THEORETICAL AND NUMERICAL RESULTS ABOUT SOME WEAKLY SINGULAR VOLTERRA-FREDHOLM EQUATIONS

. In this paper existence, uniqueness results for the solution of some weakly singular linear Volterra and Volterra-Fredholm integral equations are given. For these equations, a numerical model is proposed and its convergence and rate of convergence are analyzed. Numerical results on some polynomial test functions are given.

The aim of this paper is to present existence, uniqueness and numerical results for the solutions of the following weakly singular linear integral equations with linear modification of the argument: |x−s| α i , 0 < α i < 1, and At these problems, taking into account the kernel singularity, a particular numerical model [14] is adapted.This model is based on Nystrom collocation method, using Schoenberg variation diminishing (SVD) splines of fourth order.The order of convergence is studied and numerical results are given to test the polynomial exactness.
Section 2 and 3 are respectively devoted to the theoretical results for Volterra and Volterra-Fredholm weakly singular integral equations with linear modification of the argument.Sections 4 and 5 present the numerical model and its convergence analysis; in Section 6 numerical results are given.

VOLTERRA INTEGRAL EQUATIONS
Consider the following integral equation: By using the results given in [8] and [2] for weakly singular Volterra-Fredholm integral equations, we obtain: We have Theorem 2.3.In the conditions mentioned before, the equation ( 2 Denote We have where p > 0, q > 0,

It follows that
where We can choose τ large enough such that 0 < L T < 1.So, the proof follows from Contraction principle.

VOLTERRA-FREDHOLM INTEGRAL EQUATIONS
Consider the following Volterra-Fredholm weakly singular integral equation: We have Theorem 3.1.In the above conditions let , and we suppose that there exist p > 0, q > 0 and τ > 0, such that where

Proof. Let us consider the operators U
and T is well defined. Consider We have But from results given in Section 2, the following inequality holds: We estimate where So, the proof follows from Contraction principle.

NUMERICAL MODEL
In this section we present a numerical model suitable to (3.1 λ ) based on a global collocation method using approximating splines, in particular the so called Schoenberg variation-diminishing (SVD) splines [16].
In the following we recall the necessary background on SVD splines.Π n is assumed as mesh of the set of normalized B-splines B i,p (i = 1, . . ., n) of order p defined by the following recurrence relation: For all g ∈ C(J) we define the following spline operator: According to [9] W n is a SVD spline operator.In [9] it is shown to be a projector operator.

NUMERICAL SOLUTION OF THE PROBLEM
Let us consider the function: where α i (i = 1, 2, . . ., n) are chosen to satisfy the so called generalized Nystrom collocation system.Precisely, we introduce y n (λx) instead of y(λx) in (3.1 λ ) obtaining: (5.1) We can rewrite (5.1) as: (5.2) we choose in J a set of collocation points τ k (k = 1, 2, . . ., n), decoupled from the set of the ξ i (i = 1, 2, . . ., n).Consequently from (5.2) we obtain the following collocation system: The evaluation of the singular integrals and du is carried out by a recurrence formula analogous to (4.1).
The basis integrals du are evaluated by a closed analytical formula.
We assume y n (x) as approximated solution of (3.1 λ ).Now the problem is to analyze the convergence of y n (x) to y(x).where and 3) and let Proof.We can transform (5.4) and (5.5) in the operator form Applying the operator (4.3) to (5.7) we obtain (5.9) Analogously for (5.8) we obtain From (5.10) and (5.11) and taking account that W n is a projector operator, it follows (5.12) Then (5.6) holds.
Corollary 5.2.Let W n and K be the operators as in (5.12).Then for n sufficiently large, say n ≥ N, the operator (I − W n K) −1 from C(I) to C(I) exists.Moreover it is uniformly bounded, i.e.: Proof.From Theorem 1 and Corollary 2 in [14] it follows that K − W n K → 0 as n → ∞.Consequently, by the proof of Theorem 1 in [16], the Corollary 5.2 is proved.
Remark 5.1.From (5.12) and from the Corollary 5.2 it follows that y − y n → 0 exactly with the same rate of convergence as y − W n y does.

NUMERICAL RESULTS
In what follows we present some numerical results for some Volterra-Fredholm integral equation, by using the numerical method presented above.In particular, the exactness of the method for polynomial functions till third degree is tested.In all examples the hypotheses of existence and uniqueness of the solution are guaranteed.
We consider the following equation: where y is the unknown function and K 1 , K 2 , f are given functions.
Table 1.In Table 1 we show the results obtained with the choice K 1 (x, t) = .1 λ ) has in C[0, b] a unique solution and this solution can be obtained by the successive approximation method, starting from any element of C[0, b].Proof.Because of Lemma 2.1, we have that the operator U : C[0, b] → C[0, b], U (y)(x) := x 0 K(x, s)y(λs)ds, is well defined.So, we have that C[0, b] is an invariant set for the operator T , where T (y)(x) := f (x) + x 0 K(x, s)y(λs)ds.The equation (2.1 λ ) can be written as a fixed point problem of the form y = T (y).Consider T : (C[0, b], • B ) → (C[0, b], • B ), where • B is a Bielecki norm on C[0, b] defined by y B = max x∈[0,b] |y(x)|e −τ x , and τ > 0.

.
Then the equation (3.1 λ ) has in C[0, b] a unique solution y * and this solution can be obtained by the successive approximation method starting from any element of C[0, b].

0 K 2
s)y(λs)ds and U 2 (y)(x) := b (x, u)y(λu)du.By using Lemma 2.2 we obtain that U 1 and U 2 are well defined.The equation (3.1 λ ) is equivalent to the following fixed point problem: y = T (y), where T : C[0, b] → C[0, b], is given by