Kind Fredholm Integral Equations Research Articles

Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind

The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed, when the kernel is sufficiently differentiable into any of its two components, through its approximation by Taylor’s polynomial. In this paper, we study the case in which the kernel is not differentiable in any of its two components. So, we present a strategy that consists of approximating the kernel of the nonlinear integral equation by a Chebyshev interpolation polynomial, which is separable. This allows us to explicitly calculate the inverse of the first derivative operator in each step of the Newton method and then approximate a solution of the approximate integral equation.

A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green’s kernels

A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green’s kernels

Solving Volterra-Fredholm integral equations by non-polynomial spline functions

It depends on our information, non-polynomial spline functions have not been applied for solving Volterra- Fredholm integral equations of the second kind yet. In this paper, we want to use such functions for finding approximation solutions of Volterra-Fredholm integral equations. In our approach, the coefficients of the non-polynomial spline were found by solving a system of linear equations. Then, these functions were utilized to reduce the fredholm integral equations to the solution of algebraic equations. Analysis of convergences investigated. Finally, three examples were presented to show the effectiveness of the method. This was done with the help of a computer program that used the Python code program version 3.9.

An Iterative Method for the Approximation of Common Fixed Points of Two Mappings: Application to Fractal Functions

This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong convergence of the sequence generated by the iterative scheme to a common fixed point of two operators. The method is illustrated with some examples of application. The procedure is used to approach a common solution of two Fredholm integral equations of the second kind. In the second part of the article, the existence of a fractal function coming from two different Read–Bajraktarević operators is proved. Afterwards, a study of the approximation of fixed points of a fractal convolution of operators is performed, in the framework of Lebesgue or Bochner spaces.

Modified Successive Approximations Method for Solving Fuzzy Second Kind Fredholm Integral Equations with Separable Kernel

This paper presents an analysis, development, and application of a modified method of successive approximations to solve fuzzy second kind Fredholm integral equations with a separable kernel. The fuzziness in the equations is represented utilizing convex normalized triangular fuzzy numbers, which are based on a single and double parametric form of fuzzy numbers. In the single parametric form of the fuzzy number, a fuzzy equation converts to two crisp equations for the solution which increases the computational cost. Therefore, to reduce the computational cost and increase the solution's accuracy, the double parametric form of the fuzzy number will be used in this study. The feasibility of the proposed methods is illustrated through a numerical example. The results obtained using the presented approach demonstrate adequate agreement with relevant theoretical aspects.

Refined Unbiased Stochastic Approach for Fredholm Integral Equations

Abstract Integral equations are highly relevant across various fields such as applied mathematics, physics, engineering, geophysics, electromagnetism, the kinetic theory of gases, quantum mechanics, mathematical economics, and queuing theory. This underscores the importance of creating and examining efficient and dependable methods to solve integral equations. In the context of multidimensional issues, traditional biased stochastic algorithms, which rely on a finite set of integration points, are particularly challenged by the complexities of higher dimensionality. Therefore, it is critical to develop sophisticated unbiased algorithms to address these challenges, as discussed in this paper. We introduce and evaluate a novel unbiased stochastic approach for solving multidimensional Fredholm integral equations of the second kind.

A review of radial kernel methods for the resolution of Fredholm integral equations of the second kind

The paper presents an overview of the existing literature concerning radial kernel meshfree methods for the numerical treatment of second-kind Fredholm integral equations. More in detail, it briefly recalls radial basis function (RBF) interpolation and cubature rules to properly describe numerical methods for two-dimensional linear Fredholm equations. The RBF approach allows us to consider the case when the involved functions are not known analytically, but only as vectors of scattered data samples. The described methods do not require any background mesh and, hence, are also independent on the geometry of the domain.

Solving an internal problem for finite regular two-dimensional lattice spiral elements, excitable plate electromagnetic wave

