The Adomian Decomposition Method of Volterra Integral Equation of Second Kind

In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.


Introduction
Many problems from physics and other disciplines lead to linear and nonlinear integral equations. Several methods have been proposed for exact solution of these equations " [3]". Adomian " [1,2]" has presented and developed a so-called decomposition method for solving algebraic, differential, integro-differential, differential-delay, and partial differential equations. The solution is found as infinite series which convergence rapidly to accurate solutions. This method has proven successful in dealing with linear as well as nonlinear problems, as it yields analytical solutions and offers certain advantages over standered numerical solutions. Application of ADM to deferent types of integral equations has discussed by many authors, for examples " [4,5,7]", and " [18]". The integral equation originates from the conversion of a boundary-value problem or an initial-value problem, but many problems lead directly to integral equation and cannot be formulated in terms of the problem easier, or sometimes enable us, to prove fundamental results on the existence and uniqueness of the solution. An integral equation is an equation in which the unknown function u x to be determined appears under the integral sign " [7]"and " [13]". A typical form of an integral equation in u x is of the form where k x, t is called the kernel of integral equation, λ 0 the parameter of integral equation, α x and β x are the limits of integration. It is to be noted here that both the kernel k x, t and the function f x in equation are given function. The kernel is always defined and continuous on D x, t : α x x β x , α x t β x . When φ x 0 , the equation is of the first kind; otherwise, it is of the second kind. An integral equation can be classified as a linear or nonlinear integral equation as we know in the ordinary and partial differential equations " [10,12,13]". The most frequently used integral equations fall under two major classes, namely Volterra and Fredholm integral equations.
Definition Volterra integral equations are written in a form (1) where the upper limit of integration β x = x (independent variable). The most standard form of Volterra linear integral equations are of the form φ x u x = f x + λ k x, t u t dt

Significance of the Study
Integral equations are often easier to solve, more elegant and compact that a corresponding differential equation. Because it does not require supplementary initial or boundary conditions and the contribution of this study is that: (1) Distinguish the importance of integral equations over the differential equation.

Relationship between Integral Equations and Differential Equations
The theories of ordinary and partial differential equations are fruitful source of integral equations. The researcher shall sketch here one of the ways in which integral equations can arise from ordinary differential equations. Most ordinary differential equations can be expressed as integral equations, but the reverse is not true " [14]" and" [18]".To investigate the relationship between integral and differential equations, the researcher will need the following lemma which will allow us to replace a double integral by a single one.
Lemma1. (Replacement lemma): Suppose that u: a, b" → ℝ is continuous. Then Now considering the first order differential equation.
with the initial condition y 0 = y if we say, u(x,y) is continuous function of (x,y), integrate (2) from 0 to x, obtaining . dy dx / dx = u x, y dx y x = y + u0t, y t 1dt This illustrates the general fact that, by going over to integral equations, it includes both the differential equation and the initial conditions in a single equation. Again consider the second order differential equation.
With initial conditions y 0 = y , y -0 = y & . Then integrate (3) from 0 to x, obtaining Whence the second integration The argument is reversible; so that here again the differential equation (3), together with the initial conditions, is equivalent to the single integral equation (4).We see also that any solution of (3) satisfies an integral equation of the form.
The constants A and B being determined by the initial conditions. They may also be determined in other ways. Suppose, for instance, that y(x) is required to satisfy a two point boundary condition, say y 0 = α, y l = β, if x = l Substituting in (5), we obtain.
Hence, the function y(x) must therefore satisfy the integral equation Here k(x, t) is the kernel of the equation and the argument is again reversible, so that (6) is equivalent to (3) together with the boundary conditions. If the differential equation is linear, we are led in this way to a linear integral equation of the second kind.

Solution of Linear Volterra Integral Equation
This section deals with the Volterra integral equations and their solution techniques. The solution of this integral equation is the unknown function u(x) which satisfies that equation, but there is the problem that the upper limit of the integral is the independent variable of the equation. For thus choose a quadrature scheme that utilizes the endpoints of the interval; otherwise we will not be able to evaluate the functional equation at the relevant quadrature points. There are a host of solution techniques to deal with the Volterra integral equations.

