**Volterra** **equations** naturally appear in history-dependent problems such as population dynamics, renewal **equations**, nuclear reactor dynamics, viscoelasticity, study of epidemics, superfluidity, damped vibrations, heat conduction and diffusion [7]. **Systems** of **Volterra** **integral** **equations** have wide applications in engineering, physics, chemistry and populations growth models [14]. We consider the following system of linear **Volterra** **integral** **equations** (SLVIEs) of the second kind [7].

In this paper, the variational iteration method and its modification were successfully employed for solving **systems** of **Volterra** **integral** **equations** of the first kind. For convenient in explanation of the methods the linear **integral** **equations** were considered, but examples were investigated for non-linear system. The results shown that MVIM reduces the size of calculations and gives an accurate power series solution which converges rapidly to the closed form solution in the neighborhood of the initial point.

The pure **Volterra** **integral** **equations** with vanishing delay (VIEwND) are in- itially studied in [6] and a special form of VIEwND, proportional delay differen- tial **equations**, is widely used in practical applications, for example, electrody- namics [7] [8], nonlinear dynamical **systems** [9] [10], and also the survey papers [11] [12]. In this paper, we consider the CVIEs with a vanishing delay,

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The study of finite-dimensional linear **systems** is well developed. As an infinite-dimensional counter part of finite-dimensional linear **systems**, one can view **integral** **equations** as extensions of linear **systems** of algebraic **equations**. An **integral** equation maybe interpreted as an analogue of a matrix equation which is easier to solve. There are many different ways to transform **integral** **equations** to linear **systems**. Many different methods have been used for solving **Volterra** **integral** **equations** and Freholm-Velterra **integral** **equations** numerically.

that their kernels and right-hand sides are “good” and “smooth”) **Volterra** **integral** **equations** of the first kind belong to the class of well-posed problems and hence can be solved through any direct method based on the discretization of the unknown solutions. Both approaches seem to be a little on the extreme. The use of discretization process transforms **Volterra** **integral** **equations** of the first kind into another problem (often **systems** of linear **equations** and may be solved by the well-known singular value decomposition) that can lead to solutions which may or may not deviate strongly from the (minimum norm) solution of the original equation [3]. A complete regularization (with its accompanying difficulties and unresolved questions) turns out to be a too complicated approach as it is observed that the methods for **Volterra** **equations** of the first kind are not nearly as ill-posed as methods for Fredholm **integral** **equations** of the first kind. In this study, we investigate the regularization methods with a view to identifying the properties and types that can be used in solving **Volterra** type **equations** of the first kind, but without destroying the **Volterra** structure. The methods in view are perhaps the discrete regularization methods. The discrete approximation methods provide another approach to regularize the original problem. In this case, the regularization parameter is the discretization parameter (or step size) and coordination between this parameter and the amount of noise in the problem is required to obtain good approximations in the presence of noise.

with the iterated collocation method. Guoqiang (1993) introduced and discussed the asymptotic error expansion of a collocation-type method for **Volterra**-Hammerstein **integral** **equations**. The methods in Kumar et al. (1987) and Guoqiang [1993] transform a given **integral** equation into a system of non-linear **equations**, which has to be solved with some kind of iterative method. In Kumar et al. (1987) the definite integrals involved in the solution may be evaluated analytically only in favorable cases, while in Guoqiang (1993) the integrals involved in the solution have to be evaluated at each time step of the iteration. Orthogonal functions, often used to represent arbitrary time functions, have received considerable attention in dealing with various problems of dynamic **systems**. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic **equations**, thus greatly simplifying the problem. Orthogonal functions have also been proposed to solve linear **integral** **equations**. Runge –Kutta methods are being applied to determine numerical solutions for the problems, which are modeled as Initial Value Problems (IVP’s) involving differential **equations** that arise in the fields of Science and Engineering by Alexander and Coyle (1990), Murugesan et al. (1999; 2000; 2001; 2003), Shampine [1994] and Yaakub and Evans (1999). Runge-Kutta methods have both advantages and disadvantages. Runge-Kutta methods are stable and easy to adapt for variable stepsize and order. However, they have difficulties in achieving high accuracy at reasonable cost, which were discussed recently by Butcher (2003). Murugesan et al. (1999) have analyzed different second-order **systems** and multivariable linear **systems** via RK method based on centroidal mean. Park et al. (2004; 2005) have applied the RK- Butcher algorithm to optimal control of linear singular **systems** and observer design of singular **systems** (transistor circuits).Murugesan et al. (2004) and Sekar et al. (2004) applied the RK- Butcher algorithm to industrial robot arm control problem and second order IVP’s. In this paper, we are introducing here the STHW for finding the numerical solution of nonlinear **Volterra**-

Clearly, using this notation, U V is not always as interval. But, when we translate each interval system to two related real-valued **systems**, all these **systems** will solve distinctly. After obtaining solutions of each real-valued system, we ﬁnally check that the obtained solutions create an interval as output of original interval system or not. On the other hand, we should determine the domain the lower solution is less than or equal to upper solution for each independent argument of the solution.

