A Central Part Interpolation Scheme for Log-Singular Integral Equations

A fully discrete high order method is constructed and justified for a class of Fredholm integral equations of the second kind with kernels that may have boundary and logarithmic diagonal singularities. The method is based on the improving the boundary behaviour of the kernel with the help of a change of variables, and on “central part” interpolation by polynomials on the uniform grid.


Introduction
In the present paper we treat a fully discrete method of accuracy order O(h m ) for the integral equation a(x, y) log |x − y| + b(x, y) u(y) dy + f (x), 0 ≤ x ≤ 1, (1.1) with the logarithmic diagonal singularity in the kernel. The coefficient functions a, b ∈ C m ([0, 1] × (0, 1)) and the free term f ∈ C[0, 1] ∩ C m (0, 1) may have certain boundary singularities characterised in Lemma 1 and Lemma 2 below. The method is based on a smoothing change of variables (see for example [3,4,6,7,10]) and a product integration method based on the "central part" interpolation on the uniform grid, which has also been used in [8] for solving weakly singular integral equation. We rely on the fact that the "central part" interpolation of smooth functions by polynomials and by high order polynomial splines on uniform grids has excellent accuracy and stability properties comparable with the accuracy and stability properties of Chebyshev interpolation [16]. The use of data only on uniform grids is preferable also from the algorithmical point of view due to better numerical stability. See Section 3 for the details about the "central part" interpolation. In Section 2 we reduce problem (1.1) by the smoothing change of variables to the form with smooth data and solution.
The present paper is a complement to work [14], where the convergence and the error of the product quasi-interpolation method has been presented for the integral equation (1.1). We refer yet to [9], where another type fully discrete method for solving (1.1) has been constructed.
Denote by T the integral operator of equation (1.1) a(x, y) log |x − y| + b(x, y) u(y) dy.
Let I be the identity mapping and denote N (I−T ) = {u ∈ C[0, 1] : u=T u} . The following theorem is a consequence of Lemmas 1 and 2.
Our main results will be established under assumptions of Theorem 1.

The Smoothing Change of Variables
In the integral equation (1.1) we perform the change of variables .
We assume, that p 0 , p 1 ∈ N. If so, the integral in (2.1) can be evaluated in a stable way by an exact Gauss rule, since the integrand is a polynomial of degree p 0 + p 1 − 2. Clearly, ϕ(0) = 0, ϕ(1) = 1 and ϕ(t) is strictly increasing where v(t) = u(ϕ(t)) is the new function we look for, Let us characterise the boundary behaviour of functions in equation (2.2). Clearly, Lemma 3. Let a and b satisfy the conditions of Lemma 2. Then for j = 0, . . . , m, The proof of the inequality is based on the formula of Faà di Bruno where the sum is taken over all non-negative integers k 1 , . . . , k j such that k 1 + 2k 2 + · · · + jk j = j. The derivatives of the function Φ(t, s) have singularities at (0, 0) and (1, 1), the only zeroes of Φ(t, s) in [0, 1] × [0, 1]. It is easy to see that that together with the formula of Faà di Bruno implies the following results.
Next we present estimates for functions Lemma 5. Let a and b satisfy the conditions of Lemma 1. Then the following holds true: if p 0 , p 1 ≥ 1 satisfy Lemma 6 [see [13]]. Let the conditions of Lemma 2 be fulfilled.

