ANALYTICAL SOLUTIONS FOR LINEAR VOLTERRA AND ABEL INTEGRAL EQUATIONS OF SECOND KIND USING A POWER SERIES METHOD

Abstract: Even though they have a rather specialized structure, Volterra and Abel integral equations form an important class of integral equations in applications. This happens because completely independent problems lead to the solution of such equations. In this paper we consider the liner Volterra and Abel integral equations of second kind. Authors have been proposed a new method for constructing solutions of Abel integral equations by a power series.


Introduction and Preliminary
In this paper, we use the method of generalized power series, to solve linear Volterra integral equations of the second kind.This power series are undetermined coefficients method, or a method based on the application of the Taylor series.The result obtained in the form of generalized power series solution further converted to the inversion formula of the integral equation.One such method is the representation of the solution of the equation in the form of a power series [1], [2].We also consider Abel equation of the second order [3] where the kernel G(x, t) and the function f (x)are given real-valued functions, and G(v(x, t)) is a nonlinear function of v(t)such as u 2 (x), cos(u(x)) and e u(x) .Suppose the solution of equation ( 1) be as where a 0 = g(0) = v(0) and c 1 is a unknown parameter.By substituting Eq.( 2) into Eq.( 1) with simple calculations, we get where Q(x 2 ) is a polynomial of order greater than one.By neglecting Q(x 2 ), we have linear equation of a 1 in the form, where the parameter a 1 of x in Eq. ( 2) is then obtained.In the next step, we assume that the solution of Eq.( 1) is where a 0 and a 1 both are known and a 2 is unknown parameter.By substituting Eq.( 5) into Eq.(1), we get where Q(x 3 ) is a polynomial of order greater than two.By neglecting Q(x 3 ), we have linear equation of a 2 in the form, the unknown parameter a 2 of x 2 in Eq.( 5) is then obtained.Having repeated the above procedure for n iterations, a power series of the following form is derived: Eq.( 8) is an approximation for the exact solution v(x) of Eq.( 1) in the interval [0, 1].

Linear Volterra and Abel Integral Equations
Consider the following linear Volterra integral equation [1]: a 0 = v(0) = f (0) = 1 as the initial condition, suppose the solution of Eq.(9) be Substitute Eq.(10) in Eq.( 9) we get By integrating and solving we get 6a Suppose the solution of Eq.(10) be as Substitute Eq.(11) in Eq.( 9)we get By integrating and solving we get 6(2a 2 + 1)x 2 − 2x 3 − a 2 x 4 = 0.By neglecting a 2 x 4 , therefore 2a 2 = −1.Substitute a 2 in Eq.( 11) we get Continue by this way we will get series of the form so that gives the exact solution Let's now consider Abel integral equation of the second kind: Let's assume that the function f (x)can be represented as follows: The solution of this equation will be sought in the form of a sum of two generalized power series: Substituting the power series (13), ( 14) into equation ( 12), performing the calculations, obtain Then by equating the terms with identical powers of x we get: and this implies that our solution can be written as And as a final result: This solution identical to the solution obtained in [5].The theorem is proved by direct verification that the formula (8) is a solution of equation (6).

Conclusions
In this paper, solution is obtained by power series method.This may be used in more combinatorial way to obtain solution of higher degree non-linear integral equations.

Theorem 1 .
If f (t) ∈ C [a, b],then there exist a unique solution of equation Abel of second kind, which is expressed in the form (14).
Correspondence author a special case of the equation of Volterra.Consider the nonlinear Volterra integral equation of second kind url: www.acadpubl.eu§