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Random **integral** **equations** of the **Volterra** type occur in the general areas of biology, engineering, and physics. Specifically, the mathematical description of such phenomena as the concentration of a drug in the blood [13], [14] and the number of busy channels in telephone traffic theory [10],[15] result in such stochastic **equations**. Also, in system theory, many differential **systems** with random parameters may be reduced to stochastic **integral** **equations** of the **volterra** type [12], [3], [4]. Tsokos [5] has studied the existence of unique solution of random **integral** **equations**; in this paper we will study the solution of random **integral** **equations** using discretization methods by two approaches, namely the collocation method and the method of approximating the integrals using the trapezoidal rule. The considered random **integral** equation of **Volterra** type has the form:

Abstract This paper deals with the solution of a class of **Volterra** **integral** **equations** in the sense of the conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving conformable fractional **Volterra** **integral** **equations**.

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The method of additive decomposition which works for ordinary diﬀerential equa- tions and **integral** **equations** with continuous kernel (cf. [1, 9, 12, 13, 16]) is used here. Problems of the type (1.1) do not exhibit an exponential decay in the initial layer and therefore, the methodology developed by Angell and Olmostead, and Lange and Smith, can be improved. To emphasize the fundamental ideas and illustrate the technical diﬃculties, we only ﬁnd the leading order term U 0 (t ; ε) of the asymptotic

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In this paper, **Volterra** **integral** **equations** were first reduced to algebraic **equations** using the Laplace transform. We obtained a series that was uniformly convergent to the exact solution after applying the Taylor expansion and the inverse Laplace transforms to the mentioned algebraic **equations**. The

To illustrate the effectiveness of the proposed method, we demonstrate the method with five numerical examples which include first and second kind with regular and weakly kernels. For all examples considered, the solutions obtained by the proposed method are compared with the exact solutions available in the literature. The rate of convergence of each of the Linear **Volterra** **integral** **equations** is composed as

The theory of nonlinear fractional diﬀerential **equations** nowadays is a large subject of mathematics, which found numerous applications of many branches such as physics, en- gineering, and other ﬁelds connected with real-world problems. Based on this fact, many authors studied various results on this theory (see [–]).

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In this paper, we have worked out a computational method to approximate so- lution of nonlinear mixed **Volterra**-Fredholm **integral** **equations** of the second kind, based on the expansion of the solution as series of M3D-BFs. This method converts a nonlinear mixed **Volterra**-Fredholm **integral** equation whose answers are the coeffi- cients of M3D-BFs expansion of the solution of nonlinear mixed **Volterra**-Fredholm **integral** equation. Moreover, the numerical results show that typical convergence rate is O( 1

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in(2012)[5] studied a new approach to find the numerical solution of VIE's by using Bernsteins Approximation. Many researchers have used non- polynomial spline functions approach to find the solution of differential **equations**. Ramadan , M.A. El-Danaf , T. and Abdaal F. E.I. in(2007) [6] Presented an application of the non- polynomial spline function to find the solution of the burgers equation. Zarebnia M. Hoshyar , M. and Sedahti, M. in(2011)[7] Presented a numerical solution based on non-polynomial cubic spline function is used for finding the solution of boundary value problem.

tion f x , we can see immediately from the recurrent Formula (2.5) that the standard Adomian method will encounter computational difficulties. To see that, we ap- plied the recurrent Formula (2.5) on the nonlinear Vol- terra **integral** equation

Here, we use the Chebyshev collocation method to solve nonlinear **integral** equation of the first and second kind Fredholm–**Volterra** **integral** **equations** of the second kind by transformingour problems into a system of nonlinear algebraic **equations**. With using Chebyshev collocations points,the unknown vector is Chebyshev expansion coefficients of the solution. Numerical examples show the accuracyof this method.

Since qt < t when t ≥ 0, (1.2a) is (for t ≥ 0) a differential equation with time lag. The quantity t − qt will be called the lag and we note that the delayed argument qt satisfies qt → ∞ as t → ∞ but the lag is unbounded. Equation (1.2a) provides an example of what are frequently called delay differential **equations**. By analogy, (1.1a), which we term the stochasic pantograph equation is an example of a stochastic delay differential equation.

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In several papers among them [8]-[11], **integral** **equations** with nonsigular kernels have been studied. In [12]- [14] Darwish et al. introduced and studied the quadratic **Volterra** **equations** with supremum. Also, Banaś et al. and Darwish [13] [15]-[17] studied quadratic **integral** **equations** of arbitrary orders with singular kernels. In [18], Darwish generalized and extended Banaś et al. [15] results to the perturbed quadratic **integral** **equations** of arbi- trary orders with singular kernels.

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The Taylor polynomial solution of integro-differential **equations** has been studied by Maleknejad and Mahmoudi (2003). The use of lagrange interpolation in solving integro-differential equation was investigated by Rashed (2004), Wazwaz (2006) used the modified decomposition method and the traditional methods for solving nonlinear **integral** **equations**. A variety of powerful methods has been presented, such as the Homotopy analysis method, homotopy perturbation method, Exp-function method, and variation iteration method. By using the MADM we obtain the exact solutions for the transformed integro-differential **equations**. It is well-known that the main disadvantage of the Laplace Transform method is that it involves large computational work, in this paper instead of using the Laplace transform method we introduce a transformation method in order to reduce the integro-differential equation to a standard **integral** equation of the second kind, the solution method using MADM becomes easier and faster. Our aim in this paper is to obtain the exact solutions by using the MADM. The remainder of the paper is organized as follows: In section 2, a brief discussion of the MADM is presented. In section 3, Implementation of this method is considered by solving two examples. Section 4 ends this paper with a brief conclusion.