Background. The work is aimed at developing and researching rigorous methods for solving internal problem of electrodynamics for multi-element structures (metastructures) consisting from the final number of elements, as well as to study the physical processes occurring in them. A special case of such structures are two-dimensional lattices with a fixed interelement distance, consisting of identical elements having the same spatial orientation (regular lattices). Aim. In this work, based on an iterative approach, the internal solution is solved. problems of electrodynamics for a finite regular two-dimensional lattice of spiral elements. In order to obtain a priori information about the electrodynamic characteristics of elements lattice and justification for the choice of projection function systems are analyzed spectral characteristics of the integral operator of the internal problem for a single spiral element. Then the currents on the structure elements are calculated, their spectral characteristics are determined. The results of spectral analysis allow increase the efficiency of solving an internal problem. Methods. The research is based on a strict electrodynamic approach, within the framework of which, for the specified structure in the thin-wire approximation, an integral representation of the electromagnetic field is formed, which, when considered on the surface of conductors together with boundary conditions, is reduced to a system of Fredholm integral equations of the second kind, written relative to unknown current distributions on conductors (internal task). The solution of the internal problem within the framework of the method of moments is reduced to solving a SLAE with a block matrix. Results. A mathematical model of a finite two-dimensional lattice of spiral elements is proposed radiating structure. For the specified structure, in the case of its excitation by a flat electromagnetic wave, based on the iterative approach, the internal problem of electrodynamics was solved. The following were carried out in a wide frequency range: analysis of the convergence of the iterative process, spectral analysis of the integral operator of the internal problem for a single spiral element, as well as spectral analysis of external field and current functions functions on lattice elements. Conclusion. The feasibility of determining the spectral characteristics of integral operators is shown internal task for the elements forming the metastructure. A relationship has been identified between the frequency dependence eigenvalues of the integral operator of the internal problem of single elements, forming a metastructure, with resonance phenomena arising in the metastructure, the influence of resonances on the convergence of the iterative process was confirmed. The feasibility of considering averaged amplitude current spectra is shown. It was revealed that the averaged spectrum of current functions is close to degenerate, especially near resonant frequencies. This allows for use as projection functions a compact set of eigenfunctions that have significant amplitudes in the vicinity of the frequency under study, which significantly simplifies the solution of the internal problem.

Electrodynamic analysis of a sinusoidal antenna for small wave sizes

Background. The work is aimed at developing and researching rigorous methods for calculating thin-wire structures with a complex generatrix shape, having small wave sizes, as well as studying the physical processes occurring in them. A special case of such structures is a sinusoidal antenna operating in a standing current wave mode. Aim. In progress the solution of internal and external problems of electrodynamics is carried out for a sinusoidal antenna of small wave sizes located above an infinitely extended ideal reflector. The currents on the elements of the structure are calculated, its input resistance and radiation characteristics are determined. Methods. The research is based on a strict electrodynamic approach, within the framework of which, for the specified structure in the thin-wire approximation, an integral representation of the electromagnetic field is formed, which, when considered on the surface of conductors together with boundary conditions, is reduced to a system of Fredholm integral equations of the second kind, written relative to unknown current distributions on conductors (internal task). Results. A mathematical model of the radiating structure is proposed, determined: the input resistance of the structure and the basic characteristics of its radiation. It is shown that the operating range of a sinusoidal antenna in the standing wave mode is determined by the quality factor of the input impedance resonances; An increase in the width of the sinusoidal conductor leads to a decrease in the resonant frequencies of the input resistance with a simultaneous increase in the quality factor of the resonances. Conclusion. From a practical point of view, the use of the considered structure allows significantly reduce the dimensions in comparison with a thin electric vibrator, however in this case, the operating range will be correspondingly narrowed, which is determined, due to the weak dependence of the radiation characteristics on frequency, by the quality factor of the resonances. The current distribution on the generatrix of the structure can be considered as a «projection» standing surface wave localized in the plane of a sinusoidal conductor, and resulting from the superposition of forward and backward surface (slow) waves propagating at a speed significantly lower than the speed of light. To further clarify the physics of the processes occurring in the structure, one should use spectral analysis of current functions and study the distributions of the electromagnetic field in the near zone of the structure.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Approximate Solutions to Weakly Singular Fredholm Integral Equations of the Second Kind using Shifted Legendre Polynomials of the First Kind

In this research, we presented a simple approach to approximate the second type of linear weakly singular and non-singular Fredholm integral. Shifted Legendre polynomials of the first kind in matrix-vector forms were used to construct the approach. The singularity of the kernel was removed analytically. Theorems regarding the convergence of the estimations of the error norm and the mean were proved. The numerical examples demonstrated the method's uniqueness and precision.

