Linear Fredholm Integral Equations Research Articles

A modified algebraic method of mathematical signal processing in radar problems

A new method for approximate solution of ill-posed inverse problems of reconstructing images of objects with angular resolution exceeding the Rayleigh criterion is proposed and justified, i.e. with super resolution. Angular super-resolution allows you to obtain images of objects with increased clarity and distinguish previously invisible details of images of complex objects. In addition, on this basis, the probability of correct solutions to problems of object recognition and identification increases. Mathematically, the problem is reduced to solving the linear Fredholm integral equation of the first kind of convolution type. Solutions are sought with additional conditions in the form of restrictions on the location and size of the desired radiation source, which makes it possible to regularize the problem. The method is a development of one of the parameterization methods the algebraic method. The solution is sought in the form of a representation of the desired function in the area where the source is located in the form of a series expansion over the input sequence of orthogonal functions with unknown coefficients. Thus, the inverse problem is parameterized and reduced to searching for expansion coefficients. The presented method is based on the use of a priori information about the localization area of the radiation source, or on an estimate of the location and size of this area obtained by scanning the viewing sector with a goniometer system. Using zero values of the function describing the source outside this region, for systems based on antenna arrays it is possible to find tens and even hundreds of expansion coefficients of the desired function in a Fourier series. The solution is constructed in the form of an iterative process with a consistent increase in the number of functions used in the expansion until the solution remains stable. The adequacy and stability of the solutions was verified during numerical experiments using a mathematical model. The results of numerical studies show that the presented methods of digital processing of received signals make it possible to achieve an effective angular resolution 3–10 times higher than the Rayleigh criterion. The proposed method makes it possible to miniaturize the antenna system without degrading its characteristics. Compared to known ones, it is relatively simple, which allows it to be used by systems in real time.

A stable meshless numerical scheme using hybrid kernels to solve linear Fredholm integral equations of the second kind and its applications

The main challenge in kernel-based approximation theory and its applications is the conflict between accuracy and stability. Hybrid kernels can be used as one of the simplest and most effective tools to manage this challenge. This article uses a meshless scheme based on hybrid radial kernels (HRKs) to solve second kind Fredholm integral equations (FIEs). The method estimates the solution via the discrete collocation procedure based on a hybrid kernel that combines appropriate kernels with suitable weight parameters. The optimal parameters in the hybrid kernels can be calculated using the particle swarm optimization (PSO) algorithm based on the root mean square (RMS) error. This technique converts the problem under investigation into a system of linear equations. Moreover, the convergence of the suggested hybrid approach is studied. Lastly, some numerical experiments are included to reveal the accuracy and stability of the hybrid kernels approach. The numerical results demonstrate that the presented hybrid procedure meaningfully reduces the ill-conditioning of the operational matrices, at the same time, it maintains the accuracy and stability for all values of shape parameters.

B-spline collocation method for solving Fredholm integral equations with discontinuous right-hand side

In this paper, we propose a method for approximating the solution of the linear Fredholm integral equation of the second kind which is defined on a closed contour Г in the complex plane. The right-hand side of the equation is a piecewise continuous function that is numerically defined on a finite set of points on Г. To approximate the solution, we use a linear combination of B-spline functions and Heaviside step functions defined on Г. We discuss both theoretical and practical aspects of the pointwise convergence of the method, including its performance in the vicinity of the points where discontinuities occur.

An iterative Nyström-based method to solve nonlinear Fredholm integral equations of the second kind

In the current paper, an iterative numerical scheme based upon the quasilinearization technique and Nyström method to find the approximate solution of the nonlinear Fredholm integral equations of the second kind is provided. First, using the quasilinearization method, the nonlinear Fredholm integral equation is reduced to a sequence of linear Fredholm integral equations. Under some suitable assumptions, the exact solutions of this sequence of linear integral equations converge to the unique solution of the original problem quadratically. Then in each iteration, utilizing the linear barycentric rational quadrature, the linear Fredholm integral equation is approximated with the Nyström method. Convergence analysis of the method is investigated thoroughly. Numerical examples and comparisons with other existing methods are provided to confirm the theoretical results and demonstrate the performance and validity of the numerical method.

φ-fixed point results for nonlinear contractions with an application

In this paper, using the notion of w-distance in metric spaces, we introduce two types of nonlinear contractions, (iota ,psi ,varphi ,phi ,p) and rational-(iota ,psi , varphi ,phi ,p) contractions. Based on these contractions, we prove the existence and uniqueness of a φ-fixed point for the corresponding contraction. We also provide some examples to demonstrate the correctness and practicability of our results, along with a numerical experiment. Finally, we apply the obtained results to linear matrix equations and nonlinear Fredholm integral equations.

Iterative schemes for linear equations of the second kind and related inverse problems

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above.

Solving Fredholm integral equation based on Padé approximants using Particle swarm optimization

Several scientific and engineering applications are usually described as integral equations. A new approach for solving the type of linear and nonlinear Fredholm integral equation of the second kind is proposed. Although many methods provide an analytic solution, there are different types of integral equations are difficult to solve. Therefore, the numerical approach for solving integral equations is used. Fredholm integral equations of the second kind have been converted to unconstrained optimization problems to find their approximate solutions. This work employs particle swarm optimization combined with padé expansion to find an approximate solution of the Fredholm integral equation. This is applied by minimizing the fitness function value. The fitness function is calculated using the discrete least squares weighted function. The proposed algorithm is applied to solve linear and non-linear FIE. The results are compared to exact solutions. The stability of the proposed algorithm is also presented. The results are promising in terms of convergence , stability and accuracy of the approximate solution.

