On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

and Applied Analysis 3


Newton-Kantorovich Method for the System
Let us rewrite the system of nonlinear Volterra integral equation (4) in the operator form where  = ((), ()) and To solve (5) we use initial iteration of Newton-Kantorovich method which is of the form where  0 = ( 0 (),  0 ()) is the initial guess and  0 () and  0 () can be any continuous functions provided that  0 < () <  and () ̸ = 0.The Frechet derivative of () at the point  0 is defined as Hence, From ( 7) and ( 9) it follows that where Δ() =  1 () −  0 (), Δ() =  1 () −  0 (), and ( 0 (),  0 ()) is the initial given functions.To solve (10) with respect to Δ and Δ we need to compute all partial derivatives: Equation ( 13) is a linear, and, by solving it for Δ and Δ, we obtain ( 1 (),  1 ()).By continuing this process, a sequence of approximate solution (  (),   ()) can be evaluated from which is equivalent to the system where Thus, one should solve a system of two linear Volterra integral equations to find each successive approximation.Let us eliminate Δ() from the system (13) by finding the expression of Δ() from the first equation of this system and substitute it in the second equation to yield where and the second equation of ( 16) yields where In an analogous way, Δ  () and Δ  () can be written in the form where +  () +  ()  () . (21)

The Mixed Method (Simpson and Trapezoidal) for Approximate Solution
At each step of the iterative process we have to find the solution of ( 18) and ( 20) on the closed interval [ 0 , ].To do this the grid () of points   =  0 + ℎ,  = 1, 2, 3, . . ., 2, ℎ = (− 0 )/2 is introduced, and by the collocation method with mixed rule we require that the approximate solution satisfies ( 18) and ( 20).Hence On the grid () we set V 2 =  0 ( 2 ), suct that Consequently, the system (23) can be written in the form By computing the integral in (26) using tapezoidal formula on the first integrals and Simpson formula on the second integral, we consider two cases.
(29) Also, to compute Δ  () on the grid (), (18) can be represented in the form Let us set V 2 =  0 ( 2 ) and  2 =  −1 ( 2 ) and Then (30) can be written as and by applying mixed formula for (32) we obtain the following four cases.
To verify that Γ 0 is bounded we need to establish the resolvent kernel Γ 0 (, ) of ( 17), so we assume the integral operator  from [ 0 , ] → [ 0 , ] is given by where  2 (, ) =  1 (, )/ 0 (), and  1 (, ) is defined in (18).Due to (46), ( 17) can be written as The solution Δ * of (47) is expressed in terms of  0 by means of the formula where  is an integral operator and can be expanded as a series in powers of  [21, Theorem 1, page 378]: and it is known that the powers of  are also integral operators.In fact where  () 2 is the iterated kernel.
Substituting (50) into (48) we obtain an expression for the solution of (47): Therefore the th root test of the sequence yields Hence  = 1/lim  → ∞  √‖  ‖ = ∞ and a Volterra integral equations (17) has no characteristic values.Since the series in (51) converges uniformly (48) can be written in terms of resolvent kernel of (17): where Since the series in (57) is convergent we obtain To establish the validity of second condition, let us represent operator equation as in (41) and its the successive approximations is For initial guess  0 we have From second condition of (Theorem 1) we have In addition, we need to show that ‖  ()‖ ≤  1 for all  ∈ Ω 0 where  1 is defined in (38).It is known that the second derivative   ( 0 )(, ) of the nonlinear operator () is described by 3-dimensional array   ( 0 ) = ( 1 ,  2 )(, ), which is called bilinear operator; that is,   ( 0 )() = ( 0 , , ) where where ,  ∈ (0, 1), so we have where Table 1 shows that   () coincides with the exact  * () from the first iteration whereas only six iterations are needed for   () to be very close to  * ().Notations used here are as follows:  is the number of nodes,  is the number of iterations, and   = max ∈ [10,15] |  () −  * ()| and   = max ∈ [10,15] |  () −  * ()|.

Conclusion
In this paper, the Newton-Kantorovich method is developed to solve the system of nonlinear Volterra integral equations which contains logarithmic function.We have introduced a new majorant function that leads to the increment of range of convergence of successive approximation process.A new theorem is stated based on the general theorems of Kantorovich.Numerical example is given to show the validation of the method.Table 1 shows that the proposed method is in good agreement with the theoretical findings.