Numerical solution for the interior electric displacement of the moving DBY‐PS model for semi‐permeable cracked piezoelectric material

AbstractIn this study, we analysed a moving crack at the interface of an infinitely long piezoelectric bilayer using the Dugdale–Barenblatt yield (DBY) model and the polarisation saturation (PS) model. To model the moving crack problem, a Yoffe‐type crack moves at a constant subsonic speed on the interface of an infinitely long piezoelectric bilayer. The crack faces are assumed to be semi‐permeable, and at the boundary of the bilayer, in‐plane electrical and out‐of‐plane mechanical stresses are applied. Due to the application of electro‐mechanical loads, cracks propagate, mechanical yielding zones and electric saturation zones are developed. To arrest the crack from further propagation, mechanical yield stress and saturation electric displacement are applied at the developed zones. To address this problem analytically and numerically, the mixed boundary value problem is transformed into a set of coupled Fredholm integral equations (FIEs) of the second kind using the Fourier transform and the Copson method. The closed‐form analytical expressions for the length of the electrical saturation zone (ESZ), whether longer, shorter or equal to the mechanical yielding zone (MYZ), show dependence on external electro‐mechanical loads under semi‐permeable crack conditions. The algorithm to solve the electric crack condition parameter (ECCP) has been defined using numerical discretization and the bisection method. Illustrative examples demonstrate the proposed technique's effectiveness and suitability for Yoffe‐type moving cracks. The numerical results show the convergence of the ECCP. Furthermore, the numerical results show how mechanical and electrical zone lengths and energy release rate (ERR) are affected by electrical and mechanical loads, strip thickness and crack velocity. In addition, the size of the mechanical yielding zone is consistently promoted by electrical load, while the promotion or prevention of the electrical saturation zone by mechanical load depends on the relative sizes of the nonlinear zones.

Fredholm Integral Equations by Variational Iteration Method

In this research paper, numerical method Variational Iteration Method is used to solve Second Kind Fredholm Integral Equation .Numerical examples illustrate the accuracy of the numerical method Variational Iteration Method.

On the application of some fixed-point techniques to Fredholm integral equations of the second kind

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind.

Solution of the internal and external problems of electrodynamics for a symmetrical vibrator with a reflector made of parallel straight conductors

Background. The work is aimed at the development and research of rigorous methods for calculating multi-element emitting and re-emitting structures, consisting mainly of the same type of elements, as well as studying the physical processes occurring in them. An iterative approach to solving the internal problem is proposed, which allows minimizing the cost of computer time and computer memory. Aim. In this work, using the proposed approach, we solve the internal and external problems of electrodynamics for a symmetrical vibrator with a reflector made of parallel straight conductors. The convergence of the iterative process is studied, the currents on the elements of the structure, its input resistance and radiation characteristics are calculated. Methods. The research is based on a strict electrodynamic approach, within the framework of which, for the specified structure in the thin-wire approximation, an integral representation of the electromagnetic field is formed, which, when considered on the surface of conductors together with boundary conditions, is reduced to a system of Fredholm integral equations of the second kind, written relative to unknown current distributions on conductors (internal task). The solution of the internal problem within the framework of the method of moments is reduced to solving a SLAE with a block matrix. Results. A mathematical model of a radiating structure is proposed, which is a symmetrical vibrator with a reflector made of parallel straight conductors. The internal and external problems of electrodynamics are formulated and solved for given parameter values. An effective algorithm for calculating the block matrix of SLAEs is proposed. Recommendations are given for the selection of systems of projection functions within the framework of the method of moments. The convergence of the iterative process for solving an internal problem of electrodynamics has been studied. The input resistance of the structure and the basic characteristics of its radiation are determined. Conclusion. A rational choice of systems of projection functions, based on the properties of the structure and its constituent elements, makes it possible to significantly reduce the size of the SLAE matrix and, accordingly, reduce computational costs. Taking into account the properties of the structure and its constituent elements also makes it possible to construct effective algorithms for calculating the SLAE matrix. It is shown that the convergence of the iterative process may be absent near resonant frequencies, therefore, when solving the internal problem, a combined approach should be used, which involves the use of both rigorous and approximate methods for solving SLAEs. At non-resonant frequencies, the iterative approach demonstrates its effectiveness. It is concluded that for such structures it is advisable to distinguish between resonant and non-resonant operating modes. In the non-resonant mode, the current functions have a fairly smooth dependence on frequency; in the resonant mode, these dependences become quite sharp and difficult to predict. In this regard, the task of developing an iterative approach to solving the internal problem of electrodynamics in resonant operating modes seems relevant.

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

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

On the numerical solution of weakly singular integral equations of the first kind using a regularized projection method

This study investigates a numerical method based on the Jacobi–Gauss quadrature for solving Fredholm integral equations of the first kind with a weakly singular kernel by combining the Tikhonov regularization and projection methods. This numerical method reduces the solution of the weakly singular integral equations of the first kind to the solution of a linear system of algebraic equations. The theoretical analysis of the proposed technique is provided. Finally, several tests are presented to show the validity and efficiency of this approach.

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.