Fixed Point Results for Interpolative Kannan Type and Ćirić‐Reich‐Rus Type Cyclic Contractions in Dislocated Quasi‐Rectangular b‐Metric Spaces

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets

Integral equations, the main branch of mathematics, are widely used in physics, engineering, mechanics, control theory and other fields. Related to the application of integral equations, new fields are developing, such as economics, some sections of biology, etc. The theory of integral equations mainly developed in the late nineteenth-early twentieth century, starting with Vito Volterra (1982, 1986), Eric Ivar Fredholm (2010), David Hilbert, Erhard Schmidt, etc. scientists began to study it. Nevertheless, within the framework of mathematical concepts that existed before the first half of the twentieth century, such problems were considered incorrect because a small change in the given functions led to a greater change in the desired functions. The Volterra equation of the first kind is an integral equation that has an exact solution only in some cases. The limit of integration has been carried out in very small quantities on non-classical linear and nonlinear integral equations with variable limits and the construction of solutions in these works is based on numerical methods. Therefore, for the so-called non-classical Volterra integral equations, it is relevant to determine the conditions that ensure the uniqueness and regularization of their solutions. In this paper, the uniqueness of the solution of the non-classical nonlinear integral Volterra equation of the first kind is resolved. The aim of the study is to solve the non-classical Volterra integral equation of the first kind, that is, to determine the conditions that ensure the uniqueness of the solution of the nonlinear non-classical Volterra integral equation of the first kind. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems.

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

This dissertation explores the use of a preconditioned Richardson iterative algorithm for the solution of linear and nonlinear ill-posed integral equations of the first kind. The discussion consists of three parts, which can be roughly categorized as: numerical analysis, applications to statistical methodology, and an application to an inverse problem. In the first part, singular matrix equations that result from discretizing ill-posed integral equations of the first kind are considered. Sufficient conditions for the convergence of Richardson's algorithm to a solution are established, and necessary and sufficient conditions are proved for special cases. The inconsistent case is also discussed. A preconditioning for equations with positive kernels leads to the Conditional Expectation algorithm, which is discussed in detail. A notion of 'iterative regularization' is introduced and related to the more usual penalized least squares approach to regularization. In the second part two problems in statistical methodology are considered which involve the solution of nonlinear integral equations of the first kind. The first is the Behrens-Fisher problem. Trickett and Welch (Biometrika, 1954) determined a very nearly similar test for the Behrens-Fisher problem having reasonable power by numerically 'solving' a nonlinear integral equation. The Trickett-Welch method is examined, and a version of the Conditional Expectation algorithm for nonlinear equations is applied to the Behrens-Fisher problem. The second methodological problem that is considered is that a $\beta$-content tolerance limits involving data from a one-way balanced random effects model. The Conditional Expectation algorithm is used to approximately solve a nonlinear equation of the first kind numerically, and to thereby derive a new tolerance limit procedure which is shown to be a substantial improvement over the only other method in the statistics literature. In the third part an inverse problem is discussed in which the right hand side of the integral equation is estimated. In this example, the objective is to infer the probability density of the radii of random spheres in a two-phase medium from radii of circles in cross-sectional slices of this medium. The Conditional Expectation algorithm leads to an effective technique for solving this problem.

ABSTRACTIn this paper, a double-exponential (DE) Sinc Nyström method is utilized to solve nonlinear two-dimensional Fredholm integral equations of the second kind. Using the DE transformation, the Sinc quadrature rule for a definite integral is extended to double integral over a rectangular region. Therefore, a nonlinear Fredholm integral equation is reduced to a system of nonlinear algebraic equations, which is solved using the Newton iteration method. Convergence analysis shows that the proposed method can converge exponentially. Several numerical examples are provided to demonstrate the high efficiency and accuracy of the proposed method.

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra‐Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

Fuzzy integral equations play an important role in addressing uncertain mathematical problems. There are various techniques present in the literature to solve fuzzy linear integral equations. Different methodologies provide numerical solutions for fuzzy nonlinear integral equations. However, there are few recognized methods for finding an exact solution. The fuzzy set has limitations because it lacks a non-membership degree for investigating uncertainty. To address this limitation, we use an intuitionistic fuzzy set that considers both membership and non-membership degrees together. Using the parametric forms of an intuitionistic fuzzy number, the nonlinear Fredholm integral equation is decomposed into a set of four equations. This set of four equations is then named the intuitionistic fuzzy nonlinear Fredholm integral equation. For an exact solution to the intuitionistic fuzzy nonlinear Fredholm integral equation, we use the Direct Computational Method. We solve two different examples in detail to demonstrate the reliability, effectiveness, and applicability of the proposed methodology. Graphs made using MATLAB represent visual judgments on how uncertainty impacts solutions. The results obtained for both examples are carefully examined and discussed in detail. The proposed method is compared to different decomposition and deep learning methods to ensure its accuracy. It is concluded that the proposed method is valid and reliable to get an exact solution for an intuitionistic fuzzy nonlinear Fredholm integral equation.

ABSTRACTThe present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

In this work, we establish new fixed point theorems for w-generalized weak contraction mappings with respect to w-distances in complete metric spaces by using the concept of an altering distance function. As an application, we use the obtained results to aggregate the existence and uniqueness of the solution for nonlinear Fredholm integral equations and Volterra integral equations together with nonlinear fractional differential equations of Caputo type.