Numerical Solution of System of Three Nonlinear Volterra Integral Equations Using Implicit Trapezoidal

In this project, we will be find numerical solution of Volterra Integral Equation of Second kind through using Implicit trapezoidal and that by using Maple 17 program, then we found that numerical solution was highly accurate when it was compared with exact solution.

Volterra integral equations have been found to be effective to describe some application such as potential theory and Dirichlet problemsand electrostatics.Also, Volterra integral equations areapplied in the biology, chemistry, engineering, mathematical problems of radiation equilibrium,theparticle transport problems of astrophysics and reactor theory, and radiation heat transfer problems (Balakumar, V. & Murugesan, K., 2013;Maturi, D. A., 2014).
The systems of Volterra integral equations appear in two kinds.For systems of Volterra integral equations of the first kind, the unknown functions appear only under the integral sign in the form: The kernels   (, )and   ̃(, ), and the functions   (),  = 1,2, ⋯ ,  are given real-valued functions.
A variety of analytical and numerical methods are used to handle systems of Volterra integral equations.The existing techniques encountered some difficulties in terms of the size of computational work, especially when the system involves several integral equations (Linz, P., 1985).
In this project, we present the computation of numerical solution of systems of Volterra integral equation of the second kind.
Theorem 1.Consider the equation Where Then (3) has a unique continuous solution in 0 ≤  ≤ .

The Mathematics of the Volterra Procedure
In this section, we use the technique of the Volterra equation (Balakumar, V. & Murugesan, K., 2013;Effati, S. & NooriSkandari, M. H., 2012) to find an approximates the solution () of (1) at the equally spaced points   =  0 + ℎ for  = 1, ⋯ ,  where  0 = 0 and is the total number of steps of size ℎ.  denotes the approximation of () at  =   .
Defining   by We can rewrite (6) as Where 0denotes the zero vector.From (8), we see that   is the solution of the vector equation () = 0, (9) Where  is the vector-valued function We will obtain an approximation to the solution   of ( 9) by way of the matrix-valued function defined in (11).If () is an  by  matrix-valued function that is invertible in a neighborhood of   , then   is a fixed point of (11) Assuming the components of () have continuous first and second order partial derivatives and that the first order partial derivatives and that the first order partial derivatives at   are equal to zero, it can be shown that if () is set equal to the Jacobian matrix of the function , the iterates   () defined by (13) below will usually converge quadratically to   provided the starting value is sufficiently close to   .The Jacobian matrix of  is the  by  matrix () with the element In row  and column, where   is the Kronecker delta.Details of the statements made here follow from the discussion of Newton's method for nonlinear systems in (Balakumar, V. & Murugesan, K., 2013).Linz gives a brief outline of the trapezoidal rule and Newton's method for Volterra integral systems of the second kind in Section of (Effati, S. & NooriSkandari, M. H., 2012).

Numerical Example
In this section, we solve some examples, and we can compare the numerical results with the exact solution.
Example2.Consider the system of Volterra integral equations With the exact solution X 1 () = sin  , X 2 () = cos  and X 3 () = sin t + cos .Fig. 2 The exact and approximate solutions result of systems of three Nonlinear Volterra integral equations for example 2.
Example3.Consider the system of Volterra integral equations With the exact solution X 1 () =   , X 2 () = sin 2 and X 3 () =  2 − 1. Fig. 3 The exact and approximate solutions result of systems of three Nonlinear Volterra integral equations for example 3.
Example4.Consider the system of Volterra integral equations With the exact solution X 1 () =  , X 2 () =  2 and X 3 () =  3 .Fig. 4 The exact and approximate solutions result of systems of three Nonlinear Volterra integral equations for example 4.

Conclusion
In this project, we have studied system of three nonlinear Volterra integral equations with the implicit trapezoidal method.The basic goal of the present project is to mechanize the computing process of our implicit trapezoidal method by a Maple program and obtain more precise values of the solutions.The results showed that the implicit trapezoidal method is remarkably effective and performing is very easy.The computed values and graphics, illustrated by the results, agree well with the exact solution.

Fig. 5
Fig.5The exact and approximate solutions result of systems of three Nonlinear Volterra integral equations for example 5.

Table . 1
Numerical results and exact solution of systems of three Nonlinear Volterra integral equations for example 1.

Table . 2
Numerical results and exact solution of systems of three Nonlinear Volterra integral equations for example2.

Table . 4
Numerical results and exact solution of systems of three Nonlinear Volterra integral equations for example 4.

Table . 4
Numerical results and exact solution of systems of three Nonlinear Volterra integral equations for example 4.

Table . 5
Numerical results and exact solution of systemsof three Nonlinear Volterra integral equations for example 5.