Fuzzy Fredholm Integral Equations Research Articles

Successive approximation method to solve nonlinear fuzzy Fredholm integral equations using NC rules

Successive approximation method to solve nonlinear fuzzy Fredholm integral equations using NC rules

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.

Gauss-Legendre type quadrature iterative method for fuzzy Fredholm integral equations

In this paper we obtain an iterative method for the numerical solution of second kind fuzzy Fredholm integral equations by applying appropriate composite fuzzy quadrature formula. We approach a general fuzzy quadrature formula and show that the three-point and four-point Gauss-Legendre type fuzzy quadrature formulas are the best choice from computational cost and accuracy point of view. The convergence of the method is proved by providing the error estimates expressed in terms of the Lipschitz constant of the Picard iterations. Some numerical results confirm the obtained theoretical result and illustrate the performances of the three point Gauss-Legendre method in comparison with the techniques generated by the fuzzy Newton-Cotes and Gauss-Lobatto quadrature formulas, the results being tested on the Simpson and on the four point Gauss-Legendre and Gauss-Lobatto quadrature formulas in particular. The numerical experiment suggests that five point or more point fuzzy quadrature formulas are useless due to the accumulation of errors and the grown computational cost.

A Numerical Study for Solving the Systems of Fuzzy Fredholm Integral Equations of the Second Kind Using the Adomian Decomposition Method

In this paper, the Adomian decomposition method (ADM) is successfully applied to find the approximate solutions for the system of fuzzy Fredholm integral equations (SFFIEs) and we also study the convergence of the technique. A consistent way to reduce the size of the computation is given to reach the exact solution. One of the best methods adopted to determine the behavior of the approximate solutions. Finally, the problems that have been addressed confirm the validity of the method applied in this research using a comparison by combining numerical methods such as the Trapezoidal rule and Simpson rule with ADM.

On the Ideal Convergent Sequences in Fuzzy Normed Space

This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.

Application of Hybrid Bernstein Polynomials and Block-Pulse Functions for Solving Nonlinear Fuzzy Fredholm Integral Equations

In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of successive approximations are applied to obtain the approximate solution of nonlinear fuzzy Fredholm integral equations. The main idea of using the proposed method is that fuzzy integral in any iterative process will be reduced to the crisp integration. Some results concerning the error estimate and stability of the numerical method are presented. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

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.

Approximate solution of nonlinear fuzzy Fredholm integral equations using bivariate Bernstein polynomials with error estimation

<abstract><p>This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.</p></abstract>

Some fixed point results for <i>ξ</i>-chainable neutrosophic and generalized neutrosophic cone metric spaces with application

<abstract> <p>This manuscript is concerned for introducing novel concepts of <italic>ξ</italic>-chainable neutrosophic metric space and generalized neutrosophic cone metric spaces. We use four self-mappings to establish common fixed point theorem in the sense of <italic>ξ</italic>-chainable neutrosophic metric space and three self-mappings to establish common fixed point results in the sense of generalized neutrosophic metric spaces. Certain properties of <italic>ξ</italic>-chainable neutrosophic metric space and generalized neutrosophic metric spaces are defined and their examples are presented. An application to fuzzy Fredholm integral equation of second kind is developed to verify the validity of proposed results. These results boost the approaches of existing literature of fuzzy metric spaces and fuzzy fixed theory.</p> </abstract>

An improved numerical iterative method for solving nonlinear fuzzy Fredholm integral equations via Picard’s method and generalized quadrature rule

An improved numerical iterative method for solving nonlinear fuzzy Fredholm integral equations via Picard’s method and generalized quadrature rule

Existence and uniqueness of solution for fuzzy integral equations of product type

In this paper, we study the fuzzy Fredholm integral equations of product type. As a special case, integral equations of product type arise, e.g., in the study of the spread of an infectious disease that does not induce permanent immunity. Because of the importance and practicality of such integrals, we prove the existence and uniqueness of the solution of these integral equations with the help of the Banach fixed point theorem. Finally, we obtain an error estimation between the exact solution and the solution of the iterative method. Examples show the applicability of our results.

Bernoulli wavelet method for numerical solutions of system of fuzzy integral equations

In this paper, we have proposed an efficient numerical method to solve a system linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM). Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First, we introduce properties of Bernoulli wavelets then we used it to transform the integral equations to the system of algebraic equations, the error estimates of the proposed method are given and compared by solving some numerical examples.

