Fuzzy Integral Equations Research Articles

Numerical and analytic method for solvingproposal New Type for fuzzy nonlinear volterra integral equation

In this paper, we proved the existence and uniqueness and convergence of the solution of new type for nonlinear fuzzy volterra integral equation . The homotopy analysis method are proposed to solve the new type fuzzy nonlinear Volterra integral equation . We convert a fuzzy volterra integral equation for new type of kernel for integral equation, to a system of crisp function nonlinear volterra integral equation . We use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy volterra integral equation . Some numerical examples is given and results reveal that homotopy analysis method is very effective and compared with the exact solution and calculate the absolute error between the exact and AHM .Finally using the MAPLE program to solve our problem .

Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network

In the present paper, using generalizations of the fuzzy integral equations for interval-valued fuzzy sets, we introduce and study new generalized interval-valued fuzzy linear Fredholm integral equation concepts. The work of this paper is an expansion of the research of real fuzzy linear Fredholm integral equations. In this paper interval-valued fuzzy neural network (IVFNN) can be trained with crisp and interval-valued fuzzy data. In this paper, a novel hybrid method based on IVFNN and Newton–Cotes methods with positive coefficient for the solution of interval-valued fuzzy linear Fredholm integral equations (IVFLFIEs) of the second kind is presented. Within this paper the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Then a simple algorithm from the cost function of the interval-valued fuzzy neural network is proposed, in order to find the approximate parameters. We propose a learning algorithm from the cost function for adjusting of interval-valued fuzzy weights. Here neural network is considered as a part of a larger field called neural computing or soft computing. Finally, we illustrate our approach by some numerical examples.

Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and Bernstein Polynomials

A hybrid of regularization methodand Bernstein polynomials is used to solve the first kind fuzzy integral equations. In this paper, first the regularization method applied to convert the first kind fuzzy integral equation into the second kind fuzzy integral equation. Then by approximating Bernstein polynomials, the obtained second kind fuzzy integral equation is solved. In this method, one parameter was created in the second kind equation. When this parameter tends to zero, the solutions of integral equation of the second kind tend to solutions of the integral equation of the first kind. The obtained solutions are comparable to the solutions of the other similar methods. Performance of the mentioned method is illustrated by considering some example.

Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind.

In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.

Analysis of global solution of functional fuzzy integral equations using Krasnoselskii-Burton fixed point theorem

Fixed point theorems are very important and advantageous concepts in mathematical analysis. In this paper, we establish Krasnoselskii-Burton fixed point theorem for fuzzy Banach space. Using fuzzy Krasnoselskii-Burton and applying some notions and definitions in fuzzy functions and fuzzy normed space such as compactness and convexity of fuzzy sets, contraction fuzzy mappings and equicontinuous fuzzy functions we prove the existence of global solution of functional fuzzy integral equations.

Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions

In this paper, we propose an iterative numerical method for two-dimensional fuzzy Volterra linear integral equations. The method uses a recent fuzzy trapezoidal cubature formula for Lipschitzian fuzzy-number-valued functions applied at each iterative step. The convergence of the method is proved providing an error estimate and some numerical experiments illustrate the accuracy of this method confirming the theoretical results.

Fuzzy quasi-interpolation solution for Fredholm fuzzy integral equations of second kind

In this paper, we intend to offer a new approach to solve Fredholm fuzzy integral equation of the second kind by the concept of source distance and fuzzy quasi-interpolation. The error estimation of the proposed method is proved in terms of the fuzzy modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize acceptable accuracy of our method.

Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions

In this paper we present an efficient iterative procedure based on the triangular functions (TFs) to obtain the numerical solution of the specific nonlinear fuzzy Fredholm integral equations of the second kind. The error estimation of the proposed method is given in terms of uniform modulus of continuity. Also, we prove the stability of the method with respect to the choice of the first iteration. Finally, in order to demonstrate the accuracy and the convergence of the method, some numerical examples are included.