Existence and Uniqueness of Solutions of Linear Volterra Integral Equations of the Second Kind
Let L be an arbitrary nonempty closed subset of the real number, for > ∈ L we define the operator M ∶ L → L as follows: We note that M > ≥ > for any > ∈ L , for > ∈ L we define the operator T ∶ L → L by We note that T > ≤ > for any > ∈ L. Now we say that t is right dense if > < Q?U L and M > = > . we say that t left dense if > > OPI L and T > = >.
Definition: A function I: L → ℝ is called rd-continuous provide it is continuous at right dense points in L and its left sided limits exist (finite) at left dense points in L . Now we can state the following theorems.

The Method of Adomian Decompositions for Linear Volterra Integral Equation
The where the components ? a ; , P ≥ 0 are to be determined in a recursive manner. The decomposition method concerns itself with finding the components ? ; , ? & ; , ? d ; ,… individually. The determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated.
To establish the recurrence relation, we substitute (8)  and so on for other components. In view of (12), the components ? ; , ? & ; , ? d ; , … are completely determined. As a result, the solution u(x) of (9) in a series form is obtained by using the series (8). In the other words, the Adomian decomposition method converts the Volterra integral equation into a determin--ation of computable components. It was formally shown by many researchers that if an exact solution exists for the problem, then the obtained series converge very rapidly to that solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to confirm the rapid convergence of the resulting series.
However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms are usually used for numerical purposes. The more components we use the higher accuracy we obtain.
In recent decades, there has been a great deal of interest in the Adomian decomposition method. Recently, a modification of the Adomian decomposition method was proposed in " [11]". However, the modified method was established based on the assumption that the function f can be divided into two parts, and its success depends mainly on the proper choice of the parts I & and I d .The modified decomposition method will facilitate the computational process and further accelerate the convergence of the series solution. Therefore it cannot be used if the function I ; consists of only one term. To give a clear description of the technique, we can set In view of (13), we introduce a qualitative change in the formation of the recurrence relation (11). To minimize the size of calculations, we identify the zeroth component ? ; ? af& ; = \ e ;, > ? a > A> P ≥ 1 Note that, the proper selection of the functions I & ; andI d ; , may be obtained the exact solution ? ; by using very few iterations, and sometimes by evaluating only two components.

Numerical Examples of Linear Volteera Integral Equations
This section contains some examples of linear Volterra integral equations. Example 4.3.1 Consider the following linear Volterra integral equation Recall that the solution ? ; is assumed to have a series form given in (8). Substituting the decomposition series (8) into both sides of (15)

Solution of Nonlinear Volterra Integral Equation
In this section we will study the existence and uniqueness of the general nonlinear Volterra integral equation's which to be in the form ? ; = I ; + <0;, >; ? > 1A> ^ (16) where < and I are u d −functions. Suppose Where { d and | d are two positive constants. As in the linear case we found the solution of our equation as the limit of a sequence ? a ; of functions whose first approximation element is the given function? ; ≡ I ; ; the other approximations are evaluated by the iteration formula ? a ; = I ; + < ;, >; ? aE& > A>, P = 1,2, … ^ (20) Theorem 3: Suppose that f and F are u d − functions satisfying conditions (17), (18) and (19), then the nonlinear Volterra integral equation (16)  ; which converges absolutely and almost uniformly.

The Method of Adomian Decomposition for Nonlinear Volterra Integral Equation
In the nonlinear Volterra integral equation, The Adomian polynomials can be generated using formula (23) or formula (24). Formula (24) is programmable and the Adomian series solution can be converged faster when using it. The first four polynomials using formulas (23)   and so on. The solution in a series form is given by u x = x + 1 3 x g + 2 15 x o + 17 315 x q + ⋯ that converges to the exact solution u x = tan x

Conclusion.
In this paper we proposed of the some numerical methods to solve Volterra (linear and nonlinear) integral equations.
u x = f x + λ k x, t u t dt, a ≤ x ≤ b And u x = f x + λ k x, t F u t dt, a ≤ x ≤ b To solve these for u(x) we set u x = i u ‡ x , b ‡c plays an essential role on the speed of the convergence of the Numerical methods for solving Volterra integral equation. The sufficient condition that guarantees a unique solution to the given problem is obtained.