Kind Fredholm Integral Equations Research Articles

Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we consider integral equations featuring kernels which may be expressed as scalar multiples of conservative (i.e. Markov) kernels and we adopt a variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean–Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.

Approximate solution of Fredholm integral equations of the second kind through multinode Shepard operators

An efficient numerical method based on the Shepard method has been developed to deal with Fredholm integral equations of the second kind. The procedure involves expressing the unknown function as a linear combination of Shepard basis functions. Subsequently, a collocation method is used to get the matrix equation yielding the coefficients of such an approximate solution. The effectiveness of the proposed approach is illustrated by some examples.

The Numerical Solutions of weakly singular Fredholm integral equations of the Second kind Using Chebyshev Polynomials of the Second Kind

In this study, the second kind Chebyshev Polynomials were utilized to acquire interpolated solutions for the second kind Fredholm integral equations with weakly singular kernel. To accomplish this, the data, unknown, and kernel functions were converted into matrix form, and consequently we completely isolated the singularity of the kernel. The primary benefit of this method is the ability to change the form of integral equation to an equivalent algebraic system, which is easier to solve. The effectiveness of our technique was evaluated by applying it to three illustrated examples, and it was observed that the solutions obtained exhibit strong convergence towards the exact solutions.

Axisymmetric torsion of rigid disc parallel to penny shaped crack at the interface of layer bonded by two orthotropic half spaces

Two half-spaces and a layer of finite height of linear elastic orthotropic medium containing a penny-shaped crack and a circular rigid disk are considered such that the crack surface is parallel to the rigid disk, where both are situated at two different interfaces of layer and half-space and perpendicular to the axis of symmetry. The rigid disk is agitated by an axisymmetric torsion. The equations of equilibrium are solved with the use of Hankel’s integral transform method. Then, the tangential displacement components and shear stress components are determined. With the help of mixed-type boundary conditions, a system of integral equations is derived. After that, by utilizing a trial solution, these integral equations are transformed into a pair of simultaneous Fredholm integral equations of the second kind. The numerical results of the Fredholm integral equations are evaluated by converting it into a system of algebraic equations to interpret the physical quantity, like the stress intensity factor (SIF), which is demonstrated graphically. The obtained results are compared to the previously published article to ensure its accuracy.

Numerical Simulation of Initial Value Problem of Integro-differential Equation Models

In this article, the Volterra-Fredholm integral equation (V-FIE) is derived from an initial value problem of kind integro-differential equation (IVP). We discuss the existence and uniqueness of the solution to the problem in Hilbert space. A numerical method is used to reduce this type of equation to System of Fredholm integral equations of the second kind(SFIEs). In light of this, the clustering method and the Galerkin method to solve the system of second-order Fredholm integral equations(SFIEs) and calculate the error in each case. Finally, the approximate and exact solutions are plotted on the same coordinate plane Using MATLAB code (2022).

Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces

For the numerical solution of Fredholm integral equations on [−1,1] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods.

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.

Об особенностях идентификации переменных термомеханических характеристик функционально-градиентного прямоугольника

The inverse thermoelastic problem of identification of the variable properties of a functionally graded rectangle is studied. Unsteady vibrations are excited by applying mechanical and thermal loads to the upper side of the rectangle. To solve the direct problem in Laplace transforms, the method of separation of variables and the shooting method for harmonics are used. Transformants are inverted by expanding the origin in terms of shifted Legendre polynomials. The method proposed for solving the direct problem is verified by comparison with a finite element solution. The influence of the laws of change of variable characteristics on the boundary physical fields is analyzed. The displacement components give additional information on the mechanical loading, and the temperature measured on the upper side of the rectangle over a certain time interval – on the thermal loading. Assuming that the additional information admits expansion in Fourier series, the two-dimensional inverse problem is reduced to one-dimensional problems for various harmonics. The solution of the obtained nonlinear inverse problems is carried out on the basis of an iterative process, at each stage of which, in order to find corrections for thermomechanical characteristics, systems of Fredholm integral equations of the 1st kind are solved. The possibility of simultaneous reconstruction of several characteristics is investigated. The results of computational experiments on the phased reconstruction of thermomechanical characteristics are presented. The influence of the thermomechanical coupling parameter on the results of the thermal stress coefficient reconstruction was clarified.

Discrete Legendre spectral projection-based methods for Tikhonov regularization of first kind Fredholm integral equations

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates.

