Fredholm Integral Equation Research Articles

Homotopy Perturbation Method with Analytics for solving Bivariate type II Fuzzy Fredholm Integral Equations

A numerical scheme known as homotopy perturbation method (HPM) is a powerful tool for solving a wide range of problems arising in several scientific applications. In this manuscript, we focus on bivariate type II fuzzy fredholm integral equations (BTII-FF-IEqs) to obtain fuzzy approximate solutions using HPM. The efficiency and effectiveness of the approach is tested upon numerical example and the obtained numerical results are compared with the existing exact solutions. The results reveal that the proposed method is straightforward, accurate and convenient.

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.

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.

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.

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

Some Results of Fixed Point for Single Value Mapping in Fuzzy Normed Space with Applications

Fixed point theory is not only essential in mathematics but also to a vast array of applications. In this approach, we provide some fixed point theorems for fuzzy normed space. The triangle property of a fuzzy norm is introduced first. This property is used to demonstrate some fixed point solutions for self-mappings in fuzzy normed space. To demonstrate the importance of the obtained results, an application for the existence of a solution to the ordinary differential equation and the Fredholm integral equation is constructed.

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.

An inversion problem for optical spectrum data via physics-guided machine learning

We propose the regularized recurrent inference machine (rRIM), a novel machine-learning approach to solve the challenging problem of deriving the pairing glue function from measured optical spectra. The rRIM incorporates physical principles into both training and inference and affords noise robustness, flexibility with out-of-distribution data, and reduced data requirements. It effectively obtains reliable pairing glue functions from experimental optical spectra and yields promising solutions for similar inverse problems of the Fredholm integral equation of the first kind.

New algorithm on linearization-discretization solving systems of nonlinear integro-differential Fredholm equations

This article deals with a new strategy for solving a certain type of nonlinear integro-differential Fredholm equations with a weakly singular kernel. We build our new algorithm starting with the linearization phase using Newton's iterative process, then with the discretization phase we apply the Kantorovich's projection method. The discretized linear scheme will be approximated by the product integration method in the weak singular terms, and the other regular integrals will be approximated by the Nyström method. The process of convergence of our new algorithm is carried out under certain predefined and necessary conditions. Finally, we give practical examples where, the results show the efficiency of our new algorithm for solving systems of weakly singular nonlinear integro-differential equations.

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by x(s)=1+λx(s)∫01st+sx(t)dt, can be very often encountered in many applications, where x is the function to be determined, λ is a parameter, and t,s∈[0,1]. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form χ(l)=ϱ(l)+χ(l)∫pqk(l,z)(Vχ)(z)dz are investigated in the space Cωp,q, where χ is the unknown function to be determined, V is a given operator, and ϱ,k are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.

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.

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Existence of common fuzzy fixed points via fuzzy F-contractions in b-metric spaces

The main goal of this study is to establish common fuzzy fixed points in the context of complete b-metric spaces for a pair of fuzzy mappings that satisfy F-contractions. To strengthen the validity of the derived results, non-trivial examples are provided to substantiate the conclusions. Moreover, prior discoveries have been drawn as logical extensions from pertinent literature. Our findings are further reinforced and integrated by the numerous implications that this technique has in the literature. Using fixed point techniques to approximate the solutions of differential and integral equations is very useful. Specifically, in order to enhance the validity of our findings, the existence result of the system of non-linear Fredholm integral equations of second-kind is incorporated as an application.

On Ulam stabilities of iterative Fredholm and Volterra integral equations with multiple time-varying delays

In present study, we deal with nonlinear iterative Fredholm and Volterra integral equations (Fredholm and Volterra IEs) including variable time delays. We are interested here in the investigations of the uniqueness of solutions and Ulam type stabilities of that the iterative Fredholm and the Volterra IEs. The proofs of the new outcomes of the study with regard to these concepts are done in the light of the Banach fixed point theorem (Banach FPT) and the Bielecki metric. As for new contributions of the present study, here, first time we develop the relative outcomes that can be found in the literature to certain nonlinear iterative Fredholm and Volterra IEs including several variable time delays. Finally, a concrete example is introduced at the end of the study.

Hemi metric spaces and Banach fixed point theorems

In this work, we will define a new type metric with degree m and m+1 points which is called m-hemi metric as a generalization of two metric spaces. We will give and prove some topological properties. Also, Banach contraction mapping principle were proved and a application to Fredholm integral equation were gived in hemi metric spaces.