Application of bivariate fuzzy Bernstein polynomials to solve two-dimensional fuzzy integral equations

Numerical solutions of two-dimensional fuzzy Fredholm integral equations of the second kind (2DFFIE2) via bivariate fuzzy Bernstein polynomials are proposed in this paper. We introduce a numerical iterative procedure using successive approximations method to approximate the solution of 2DFFIE2. We prove the convergence of the method and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration. This numerical stability is proved and finally, some numerical experiments confirm the theoretical results and illustrate the efficiency and accuracy of the method.

Two hybrid and non-hybrid methods for solving fuzzy integral equations based on Bernoulli polynomials

In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orthogonal Bernoulli polynomials to approximate solution of linear fuzzy Fredholm integral equations. At first, using Bernoulli basis polynomials and also combining them with known block-pulse functions, we convert the fuzzy integral equations to two algebraic systems. The convergence and error estimates of the methods is also given. Finally, we present some illustrative examples and compare the numerical computational results to confirm the theoretical topics and demonstrate the convergence rate of the methods.

Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method

In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem.

New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein–Fredholm integral equations

In this paper, we obtain the error estimation of the iterative method based on quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind given in Fuzzy Sets & Syst. 245 (2014) 1–17, in terms of uniform and partial modulus of continuity. Moreover, we extend in the context of using the modulus of continuity, the notion of numerical stability of the solution with respect to the first iteration.

Fuzzy trapezoidal cubature rule and application to two-dimensional fuzzy Fredholm integral equations

In this paper, we construct the fuzzy trapezoidal cubature rule providing its remainder estimate for the case of Lipschitzian functions. As an application, we propose an iterative numerical method in order to approximate the solution of nonlinear fuzzy Fredholm integral equations in two variables, the fuzzy cubature rule being used in the construction of the numerical method. The convergence of the method is proved and tested through a numerical experiment, that confirms the obtained theoretical results.

Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula

In this paper, our aim is to provide an efficient iterative method of successive approximations to approximate solution of linear and nonlinear two-dimensional Hammerstein fuzzy integral equations by defining and developing an optimal quadrature formula for classes of two-dimensional fuzzy-number-valued functions of Lipschitz type. After the introduction of the optimal formula, we prove the convergence of the method of successive approximations used to approximate the solution of two-dimensional Hammerstein fuzzy integral equations and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, some illustrative numerical experiments confirm the theoretical results and demonstrate the accuracy of the method.

NUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND

In this article some numerical methods, namely: the Taylor expan- sion method and the Trapezoidal method, have been investigated and imple- mented to solve a fuzzy Fredholm integral equation of the second kind. Conse- quently, we convert a linear fuzzy Fredholm integral equation of the second kind into a linear system of integral equations of the second kind in crisp case. To demonstrate the credibility of these numerical schemes we consider a numerical test example. The numerical results show to be in a close agreement with the exact solution.

Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using triangular functions

The main purpose of this paper is to approximate the solution of linear two-dimensional fuzzy Fredholm integral equations of the second kind (2D-FFIE-2). We use fuzzy two-dimensional triangular functions (2D-TFs) to reduce the 2D–FFIE–2 to a system of linear Fredholm integral equations of the second kind with three variables in crisp. More over, we prove the convergence of the method. Finally we illustrate this method with some numerical examples to demonstrate the validity and applicability of the technique.

Existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations

In this paper the fixed point theorem of Schauder is used to prove the existence of a continuous solution of the nonlinear fuzzy Volterra integral equations. Then using some conditions the uniqueness of the solution is investigated.

Existence and uniqueness of the solution of fuzzy-valued integral equations of mixed type

In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed type (FVFIEMT) involving fuzzy number valued mappings have been investigated. Then, by using Banach's contraction principle, sufficient conditions for the existence of a unique solution of FVFIEMT are given. Finally, illustrative examples are presented to validate the obtained results.

Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series

The main purpose of this paper is to introduce a method of successive approximations in terms of a hybrid of Taylor series and a block-pulse function, which is given for solving nonlinear fuzzy Fredholm integral equations of the second kind and the error estimate of the approximation solution. Finally, two numerical examples are presented to show the accuracy of the proposed method.

Fuzzy fractional functional integral and differential equations

This paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle. Moreover, the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives. Finally, the solution to fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method (MFEM) is presented. The method is illustrated by solving some examples.