“Spectral Method” for Determining a Kernel of the Fredholm Integral Equation of the First Kind of Convolution Type and Suppressing the Gibbs Effect

A set of one-dimensional (as well as one two-dimensional) Fredholm integral equations (IEs) of the first kind of convolution type is solved. The task for solving these equations is ill-posed (first of all, unstable); therefore, the Wiener parametric filtering method (WPFM) and the Tikhonov regularization method (TRM) are used to solve them. The variant is considered when a kernel of the integral equation (IE) is unknown or known inaccurately, which generates a significant error in the solution of IE. The so-called “spectral method” is being developed to determine the kernel of an integral equation based on the Fourier spectrum, which leads to a decrease of the error in solving the IE and image improvement. Moreover, the authors also propose a method for diffusing the solution edges to suppress the possible Gibbs effect (ringing-type distortions). As applications, the problems for processing distorted (smeared, defocused, noisy, and with the Gibbs effect) images are considered. Numerical examples are given to illustrate the use of the “spectral method” to enhance the accuracy and stability of processing distorted images through their mathematical and computer processing.

TRANSFER OF LOADS FROM THREE HETEROGENEOUS ELASTIC STRINGERS TO AN INFINITE SHEET THROUGH ADHESIVE LAYERS

This paper considers the problem for an elastic infinite plate (sheet), which on parallel finite parts of its upper surface is strengthened by three finite stringers, two of which are located on the same line, having different elastic properties. The stringers are deformed under the action of horizontal forces. The interaction between infinite sheet and stringers takes place through thin elastic adhesive layers having other physical-mechanical properties and geometric configuration. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on three finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated. Further, for various values of changing characteristic parameters of the problem the multiple numerical results and its analysis are presented.

Averaged Nyström interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nyström methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose and introduces new stability properties of them.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Advanced Stochastic Approaches for Fredholm Integral Equations

Integral equations are of high applicability in different areas of applied mathematics, physics, engineering, geophysics, electricity and magnetism, kinetic theory of gases, quantum mechanics, mathematical economics, and queuing theory. That is why it is reasonable to develop and study efficient and reliable approaches to solve integral equations. For multidimensional problems the existing biased stochastic algorithms based on evaluation of finite number of integrals will suer more from the effect of high dimensionality, because they are based on quadrature points. So we need advanced unbiased algorithms for solving the multidimensional problem which is developed in this paper. A new unbiased stochastic method for solving multidimensional Fredholm integral equations of second kind is proposed and analysed. We compared the newly proposed unbiased algorithm with the old unbiased stochastic algorithm for the one dimensional problem and multidimensional problem.

Splines of the Second and Seventh Order Approximation and the Stability of the Solution of the Fredholm Integral Equations of the Second Kind

This work is a continuation of a series of works on the use of continuous local polynomial splines for solving interpolation problems and for solving the Fredholm integral equation of the second kind. Here the construction of a numerical solution to the Fredholm integral equation of the second kind using local spline approximations of the second order and the seventh order of approximation is considered. This paper is devoted to the investigation of the stability of the solution of the integral equation using these local splines. Approximation constants are given in the theorem about the error of approximation by the considered splines. Numerical examples of the application of spline approximations of the second and seventh order of approximation for solving integral equations are given.

Adaptive spectral solution method for Fredholm integral equations of the second kind

Adaptive spectral solution method for Fredholm integral equations of the second kind

Efficient Numerical Methods of Inverse Coefficient Problem Solution for One Inhomogeneous Body

In the present paper, the problems of longitudinal and flexural vibrations of an inhomogeneous rod are considered. The Young’s modulus and density are variable in longitudinal coordinate. Vibrations are caused by a load applied at the right end. The proposed method allows us to consider a wider class of inhomogeneity laws in comparison with other numerical solutions. Sensitivity analysis is carried out. A new inverse problem related to the simultaneous identification of the variation laws of Young’s modulus and density from amplitude–frequency data, which are measured in given frequency ranges, is considered. Its solution is based on an iterative process: at each step, a system of two Fredholm integral equations of the first kind with smooth kernels is solved numerically. The analysis of the kernels is carried out for different frequency values. To find the initial approximation, several approaches are proposed: a genetic algorithm, minimization of the residual functional on a compact set, and additional information about the values of the sought-for functions at the ends of the rod. The Tikhonov regularization and the LSQR method are proposed. Examples of reconstruction of monotonic and non-monotonic functions are presented.

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.

Jacobi spectral projection methods for Fredholm integral equations of the first kind

Jacobi spectral projection methods for Fredholm integral equations of the first kind

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method