Central Part Interpolation by Polynomials and Piecewise Polynomials
Given an interval [a, b] and m ∈ N, introduce the uniform grid consisting of m points Denote by Π m the Lagrange interpolation projection operator assigning to any u ∈ C[a, b] the polynomial Π m u ∈ P m−1 that interpolates u at points (3.1); P m−1 is the set of polynomials of degree not exceeding m − 1.
Lemma 7 [see [16]]. In the case of interpolation knots In the central parts of [a, b], the interpolation process on the uniform grid has good stability properties as m increases: in contrast to an exponential growth of Π m C[a,b]→C[a,b] as m → ∞, it holds by the Runck's theorem (see [12]) where the constant c r depends only on r > 0.
Introduce in R the uniform grid {jh : j ∈ Z} where h = 1/n, n ∈ N. Let m ≥ 2 be fixed. Given a function f ∈ C[−δ, 1 + δ], δ > 0, we define a piecewise polynomial interpolant Π h,m f ∈ C[0, 1] as follows. On every subinterval [jh, (j + 1)h], 0 ≤ j ≤ n − 1, the function Π h,m f is defined independently from other subintervals as a polynomial Π [j] h,m f ∈ P m−1 of degree ≤ m − 1 by the conditions h,m f (lh) = f (lh), for l ∈ Z such that l − j ∈ Z m , where Z m = k ∈ Z : − m 2 < k ≤ m 2 . The Π h,m f is uniquely defined at interior knots and Π h,m f is continuous on [0, 1] (see [8]); the one side derivatives of the interpolant Π h,m f at the interior knots may be different.
Introduce the Lagrange fundamental polynomials L k,m ∈ P m−1 , k ∈ Z m satisfying L k,m (l) = δ k,l for l ∈ Z m , where δ k,l is the Kronecker symbol, δ k,l = 0 for k = l and δ k,k = 1. An explicit formula for L k,m is given by Then (see [8]) h,m f (t) = k∈Zm f (j + k)h L k,m (nt − j), j = 0, . . . , n − 1.
maintains the smoothness of f . The operator is well defined and P 2 h,m = P h,m , i.e., P h,m is a projector in C[0, 1]. For w h ∈ R(P h,m ) (the range of P h,m ) we have w h = P h,m w h = Π h,m E δ w h , and due to (3.8) Thus w h ∈ R(P h,m ) is uniquely determined on [0, 1] by its knot values w h (ih), i = 0, . . . , n. We conclude, that dim R(P h,m ) = n + 1. It is also clear, that for a w h ∈ R(P h,m ) we have w h = 0 if and only if w h (ih) = 0, i = 0, . . . , n. On h,m f coincides with the polynomial interpolant Π m f constructed for f on the interval [a j , b j ] where a j = (j − m−1 2 )h, b j = (j + m+1 2 )h in the case of even m and a j = (j − m 2 )h, b j = (j + m 2 )h in the case of odd m; moreover, [jh, (j + 1)h] is a "central" part of [a j , b j ] on which the interpolation error can be estimated by (3.3) and (3.4).
In this way we obtain the following result.
with ϑ m defined in (3.3) and (3.4) respectively for even and odd m.

A Product Integration Method Based on the Central Part Interpolation
We determine the approximate solution v h of equation ( Here P h,m (see (3.9)) is applied to the products A(t, s)v h (s) and B(t, s)v h (s) as functions of s, treating t as a parameter. This is the operator form of a product interpolation method corresponding to the piecewise polynomial "central part" interpolation on the uniform grid {ih : i = 0, . . . , n} . Below we use the following notations for the integral operators of equations (1.1), (2.2) and (4.1) respectively: where v ∈ C[0, 1] is the unique solution of equation ( Then constant c in (4.3) is independent of n and f .
(ii) Let us prove the error estimate (4.3) under conditions (2.5) on p 0 and p 1 , and p 0 , p 1 > m. For the solution u of (1.1) we have by the Theorem 1 u ∈ C m,λ0,λ1 (0, 1). On the basis of (2.7)-(2.9) and (3.10) we find that This proves (4.3) and completes the proof of the Theorem.
Let us derive the matrix form of the product interpolation method (4.1). From the definition of the operator P h,m (see (3.8) We obtain the algebraic system of linear equations with respect to the grid values v h (ih), i = 0, . . . , n by collocating at the points ih: + g(ih), i = 0, . . . , n; note that A(ih, (j + k)h) = 0 and B(ih, (j + k)h) = 0 for j + k ≤ 0 and for j + k ≥ n, thus in the r.h.s the values v h (lh) with l ≤ 0 and l ≥ n actually are not exploited.
With the change of variables ns − j = σ we see that We took into account that A(ih, lh) = 0 and B(ih, lh) = 0 for l ≤ 0 and l ≥ n.