ON PROJECTIVE METHODS OF APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS. Introduction Let us consider an operator equation of second kind [1]

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, ON PROJECTIVE METHODS OF APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS A. JISHKARIANI AND G. KHVEDELIDZE Abstract. The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov Galerkin and collocation methods are investigated. Introduction Let us consider an operator equation of second kind [1] u T u = f, u E, f E, (1 where E is a Banach space and T : E E is a linear bounded operator. Let the sequences of closed subspaces {E n }, E n E, and of the corresponding projectors {P n } be given so that D(P n E, E n D(P n, P n (D(P n = E n, T E D(P n, f D(P n, n = 1, 2,..., where D(P n denotes the domain of definition of P n. Applying the Galerkin method to equation (1, we obtain an approximate equation [1] u n P n T u n = P n f, u n E n. (2 It is known [1] that if the operator I T is continuously invertible, and P (n T 0 for n, where P (n I P n, then for sufficiently large n the approximating equation (2 has a unique solution u n, and the estimate u u n = O( P (n u is valid. Assume that we have found an approximate solution u n of equation ( Mathematics Subject Classification. 65R20. Key words and phrases. Singular integral operator, approximate projective method, iteration, rate of convergence. * This author is also known as A. V. Dzhishkariani X/96/ $12.50/0 c 1997 Plenum Publishing Corporation

2 458 A. JISHKARIANI AND G. KHVEDELIDZE Take one iteration (see [2] ũ n = T u n + f. (3 The element ũ n E, being the approximate solution of equation (1 by the Galerkin method, satisfies the equation From (1 and (4 we have ũ n T P n ũ n = f. (4 (I T P n (u ũ n = T P (n u. If the operator I T is continuously invertible, and T P (n 0 for n, then for sufficiently large n there exists the inverse bounded operator (I T P n. Therefore u ũ n (I T P n T P (n u, n n 0. (5 Since T P (n u T P (n P (n u, the rate of convergence u ũ n compared to u u n can be increased by means of a good estimate T P (n. In the present paper we consider in the weighted space a singular integral equation of the form Su + Ku = f, (6 where Su 1 u(tdt t x, < x < 1, is a singular integral operator, and Ku 1 1 K(x, tu(tdt is an integral operator of the Fredholm type (see [3], [4]. For the singular integral equation (6 we may have three index values: κ =, 0, 1. Our aim is to derive, for (6, an estimate of the convergence rate of the projective Bubnov Galerkin and collocation methods with one iteration when Chebyshev Jacobi polynomials are taken as a coordinate system. Note that the results described below are also valid with required modifications for the singular integral equation of second kind (a + bs + Ku = f, where a and b are real numbers, a 2 + b 2 > 0.

3 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS The Bubnov Galerkin Method with One Iteration 1.1. Index κ = 1. We take a weighted space L 2,ρ [, 1], where the weight ρ = ρ 1 = (1 x 2 1/2. The scalar product [u, v] = ρ 1uv dx. For the index κ = 1 we have the additional condition u(tdt = p, (7 where p is a given real number. The operator S is bounded in L 2,ρ (see [4]. We require of the kernel K(x, t that the operator K be completely continuous in L 2,ρ. The homogeneous equation Su = 0 in the space L 2,ρ has a nontrivial solution u = (1 x 2 /2. In the space L 2,ρ the following two systems of functions are orthonormalized and complete: (1 ϕ k (x (1 x 2 /2 Tk (x, k = 0, 1,..., ( 1 1/2T0 ( 2 1/2Tk+1 T 0 =, Tk+1 =, k = 0, 1,..., where T k, k = 0, 1,..., are the Chebyshev polynomials of first kind, and ( 1/2Uk 2 (2 ψ k+1 (x (x, k = 0, 1,..., where U k, k = 0, 1,..., are the Chebyshev polynomials of second kind. Denote Φ u p (1 x 2 /2. Then problem (6 (7 can be written in the form (see [5] SΦ + KΦ = f 1, Φ L (2 2,ρ, f 1 L 2,ρ, (8 Φ(tdt = 0, (9 where f 1 f p K(1 t 2 /2, L 2,ρ = L (1 2,ρ L(2 2,ρ decomposition, L (1 L (2 2,ρ is the orthogonal 2,ρ is the linear span of the function ϕ 0 = (1 x 2 /2, and is its orthogonal complement. In the sequel, under S we shall mean its restriction on L (2 2,ρ. Then S(L(2 2,ρ = L 2,ρ and S (L 2,ρ = L (2 2,ρ. The relations Sϕ k = ψ k, k = 1, 2,..., (10

