Power Series Method For Solving Nonlinear Volterra Integro-Differential Equations of The Second Kind

In this work, we present the power series method for solving special types of the first order nonlinear Volterra integro-differential equations of the second kind. To show the efficiency of this method, we solve some numerical examples.


Introduction
It is known that the integrodifferential equations arise in a great many branches of sciences, for example, in potential theory, acoustics, elasticity, fluid mechanics, theory of population, [4], [3].
Many researchers studied the integro-differential equations, see [4] discussed the existence of the solutions for special types of integrodifferential equations, [5], devoted some analytic methods for solving linear Volterra integro-differential equations, [6], [1] gave some numerical methods for solving linear and nonlinear Volterra integrodifferential equations, [5] used some approximated methods for solving linear Volterra integro-differential equations.
The power series method is one of the important methods that can be used to solve the initial value problem of the linear Volterra integro-differential equations of the second kind, [2].
In [7], the power series method is used to solve the nonlinear Volterra integral equations of the second kind of the form: where f and k are known functions, λ is a scalar parameter and u is the unknown function that must be determined.
Here we use the same method to solve the initial value problem that consists of the first order non-linear Volterra integrodifferential equations of the second kind of the form: where f and k are known functions, α is a known constant, λ is a scalar parameter and u is the unknown function that must be determined.

Power Series Method for Solving Equations (1):
Consider the initial value problem given by equations (1).Assumed the solution of equations (1) takes the form: Thus the approximated solution takes the form: where 2 e is the unknown parameter that must be determined.To do this, we expand k(x,y) and f(x) as a power series.That is, ) )   7) is an approximated solution of the initial value problem given by equations (1).

Numerical Examples:
In this section we present two examples that are solved by using power series method.These examples shows the efficiency of this method.Example (1): Consider the first order nonlinear integro-differential equation of the second kind: By repeating the above argument for the approximated solution: 9    x ! 4 By continuing in this manner, one can get: Note that this approximated solution is the exact solution of the initial value problem given by equations (8).

≅
By using the initial condition given by equation (1.b), one can get: .e 0 α = Then by differentiating equation (2) with respect to x and setting x=0 in the resulting equation on can have: On the other hand, from equation (1.a), one can have: equation (1.a) one can get:

2 Q
is a polynomial of degree greater than or equal two.By neglecting PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.

Example ( 2 .
Assume the solution of the above initial value problem takes the form: PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No. Thus PDF created with pdfFactory Pro trial version www.pdffactory.com