This paper describes an approximating solution, based on Lagrange interpolation and spline functions, to treat Functional integral equations of Fredholm type and Volterra type.This method can be extended to Functional differential and integro-differential equations. For showing efficiency of the method we give some numerical examples.

introduction: In this paper, we propose a numerical method for approximating the solution of the fuzzy Functional integral equations of Volterra and Fredholm type by using Lagrange interpolation. For this purpose, we convert the fuzzy Fredholm and Volterra integral equations to the crisp systems of integral equations. The proposed method is illustrated by various fuzzy numerical examples.Aim: In almost, the integral equations cannot be solved analytically or simply.Therefore, we propose the numerical methods for solving the integral equations.Material and Method: For solving fuzzy Functional integral equations, at first, we introduce the parametric form of them and then we obtained (2n+2) x (2n+2) systems via Lagrange interpolation. In the following, this system convert to two (n+1) x (n+I) systems.By solving them, we have the support points which can be approximate the exact solution.Results: In this work, for solving Fredholm or Volterra fuzzy Functional integral equation, we replaced each of this fuzzy integral equation by two crisp integral equations. For numerical solution of these equations, we applied the Lagrange interpolation with different r_cuts that is between zero and one. At last, by substituting x in the crisp equations by Xj for j = 0,1,...,n we obtained a linear system of equations with 2n + 2 equations and 2n + 2 unknown. By solving two (n+l) x (n+l) systems, y(xj.;r) and ȳ (xj;r) for j=0,1,...,n and 0£r£1 are calculated. Consequently, the approximation for exact solution is given by putting y(xj .;r) and . ȳ (xj;r) for j = 0, 1,...,n in Lagrange interpolation function. The advantage of this method in comparison with the other methods is that the solution of the integral equation is approximated by having the supported points. Also, the proposed method in comparison with Freidman et al.'s method converges to the exact solution with less number of the iterations and node points.Conclusion: This By the proposed method, we can numerically solve the fuzzy Functional integral equations of Volterra and Fredholm of the second kind, Also, the proposed method in comparison with Freidman et al.' s method converges to the exact solution with less number of the iterations and node points.

The purpose of this paper is to analyze the solvability of a class of stochastic Functional integral equations by utilizing the measure of non-compactness with Petryshyn’s fixed point theorem in a Banach space. The results obtained in this paper cover numerous existing results concluded under some weaker conditions by many authors. An example is given to support our main theorem.

In this paper, the Functional Volterra integral equations of the Hammerstein type are studied. First, some conditions that ensure the existence and uniqueness of the solutions to these equations within the space of square-integrable functions are established and then the Euler operational matrix of integration is constructed and applied within the collocation method for approximating the solutions. This approach transforms the integral equation into a set of nonlinear algebraic equations, which can be efficiently solved by employing standard numerical methods like Newton's method or Picard iteration. One significant advantage of this method lies in its ability to avoid the need for direct integration to discretize the integral operator. Error estimates are provided and two illustrative examples are included to demonstrate the method’s effectiveness and practical applicability.

This article is an attempt to obtain the numerical solution of Functional linear Voltrra two-dimensional integral equations using Radial Basis Function (RBF) interpolation which is based on linear composition of terms. By using RBF in Functional integral equation, first a linear system 􀀀, C = G will be achieved, then the coefficients vector is defined, and finally the target function will be approximated. In the end, the validity of the method is shown by a number of examples.

In 1930, Kuratowski introduced the concept of measure of noncompactness. Later, Banas and Goebel generalized this concept axiomatically, which is more convenient in applications. The principal application of measures of noncompactness in fixed point theory is contained in the Darbo's fixed point theorem. This is a tool to investigate the existence and behaviour of solutions of many classes of integral equations such as Volterra, Fredholm and Uryson types. The technique of measure of noncompactness is applicable in several branches of nonlinear analysis. In particular, it is a very useful tool for several types of integral and integral-differential equations. In addition, the measure of noncompactness is also used in Functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory. The purpose of this article is to introduce a new measure of noncompactness in the Sobolev space W^(k, ∞ ) (R^n). The results are obtained to solve integral-differential equations. Finally, by providing an example to show the efficiency of our results.

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.