4 460 A. JISHKARIANI AND G. KHVEDELIDZE (see [6] are valid. An approximate solution of equation (8 is sought in the form Φ n = a k ϕ k. Owing to (10, the algebraic system composed of the conditions yields [SΦ n + KΦ n f 1, ψ i ] = 0, i = 1, 2,..., n, a i + a k [Kϕ k, ψ i ] = [f 1, ψ i ], i = 1, 2,..., n. (11 It is known [5] that if there exists the inverse operator (I + KS mapping L 2,ρ onto itself, then for sufficiently large n the algebraic system (11 has a unique solution (a 1, a 2,..., a n, and the sequence of approximate solutions u n = Φ n + p (1 x 2 /2 converges to the exact solution u in the metric of the space L 2,ρ. Similar results are valid for κ =, 0. With the help of the orthoprojector P n which maps L 2,ρ onto the linear span of the functions ψ 1,..., ψ n we can rewrite the algebraic system (11 as w n + P n KS w n = P n f 1, w n SΦ n = From the initial equation (8 we have a k ψ k. (12 w + KS w = f 1, w L 2,ρ, f 1 L 2,ρ, w SΦ. (13 Equation (12 is the Bubnov Galerkin approximation for (13. As in [2], let us introduce the iteration where w n satisfies the equation For n n 0 we obtain w n = KS w n + f 1 = KΦ n + f 1. (14 w n = KS P n w n + f 1. w w n C KS P (n P (n w.

5 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 461 Let Φ n S w n. To find Φ n, it is necessary to calculate the integral We have S w n = (1 t2 /2 (1 x 2 1/2 w n(xdx t x. u ũ n = Φ Φ n = S (w w n = w w n where ũ n = Φ n + p (1 x 2 /2. C KS P (n P (n w, Theorem 1. If there exists the inverse operator (I + KS mapping L 2,ρ onto itself, and the conditions w (n Lip M α, 0 < α 1, and K (l (x, t Lip M α 1, 0 < α 1 1, x [, 1], are fulfilled for the derivatives, then the estimate is valid. Proof. We have P (n w 2 = = u ũ n = O(n (m+α (l+α1 P n w = [w, ψ k ]ψ k. (1 x 2 1/2 (w (1 x 2 1/2 (w P n w 2 dx = [w, ψ k ]ψ k 2 ds (1 x 2 1/2 (w P n 2 dx 2 w P n 2 C, where P n is the polynomial of the best uniform approximation. By Jackson s theorem [7] we have w P n C C(w, n > 1, (n 1 m+α with a constant C(w depending on w and its derivatives, i.e., P (n w = O(n (m+α.