The Use of Homotopy Regularization Method for Linere and Nonlinner Fredholm Integral Equations of the First Kind

Recently, Wazwaz has studied the regularization method to the one-dimensional linear Fredholm integral equations of the first kind [Wazwaz, 2011]. In this work, we develop this method for the linear and nonlinear two-dimensional Fred-holm integral equations of the first kind. Indeed, the regularization method is used for linear integral equations directly. But nonlinear integral equations of the first kind are transformed to linearintegral equations of the first kind by a change of variable; then, The Regularization-Homotopy Method is applied. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method, is used to find a solution to the equation. Some examples will be used to highlight the reliability of the generalized of Regularization-Homotopy Method.

Convergence Analysis for Linear Fredholm and Nonlinear Fredholm Hammerstein Integral Equations

In this article, we consider the linear Fredholm integral equations and Fredholm-Hammerstein’s integral equations. We propose the Legendre polynomial based degenerate kernel method to solve linear Fredholm and Fredholm-Hammerstein integral equations. We discuss the convergence and error analysis of the proposed method and also obtain the superconvergence results for iterated degenerate kernel method.

Using (Direct Computation, Variation Iteration, Successive Approximation and Regularization) Methods to Solve Linear Fredholm Integral Equation and Comparison of These Methods

Integral equation is the equation in which the unknown function to be determined, appears under integral sign as it presented in introduction. I discussed about linear Fredholm integral equation in which it is one kind of integral equation and solved this equation by different methods (Direct Computation, Variational Iteration, Successive approximation and the Regularization methods and comparison of these methods in order to solve linear Fredholm integral equation). This paper has three parts: First part: I introduced the Fredholm integral equation Second part is methods that is written above and on third part, I solved one example by these different methods and compare the methods.

Gelfand-Levitan integral equation for solving coefficient inverse problem

In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

A Specific Numerical Method for Two-Dimensional Linear Fredholm Integral Equations of the Second Kind by Boubaker Polynomial Bases

A Specific Numerical Method for Two-Dimensional Linear Fredholm Integral Equations of the Second Kind by Boubaker Polynomial Bases

Two Singularity Subtraction Schemes for a Class of Nonlinear Weakly Singular Integral Equations

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as three numerical experiments confirm.

Ten non-polynomial cubic splines for some classes of Fredholm integral equations

In this paper, a new general form for the non-polynomial cubic spline function is introduced. This form enables us to generate about ten different formulas of the proposed spline function. Although some of these functions have been used before to solve some types of differential and integral equations, the others are considered new. These ten functions are used to approximate the solution of linear, nonlinear, and fuzzy Fredholm integral equations of second kind. Moreover, the convergence analysis of the proposed method is established which indicates that its order of convergence is four. Also, some examples are solved and compared with the previous methods which showed that the present method is more accurate.

BOUNDARY CONTROL OF ROD TEMPERATURE FIELD WITH A SELECTED POINT

In this paper, we study the issue of boundary control of the temperature field of a rod with a selected point. The main purpose of the work is to clarify the conditions for the existence of a boundary control that ensures the transition of the temperature field from the initial state to the final state. Relations connecting the boundary controls with the initial and final states, as well as with the external temperature field are found. Such boundary controls, generally speaking, constitute an infinite set. For an unambiguous choice of the boundary control, a strictly convex objective functional is chosen. We are looking for a boundary control that minimizes the selected target functional. To do this, we first investigate the existence and uniqueness of solutions to the initial boundary value problem and the conjugate problem. And also, we present the derivation of a system of linear Fredholm integral equations of the second kind, which are satisfied by an optimal boundary control that minimizes a strictly convex target functional on a convex set. Along the way, the linear part of the increment of the target functional is highlighted. Necessary and sufficient conditions for the minimum of a smooth convex functional on a convex set are established. The difference between the results of this work and the available ones is that in the proposed work, the temperature field is given by the heat conduction equation with a loaded term. As a result, the conjugate problem has a slightly different domain of definition than the domain of the conjugate problem in the case of no load.

Haar Wavelet Collocation Method for Solving Linear Volterra and Fredholm Integral Equations

Haar Wavelet Collocation Method for Solving Linear Volterra and Fredholm Integral Equations

Haar Wavelet Collocation Method for Solving Linear Volterra and Fredholm Integral Equations

Haar Wavelet Collocation Method for Solving Linear Volterra and Fredholm Integral Equations

A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind

A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind

A Combination of Bernstein and Improved Block-Pulse Functions for Solving a System of Linear Fredholm Integral Equations

The present paper illustrates a new numerical technique to solve a system of linear Fredholm integral equations of the second kind. The current work introduces a coupling between hybrid Bernstein functions and improved block-pulse functions (HBI). The current method transforms the linear second kind Fredholm integral equations system into an algebraic system that can be solved by using classical methods. Some numerical examples are introduced to validate the new approach. The results showed that the method is promising, powerful, ultimately accurate, and highly efficient.

Application of numerical solution of linear Fredholm integral equation of the first kind for image restoration

Application of numerical solution of linear Fredholm integral equation of the first kind for image restoration