An analytical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper, we use the parametric form of a fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case. For fuzzy Fredholm integral equations with kernels, the sign of which is difficult to determine, a new parametric form of the fuzzy Fredholm integral equation is introduced. We use the homotopy analysis method to find the approximate solution of the system, and hence, obtain an approximation for fuzzy solutions of the linear fuzzy Fredholm integral equation of the second kind. The proposed method is illustrated by solving some examples. Using the HAM, it is possible to find the exact solution or the approximate solution of the problem in the form of a series.     

Solving fuzzy differential equations by differential transformation method

An extension of the differential transformation method (DTM), which is an analytical-numerical method for solving the fuzzy differential equation (FDE), is given. The concept of generalised H-differentiability is used. This concept is based on an enlargement of the class of differentiable fuzzy mappings; to define this, the lateral Hukuhara derivatives are considered. The proposed algorithm is illustrated by numerical examples, and some error comparisons are made with other methods for solving a FDE.     

Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method

In this paper rationalized Haar functions are developed to approximate the solutions of the linear Fredholm integral equations system. Properties of rationalized Haar functions are first presented, the operational matrix of the product of rationalized Haar functions vector is utilized to reduce the computation of Fredholm integral equations system to some algebraic equations. Finally, numerical result are given which support the theoretical results.     

The numerical solution of linear fuzzy Fredholm integral equations of the second kind by using finite and divided differences methods

In recent years, many numerical methods have been proposed for solving fuzzy linear integral equations. In this paper, we use the divided differences and finite differences methods for solving a parametric of the fuzzy Fredholm integral equations of the second kind with arbitrary kernel and present some examples to illustrate this method.     

Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method

In this paper, the fuzzy variational iteration method is proposed to solve the nonlinear fuzzy differential equation (NFDE). The convergence and the maximum absolute truncation error of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method.     

The -method and Fredholm integral equations

Instead of using approximate methods on the equation

$\text{f(x) = g(x) + λ}\text{∫}\text{0}\text{1}\text{K(x,t)f(t) dt}$

, the τ-method is employed to obtain the exact solution of the equation

$\text{h(x) = g(x) + λ}\text{∫}\text{0}\text{1}\text{K(x,t)h(t) dt + R(x,λ)}$

,The analytical from of R(x, λ) determines the type of approximation which results. In this paper R(x, λ) is chosen to be a function which is rational in the parameter λ in which case, the approximation to the true solution has the same character. An example is given.     

Solving parametric fuzzy systems of linear equations by a nonlinear programming method

Linear systems of equations, with uncertainty on the parameters, play a major role in various problems in economics and finance. In this paper parametric fuzzy linear systems of the general form A ₁ x + b ₁ = A ₂ x + b ₂, with A ₁, A ₂, b ₁ and b ₂ matrices with fuzzy elements, are solved by means of a nonlinear programming method. The relation between this methodology and the algorithm proposed in Muzzioli and Reynaerts [(2006) Fuzzy Sets and Systems, in press] is highlighted. The methodology is finally applied to an economic and a financial problem.     

Simulation and evaluation of system of fuzzy linear Fredholm integro-differential equations with fuzzy neural network

Neural Computing and Applications - In this paper, fuzzy neural network (FNN) can be trained with crisp and fuzzy data. The work of this paper is an expansion of the research of fuzzy linear...     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

The regularization method for Fredholm integral equations of the first kind

In this work, the Fredholm integral equations of the first kind will be examined. The regularization method combined with the existing techniques are applied to handle the ill-posed Fredholm problems. Examples will be used to highlight the reliability of the regularization method.     

Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations

Using a fixed point technique, the sequence of successive approximations and a recent quadrature formula for fuzzy-number-valued functions, it is constructed a numerical method for the solution of nonlinear fuzzy Fredholm integral equations. Moreover, the error estimate of the method and a criterion to stop the corresponding algorithm are given.     

Solving fuzzy equations using evolutionary algorithms and neural nets

 In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation AˉXˉ + Bˉ= Cˉ; for Xˉ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve AˉXˉ + Bˉ= Cˉ for Xˉ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.     

Exponential difference schemes with double integral transformation for solving convection-diffusion equations

Convection-diffusion equations are studied. These equations are used for describing many nonlinear processes in solids, liquids, and gases. Although many works deal with solving them, they are still challenging in terms of theoretical and numerical analysis. In this work, the grid approach based on the method of finite differences for solving equations of this kind is considered. In order to make it easier, the one-dimensional version of such an equation was chosen. However, the equation preserves its principal properties; i.e., it is non-monotonic and non-linear. To solve boundary-value problems for such equations, a special variant of the non-monotonic sweep procedure is proposed.     

Solving fuzzy linear systems

In this paper, the main aim is to develop a method for solving an arbitrary general fuzzy linear system by using the embedding approach. Considering the existing and uniqueness of fuzzy solution to n × n linear fuzzy system is done. Numerical examples are presented to illustrate the proposed model.     

The multi-projection method for weakly singular Fredholm integral equations of the second kind

In this paper, we propose a multi-projection method and its re-iterated algorithm for solving weakly singular Fredholm integral equations of the second kind. We apply our methods to Petrov–Galerkin versions to establish excellent superconvergence results, and we illustrate our theoretical results with a numerical example.     

An analog computer method to solve Fredholm integral equations of the first kind

Many problems of contemporary interest are characterized by Fredholm type integral equations of the first kind. These equations are inherently ill-posed and difficult to solve. It is customary to convert the equation into a set of m algebraic equations Af = g in n unknowns with m not necessarily equal to n. Then one can solve these m equations in a least square sense. Among the class of vectors f that minimize the Euclidean norm of the error, there exists a unique vector A⁺g which is of least norm where A⁺ is the generalized inverse of A. One method of finding the generalized inverse of A is to reformulate the problem into an equivalent system of first order ordinary differential equations with specified initial conditions. The steady state solution of this system is A⁺g, the required value of f. This procedure was implemented on an analog computer and the results presented.     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

Numerical solution of fuzzy Fredholm integral equations by the Lagrange interpolation based on the extension principle

In this paper, a numerical procedure is proposed for solving the fuzzy linear Fredholm integral equations of the second kind by using Lagrange interpolation based on the extension principle. For this purpose, a numerical algorithm is presented, and two examples are solved by applying this algorithm. Moreover, a theorem is proved to show the convergence of the algorithm and obtain an upper bound for the distance between the exact and the numerical solutions.