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 . Systems of Volterra integral equations have wide applications in engineering, physics, chemistry and populations growth models . We consider the following system of linear Volterra integral equations (SLVIEs) of the second kind .
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  and a special form of VIEwND, proportional delay differen- tial equations, is widely used in practical applications, for example, electrody- namics  , nonlinear dynamical systems  , and also the survey papers  . 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 . 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  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  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 ,  and the number of busy channels in telephone traffic theory , 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 , , . Tsokos  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) 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)  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) 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 -, integral equations with nonsigular kernels have been studied. In -  Darwish et al. introduced and studied the quadratic Volterra equations with supremum. Also, Banaś et al. and Darwish  - studied quadratic integral equations of arbitrary orders with singular kernels. In , Darwish generalized and extended Banaś et al.  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.