6 462 A. JISHKARIANI AND G. KHVEDELIDZE Furthermore, KS P (n v 2 = KS 1 2 { = v 2 2 = v 2 2 = = = 1 2 v 2 2 [v, ψ k ] 2} 1/2 { (1 x 2 1/2( (1 x 2 1/2( [v, ψ k ]ψ k 2 = K [v, ψ k ] ( K(x, t, ϕ k (t 2 (1 x 2 1/2 (1 t 2 1/2( [ K(x, t, Tk (t ] Lr,ρ (1 t 2 1/2( K(x, t [v, ψ k ]ϕ k 2 = ( K(x, t, ϕk (t 2 } 1/2 2 ( K(x, t, ϕk (t 2 dx = ( K(x, t, (1 t 2 /2 Tk (t 2 dx = [ K(x, t, Tk (t ] Lr,ρ T k (t 2 = L 2,ρ 2 T k (t dx, L 2,ρ [ K(x, t, Tk (t ] 2dt T L2,ρ k (t = [ K(x, t, Tk (t ] 2dt T L2,ρ k (t k=0 (1 t 2 1/2( K(x, t P n (x, t 2 dt (E t n( K(x, t 2, where x is a parameter, P n (x, t is the polynomial of the best uniform approximation with respect to t, and En(K(x, t t the corresponding deviation K(x, t P n (x, t En t ( K(x, t, < x, t < 1. If K (l t (x, t Lip M1 α 1, 0 < α 1 1 x [, 1], and is continuous with respect to x in [, 1], then (see [8, Ch.XIV, 4] E t n( K(x, t = O ( n (l+α 1.

7 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 463 Furthermore, KS P (n v 2 v 2 2 (1 x 2 1/2 ( E t n(k(x, t 2 dx = = v 2 2 ( E t 2 n (K(x, t 2. Thus KS P (n = O ( n (l+α 1. Finally, we get u ũ n KS P (n w KS P (n P (n w = O(n (m+α (l+α 1. For the approximate solution u n we have u u n C P (n w = O(n (m+α, while for one iteration ũ n performed over u n when l = m, α 1 = α, we obtain the estimate u ũ n = O(n 2(m+α Index κ =. The operator S is bounded in the weighted space L 2,ρ [, 1], where ρ = ρ 2 = (1 x 2 /2 (see [4]. We require of the kernel K(x, t that the operator K be completely continuous in L 2,ρ. The equation Su = 0 in L 2,ρ has the zero solution only, while the equation S u = 0 has the nonzero solution u = 1. If in the weighted space L 2,ρ the equation Su + Ku = f has a solution u, then [Ku f, 1] = 0. This condition will be fulfilled if K(L L2,ρ 1 and [f, 1] = 0, which can be achieved by specific transform [9]. In the space L 2,ρ the following two systems of functions are complete and orthonormal: ( 2 1/2Uk (1 ϕ k+1 (x (x, k = 0, 1,..., where U k, k = 0, 1,... are Chebyshev polynomials of second kind, and ( 2 (2 ψ k+1 (x T k+1 (x, k = 0, 1,..., where T k+1, k = 0, 1,... are Chebyshev polynomials of first kind. Relations Sϕ k = ψ k, k = 1, 2,... (see [6] are valid. An approximate solution is again sought in the form u n = a k ϕ k.

8 464 A. JISHKARIANI AND G. KHVEDELIDZE The Bubnov Galerkin method results in the algebraic system a i + a k [Kϕ k, ψ i ] = [f, ψ i ], i = 1, 2,..., n. (15 Denote w n Su n = n a kψ k. Then using the orthoprojector P n mapping L 2,ρ onto the linear span of the functions ψ 1, ψ 2,..., ψ n the algebraic system (15 can be rewritten as Let the approximate solution w n be found. Taking one iteration w n + P n KS w n = P n f. (16 w n = KS w n + f = Ku n + f, we find that ũ n = S w n = (1 t2 1/2 (1 x2 /2 w n(xdx t x. Theorem 2. If there exists the inverse operator (I + KS mapping L (2 2,ρ onto itself, and the conditions w(m Lip M α, 0 < α 1, K (l t (x, t Lip M α 1, 0 < α 1 1, x [, 1], are fulfilled for the derivatives, then the estimate u ũ n = O(n (m+α (l+α1 is valid. This theorem as well as Theorem 3 which will be formulated in the next subsection can be proved similarly to Theorem Index κ = 0. Here we may have two cases: (1 α = 1 2, β = 1 2 and (2 α = 1 2, β = 1 2. Let us consider the first case. The second one is considered analogously. The operator S is bounded in the weighted space L 2,ρ [, 1] with the weight ρ = ρ 3 = (1 x 1/2 (1 + x /2 [4]. We require of the kernel K(x, t that the operator K be completely continuous in L 2,ρ [, 1]. In the space L 2,ρ the equations Su = 0 and S u = 0 have only trivial solution u = 0, S(L 2,ρ = L 2,ρ, where S is the unitary operator. We have the equation Su + Ku = f, u L 2,ρ, f L 2,ρ. (17 In L 2,ρ we take two complete and orthonormal systems of functions (see [10]: (1 ϕ k c k (1 x 1/2 (1 + x /2 P (1/2,/2 k, k = 0, 1,..., c 0 =, c k = (h (/2,1/2 k /2, k = 1, 2,...,