Modal analysis of a submerged elastic disk: A hypersingular integral equation approach

A method based on the hypersingular integral equation approach and the modal analysis is presented to consider the effects of the motion of a submerged elastic disk on the incoming waves. Initially, the governing boundary value problem is reduced to a two-dimensional integral equation with a hypersingular kernel. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind with the help of Fourier series expansions and a newly defined function. As a part of modal analysis, eigenfunction expansion based on natural modes of structural motion is considered to describe the motion of a thin circular elastic disk. Physical quantities, such as hydrodynamic force, added mass, damping coefficient, and surface elevation, are numerically evaluated. The computed numerical results are verified by comparing them with those for the rigid disk horizontally submerged in deep water. Apart from this, as a part of the analytical verification of our present analysis, the reciprocity relation has been included. The effects of different parameters (disk's rigidity, radius, submergence depth, and mode of vibrations) on the aforementioned physical quantities have been studied. The maximum hydrodynamic force occurs around Ka = 0.5, while the maximum added mass and damping coefficient occur around the wavenumber Ka = 0.3 and Ka = 0.5, respectively. The peaks of the hydrodynamic force and free surface elevation become sharper with the increasing values of the disk's size. The numerical results emphasize that the wave focusing can be controlled by changing the submergence depth, size, and rigidity of the disk.

Обобщённая математическая модель основных задач теории упругости для анизотропных тел

Introduction. The assessment of the strength of underground structures is a complex task that requires a comprehensive approach. In many cases empirical methods based on experience are not sufficient. For effective application of analytical methods, it is necessary to use mathematical modelling. This allows to simplify the problem considerably, to consider different variants, to identify cases to obtain a solution in a closed form. Materials and methods. The model is based on a system of singular integral equations with a shift and complex conjugate terms. The solution of the system is the complex elastic potential. Systems of singular equations with a Carleman shift are considered on the space of random functions converging in the mean square to expand the possibility of practical use of the work results. Results. The results of the study are: a mathematical model formulation, the development of an algorithm for reducing a system of singular equations to a system of known Fredholm integral equations of the second kind, a method for determining the solvability conditions and the number of solutions. Discussion. As a result of the study: (1) a general algorithm for finding a solution to a mathematical model is obtained; (2) the methodology for determining the number of solutions and the type of solvability conditions is given; (3) The solution algorithm is extended for the stochastic case; (4) The possibility of wider application of the mathematical apparatus of functions of complex variable for research of strength properties of rocks is proved. Conclusion. The proposed comprehensive approach to assessing the strength characteristics of underground structures, combining a set of analytical methods in conjunction with mathematical modelling, expands the variability of calculation and allows to simplify the problem significantly and obtain a solution in closed form. Resume. The paper develops a mathematical model of the stress state of an elastic body, which allows solving the basic problems of elasticity theory for a variety of media. Suggestions for practical applications and directions for future research. The results of the study make it possible to improve the efficiency of analytical methods in the study of the strength of underground structures. The obtained results are applicable for predicting the strength and safety of underground structures.

Common Fixed Point Theorems in Fuzzy Normed Space

. Fixed point theory is an intriguing topic with numerous applications in several branches of mathematics. In addition, fixed point theory offers useful methods for problem-solving that may be used across many subfields of mathematical analysis. Fixed point theorems are concerned with mappings f of a set X into itself that, under particular conditions, permit a fixed point, that is, a point such that . In this paper, we use the triangle property on fuzzy normed space(Fn-space) to show a common fixed point (CF-point) without continuity. First, we review some of the fundamental terms used in the fuzzy context. After that the notion of triangle fuzzy norm is given then we show that the self mappings Γ and Λ have a CF-point in the setting of Fn-space. Additionally, certain applications of our main results to the Fredholm integral equation are investigated.

Numerical approximation of Fredholm integral equation by the constrained mock-Chebyshev least squares operator

In this paper, we propose two numerical approaches for approximating the solution of the following kind of integral equation f(y)−μ∫−11f(x)k(x,y)w(x)dx=g(y),y∈[−1,1],where f is the unknown solution, μ∈R∖{0}, k,g are given functions not necessarily known in the analytical form, and w is a Jacobi weight. The proposed projection methods are based on the constrained mock-Chebyshev least squares polynomials, and starting from data known at equally spaced points, provide a fine approximation of the solution. Such peculiarity can be helpful in all cases we deal with experimental data, typically measured at equispaced points. We prove the introduced methods are stable and convergent in some Sobolev subspace of C[−1,1]. Several numerical tests confirm the theoretical estimates and numerical effectiveness of the proposed methods.