Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel

In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel; then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.


Introduction
Integral equations of various types and kinds play an important role in many branches of linear and nonlinear function analysis and their applications in the theory of elasticity, engineering, mathematical physical and contact mixed problems. Therefor, many different methods are used to obtain the solution of the Volterra integral equation. In [1] Linz, studied analytical and numerical methods of Volterra equation. In [2], Mirzaee and Rafei used the BBM for the numerical solution of the nonlinear two-dimensional Volterra integral equations.
In the references [3]- [8] the authors considered many different methods to solve linear and nonlinear system of Volterra integral equations numerically with continuous and singular kernels. In [9], Al-waqdani studied linear F-VIE with continuous kernel and solved the linear SVIEs numerically with continuous kernel.

Existence of Solution of NF-VIE
To prove the existence of a unique solution of Equation (1) using fixed point theorem.
We write it in the integral operator form: Then, we assume the following conditions: i) The kernel of Fredholm integral term satisfies: ii) The kernel of Volterra integral term satisfies: iii) The given function ( ) , f x t with its partial derivatives is continuous in iv) The known continuous function Theorem 1: If the condition i)-iv) are verified, then Equation (1)  The provement of this theorem depends on the following two lemmas: Under the conditions i)-iv-a), the operator W defined by (2), maps the space Proof: In view of Formula (2) and (3) we get: Using the conditions (i)-(iii), then applying Cauchy-Schwarz inequality, we have: In the light of the condition (iv-a), the above inequality take the form: The lost inequality (5) shows that, the operator W maps the ball r S into itself, where ( ) Since 3 0, 0 r A > > , therefore we have 1 α < . Moreover, the inequality (5) involves the boundedness of the operator W of Equation (2) where: Also, the inequalities (5) and (7) define the boundedness of the operator W . Lemma 2: If the conditions (i),(ii) and (iv-b) are satisfied, then the operator W is con- Proof: Using the condition (iv-b), then apply Cauchy-Schwarz inequality we have: In equality (8) shows that, the operator W is continuous in the space

RKM
In this section, the RKM is used to solve NF-VIE of the second kind. By divide using the quadrature formula, the integral Equation (1) represent a NSVIEs as: To solve the NSVIEs: Now, applying the RKM for solve (15): Suppose that: Substituting from (16) into (15), where Journal of Applied Mathematics and Physics By derivative (18), we have, Now, apply the RKM to this system of equations to give,

t G t F t t p h r t G t F t t q s t G t hF t t r
is the approximate solution at ( )

Conclusions
From the previous discussions we conclude the following: 1) As N is increasing the errors are decreasing.
2) As x and t are increasing in [ ] [ ] 0,1 0,1 × , the errors due to RKM and BBM are also increasing.
3) The errors due to the BBM are less than the errors due to RKM (i.e. BBM the better than RKM to solve NF-VIE with continuous kernel).