basis in Galerkin method to reduce the solution of linear integral equations to a system of algebraic equations. Illustrative examples are included to demonstrate the efficiency and the application of the technique.

Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets

An efficient method based on the Legendre wavelet approach is proposed for approximate the solution of Fredholm integral equations of the first kind. The continues Legendre wavelets constructed on [0, 1] are utilized as a basis in Galerkin method to reduce the solution of linear integral equations to a system of algebraic equations. For solving this system, we use the CG method. Furthermore, we suggest a convergence analysis and error estimation for this method. For showing efficiency of method some test problems, for which the exact solution is known, are considered.

An iterative solution for the second kind integral equations using radial basis functions

The purpose of this paper is to present a new method based on radial basis functions (RBFs) and iterative procedure for the solution of linear integral equations of Fredholm and Volterra types of the second kind. To obtain a primary approximate solution, the collocation method with a small number of nodes, based on the RBFs and their first-order derivatives, can be adapted. Moreover, we present an iterative procedure from the integral equation to improve the accuracy of solution. Numerical examples show that our approach has the potentiality to become an efficient method.