9 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 465 h (/2,1/2 k = h (1/2,/2 2Γ(k + 1/2Γ(k + 3/2 k = (2k + 1(k! 2, where P (1/2,/2 k, k = 0, 1,..., are the Jacobi polynomials; (2 ψ k c k P (/2,1/2 k. The relations Sϕ k = ψ k, k = 0, 1,... (18 (see [6] are valid. We seek an approximate solution of equation (17 in the form u n = a k ϕ k. With regard to (18 the Bubnov Galerkin method yields the algebraic system [Su n + Ku n f, ψ i ] = 0, i = 0, 1,..., n, a i + a k [Kϕ k, ψ i ] = [f, ψ i ] i = 0, 1,..., n, (19 k=0 which, by means of the orthoprojector P n mapping L 2,ρ onto the linear span of the functions ψ 0, ψ 1,..., ψ n, can be written in the form w n + P n KS w n = P n f, w n Su n = Let the approximate solution w n be found. Taking one iteration we find a k ψ k. (20 k=0 w n = KS w n + f = Ku n + f, ũ n = S w n = (1 + t1/2 (1 t /2 Then (1 x 1/2 (1 + x /2 w n(xdx t x. u ũ n = S (w w n = w w n C KS P (n P (n w.

10 466 A. JISHKARIANI AND G. KHVEDELIDZE Theorem 3. If there exists the inverse operator (I + KS mapping L 2,ρ onto itself, and the conditions w (m Lip M α, 0 < α 1, K (l t (x, t Lip M α 1, 0 < α 1 1, x [, 1], are fulfilled for the derivatives, then the estimate u ũ n = O(n (m+α (l+α1 is valid. 2. Method of Collocation with One Iteration Using the collocation method, let us now consider the solution of equation (6. Assume that the kernel K(x, t and f(x are continuous functions Index κ = 1. As in Subsection 1.1 we seek an approximate solution of problem (8 (9 in the form Φ n = a k ϕ k. By the collocation method the residual SΦ n +KΦ n f 1 (f 1 is introduced above by (8 at discrete points will be equated to zero, [SΦ n + KΦ n f 1 ] xj = 0, j = 1, 2,..., n. This, owing to (10, results in the algebraic system a k ψ k (x j + a k (Kϕ k (x j = f 1 (x j, j = 1, 2,..., n. (21 As is known [11], if there exists the operator (I + KS mapping L 2,ρ onto itself, and as the collocation nodes are taken the roots of the Chebyshev polynomials of the second kind U n, then for sufficiently large n the algebraic system (21 has a unique solution, and the process converges in the space L 2,ρ. Analogous results are valid for the index κ =, 0. Let us take the space of continuous functions C[, 1]. Let Π n be the projector defined by the Lagrange interpolation polynomial Π n v = L n v. With the help of this projector the algebraic system (21 can be rewritten in the form w n + Π n KS w n = Π n f 1, w n L (n 2,ρ, where L (n 2,ρ is the linear span of functions ψ 1, ψ 2,..., ψ n (ψ k is a polynomial of degree k 1. Let the approximate solution w n be found. As in the Bubnov Galerkin method we take one iteration w n = KS w n + f 1 = KΦ n + f 1.