Efficient Numerical Solution to a Bivariate Nonlinear Fuzzy Fredholm Integral Equation

In this article, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well-proven approaches-successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that, by targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove the stability and the method is performed on test problems to verify our theoretical results; numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method.

Computation of semi-analytical solutions of fuzzy nonlinear integral equations

In this article, we use a fuzzy number in its parametric form to solve a fuzzy nonlinear integral equation of the second kind in the crisp case. The main theme of this article is to find a semi-analytical solution of fuzzy nonlinear integral equations. A hybrid method of Laplace transform coupled with Adomian decomposition method is used to find the solution of the fuzzy nonlinear integral equations including fuzzy nonlinear Fredholm integral equation, fuzzy nonlinear Volterra integral equation, and fuzzy nonlinear singular integral equation of Abel type kernel. We also provide some suitable examples to better understand the proposed method.

Study of hybrid orthonormal functions method for solving second kind fuzzy Fredholm integral equations

The approximate numerical solution of the linear second kind of fuzzy integral Fredholm equations is discussed in this article. A new approach uses hybrid functions, and some useful properties of these functions are proposed to transform linear second type fuzzy integral Fredholm equations into an algebraic equation. The new approach is a mixture of Bernstein polynomials (BPs) and enhanced block-pulse functions (IBPFs) at interval [0, 1). The approach is appealing and very easy to implement computationally. Some numerical tests show the reliability and exactness of the suggested scheme.

A Highly Efficient and Accurate Finite Iterative Method for Solving Linear Two-Dimensional Fredholm Fuzzy Integral Equations of the Second Kind Using Triangular Functions

This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two m-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions.

An Accurate and Efficient Technique for Approximating Fuzzy Fredholm Integral Equations of the Second Kind Using Triangular Functions

An Accurate and Efficient Technique for Approximating Fuzzy Fredholm Integral Equations of the Second Kind Using Triangular Functions

Solving two-dimensional fuzzy Fredholm integral equations via sinc collocation method

In this paper, we present a numerical method to solve two-dimensional fuzzy Fredholm integral equations (2D-FFIE) by combing the sinc method with double exponential (DE) transformation. Using this method, the fuzzy Fredholm integral equations are converted into dual fuzzy linear systems. Convergence analysis is performed in terms of the modulus of continuity. Numerical experiments demonstrate the efficiency of the proposed method.

Classical approximation for fuzzy Fredholm integral equation

We are focused on fuzzy Fredholm integral equations of the second kind. To solve these equations with respect to fuzzy valued functions, we apply the ordinary approximation technique. We propose to use Chebyshev polynomials due to their smoothness and reasonable behavior near boundaries. This trick is our principal innovation, because we combine a classical technique with a fuzzy problem. This approach allows to use the Clenshaw-Curtis representation and transform a fuzzy Fredholm integral equation to the system of algebraic equations. A solution to this algebraic system gives an approximate (functional) solution to the original problem. We discuss the existence and uniqueness of a solution. The proposed method is illustrated by numerical examples, which confirms the theoretical convergence estimate.

Valid implementation of Sinc-collocation method to solve the fuzzy Fredholm integral equation

The main purpose of this work is to validate the numerical solution of fuzzy Fredholm integral equation (IE) by applying the Sinc-collocation method (S-CM) based on the double exponential (DE) and single exponential (SE) decays. This approach is based on the discrete stochastic arithmetic (DSA) which is able to validate the results with optimal solution and rely the proposed algorithm in comparison with the floating-point arithmetic (FPA). In order to achieve this goal, the fuzzy CESTAC11Controle et Estimation Stochastique des Arrondis de Calculs. method and the CADNA22Control of Accuracy and Debugging for Numerical Applications. library are applied. The accuracy of the S-CM in both SE and DE cases is discussed by proving the theorems. Also, two numerical algorithms based on the S-CM in SE and DE decays by applying the fuzzy CESTAC method and CADNA library are presented. The termination criterion in each algorithm is based on the Hausdorff distance to be an informatical zero. By solving the examples, not only the optimal approximation and the optimal iteration of the S-CM can be found but also similar to the non-fuzzy case, it is shown that the DE precision is more accurate with faster convergence rate than the SE.