Collocation Method for Nonlinear Volterra-Fredholm Integral Equations

A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of VolterraFredholm integral equations.


Introduction
We shall consider the nonlinear Volterra-Fredholm integral equation , 0 , y t g t y t Fy t t I T The Volterra integral operators given by where and and Fredholm integral operators given by where , 1, r r 2 denotes (real or complex) parameters and and let be a given function.In particular, they turn out to be fundamental when ordinary differential equations based model fail.These equations arise in industrial applications and in studies based on biology, economy, control and electro-dynamic.
Collocation method is a widely popular numerical technique in solving integral equations, differential equations, etc.When collocation method is used to solve complicated engineering problems, it has several disadvantages, that is, low efficiency, ill-conditioned, etc.Thus, different types of techniques were proposed to improve the computational performance of collocation method.
Recently, Chelyshkov has introduced sequences of polynomials in [1], which are orthogonal over the interval   have properties, which are analogous to the properties of the classical orthogonal polynomials.These polynomials can also be connected to a fixed set of Jacobi polynomials .Precisely Investigating more on (4), we deduce that in the family
We discuss existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of linear Volterra-Fredholm integral equations and finally, some numerical results are presented in the final section, which support the theoretical results obtained in this paper.

Let
denote the Banach space continuous realvalued functions, such that Lemma 2.1.Assume H is a nonempty closed set in a Banach space V, and that is continuous.Suppose is a contraction for some positive integer m.Then, T has a unique fixed-point in H.
Proof: For proof see [7]. Here, in integral Equation (1), we assume that for some constants i i k M , satisfies a Lipschitz condition with respect to its third argument : , , 0, Let us show that for m sufficiently large, the operator is a contraction on .For

Ty t Ty t k t s y s k t s y s s k t s y s k t s y s s
.

T y t T y t k t s Ty s k t s Ty s s k t s Ty s k t s Ty s s
By a mathematical induction, we obtain .

Collocation Method
with a given mesh N  we associate the set of its interior points, For a fixed and, for given integers and the piecewise polynomial space where π m d  denotes the set of (real) polynomials of a degree not exceeding m d  .The dimension of this space is given by dim For integral equation, we have hence, the collocation space will be From ( 15) we see that an element  is well defined when we know the coefficients In order to compute these coefficients, we consider the set of collocation parameters 0, , 1.
, where and define the set of collocation points by The collocation solution will be determined by imposing the condition that satisfies the integral Equation (1) on the finite set Thus, for , n j the collocation Equation (16) assumes the form From this equation and after some computations, we obtain Now, by using the local Lagrange basis functions for approximating the integral terms, we use the Lagrange interpolating polynomial to approximate and , we obtain Defining the quadrature weights the fully discretized collocation equation corresponding to (20)-( 22) is thus given by

k t t c h u t c h h w k t t c h u t c h h wk t t chu t ch
Note that, and Equation (23) represent for each a recursive system of m nonlinear algebraic equations with the unknowns   , n j u t  .

Global Convergence
Let denote the (exact) collocation solution to (1) defined by (16).In our convergence analysis we examine the linear test equation where of the Fredholm integral operator F. A comment of the convergence results to the nonlinear Equation (1) can be found at the end of this section.
Theorem 4.1.Assume that the given function in (24) satisfy Then for all sufficiently small h T N  the constrained mesh collocation solution to (24), for all where m are positive constants not depending on h.This estimate holds for all collocation parameters , the exact solution y of ( 24) is m times continuously differentiable.This follows from the smoothness hypotheses we have imposed on g k k and from the expressions for   y t .From this it is obvious that both the left and right limits of   y t , as t tends to , exist and are finite.We will prove the estimate (25) by using the Peano's Theorem to write Here, we have and Thus, it follows from (15) that the collocation error : y u    possesses to the local representation with and it satisfies the equation .
By substituting the (29) in the (30) and after some computations, we obtain , and the vectors in by , , , , , , by substituting the Equations ( 32)-(38) in Equation ( 31) we obtain this linear algebraic system may be written more concisely as Then we have .
Since the kernel i K is continuous on their domains, the elements of the matrixes 2, , 0,1, , for some matrix norm.This clearly holds whenever h is sufficiently small.In other words, there is an 0 h  so that for any mesh N  with , h h  each matrix n  has a uniformly bounded inverse.Therefore, matrix  has a uniformly bounded inverse.
Also, the invertibility of the m m block matrix now depends not only on h but also on and the elements of the matrixes Q are all bounded.Thus from (44), we get where and .
It is clear that, matrix has a uniformly bounded inverse and the elements of the matrixes are all bounded.Note that, from these assumptions and   there exists a constant so that for all mesh di- From ( 46) and (48) we have , and hence where Then, from (52) and (54), we have .
Now, by using the discrete Gronwall inequality, we have    Now, by using the local error representation (29) this yields, setting : max , The is equivalent to the estimate We conclude this section with a comment regarding the extension of the results of Theorem 1 to the nonlinear Equation (1).Under the assumption of the existence of a (unique) solution   y t on I, the nonlinear analogue of the error Equation ( 30) is are continuous and bounded on with for some and if is sufficiently small, then (55) may again be written in the form (30).The roles of are now assumed by . Hence, the above proof is easily adapted to deal with the nonlinear case (1), and so the convergence results of Theorem 1 remain valid for nonlinear Volterra-Fredholm integral equations.

Presentation of Results
In this section, we report on the numerical result of test problem solved by the proposed method of this article.
Typical forms of collocation parameters j c are: Gauss points: Zeros of   has the following analytical solution   y t  t therefore, provides an example to verify the accuracy of this method.
Table 1 shows the maximum errors involved presented method with 1 1 1 , , , 2 4 8 h  along with the exact solution.
For computational purposes, in the test problem different forms of kernels are considered.All the computations were carried out with Maple.In each cases of Example the obtained nonlinear equations was solved by the Newton's method.
The result for collocation points j c are presented in

Conclusion
We have shown that the collocation method yields an efficient and very accurate numerical method for the approximation of solutions to Volterra-Fredholm integral equations.Also we have shown that, if the roots of   0 m P t are chosen as collocation points, then we can obtain an accurate numerical quadrature.
The mentioned equations are characterized by the presence of a linear functional argument and play an important role in explaining many different phenomena.

0, 1
with the weight function 1.These polynomials are explicitly defined by of orthogonal polynomials on   0,1 which possesses all the properties of other classic orthogonal polynomials e.g.Legendre or Chebyshev polynomials.Therefore, if the roots of are chosen as collocation points, then we can obtain an accurate numerical quadrature. i chosen sufficiently large.By the Lemma 2.1, the operator T has a unique fixed-point in   I C n h are all bounded.By using the Neumann Lemma the inverse of the d

Table 1
which indicates that the numerical solutions ob-These results indicate that, if we use the Chelyshkov points, then we obtain the numerical solutions of minimum error.