11 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 467 where w n satisfies the equation By means of (13 and (22 we obtain w n + KS Π n w n = f 1. (22 w w n + KS w KS Π n w n + KS w n KS w n = 0, (I + KS (w w n = KS Π (n w n, Π (n I Π n, w w n = (I + KS KS Π (n ( Kφ n + f 1, w w n C ( KS Π (n K Φ n + KS Π (n f 1. For sufficiently large n we have [11] w n = (I + Π n KS Π n f 1, Φ n = S w n C Π n f 1, Π n f 1 f 1, for n, f 1 C[, 1], i.e., Φ n are uniformly bounded owing to the Erdös Turan [10] and Banach Steinhaus [8] theorems. Therefore Then We find that w w n C ( KS Π (n K + KS Π (n f 1. Φ n S w n = 1 1 (1 t2 /2 (1 x 2 1/2 w n (xdx. t x u ũ n = Φ Φ n = S (w w n = w w n C ( KS Π (n K + KS Π (n f 1, (23 where ũ n Φ n + p (1 x 2 /2. It is known [12] that Π (n v(t = ω(tδ (n v(t, v C[, 1], where ω(t n i=1 (t t i and δ (n v(t is the divided difference of the continuous function v(t. If the roots of the Chebyshev polynomial of second kind U n are taken as interpolation nodes, then (see [13] ( 1/2 Π (n Ûn (t v(t = 2 2 n δ(n v(t Denote K 1 (x, t (1 t 2 /2 K(x, t. ( 2 (Ûn (t = 1/2Un (t.

12 468 A. JISHKARIANI AND G. KHVEDELIDZE Theorem 4. If there exists the inverse operator (I + KS mapping L 2,ρ, ρ = ρ 1, onto itself, the roots of the second kind Chebyshev polynomial U n are taken as collocation nodes, (SK 1 (x, t m L PM α, 0 < α 1, x [, 1], and the divided differences of all orders of the functions f 1 (x and K 1 (x, t with respect to x are uniformly bounded t [, 1], then the estimate is valid. ( 1 u ũ n = O 2 n 1 (n 1 m+α Proof. Let us estimate the norms on the right-hand side of inequality (23. We have KS Π (n Kv L2,ρ = 1 ( K(x, t, S Π (n (K(x, τ, v(τ = 1 2 ( (1 t 2 1/2 (1 t 2 /2 K(x, t, S Π (n (1 τ 2 1/2 (1 τ 2 /2 (K(x, τ, v(τ = = 1 2 [ SK1 (x, t, Π (n [K 1 (x, τ, v(τ] ] = = 1 2 [ SK1 (x, t, [Π (n K 1 (x, τ, v(τ] ] = = 1 2 [ [SK1 (x, t, Π (n K 1 (x, τ], v(τ] ] = = 1 ( 1/2 [ Û n (t [SK1 2 (x, t, 2 2 n δ(n K 1 (x, τ, v(τ ] = = 1 ( 1/2 1 [ δ (n n K 1 (x, τsk 1 (x, t, P (n Û n (t]v(τ ] = = 1 ( 1/2 1 [ [P (n n δ (n K 1 (t, τsk 1 (x, t, Ûn(t]v(τ ] 1 ( 1/2 1 [ P (n n δ (n K 1 (t, τsk 1 (x, t, Ûn(t] v(τ v ( 1/2 1 P (n n δ (n K 1 (t, τsk 1 (x, t Ûn(t v ( 1/2 1 P (n n δ (n K 1 (t, τsk 1 (x, t. Under the conditions of the theorem (δ (n K 1 (t, τsk 1 (x, t (m Lip M α, 0 < α 1, x, τ [, 1]. By Jackson s theorem (see [7], [8] P (n δ (n K 1 (t, τsk 1 (x, t c m2 m+α M (n 1 m+α,

13 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 469 ( where c m 12 6m m m m+1 α. m! 2 Therefore we obtain ( 1 KS Π (n K = O 2 n 1 (n 1 m+α. (24 Furthermore, KS Π (n f 1 L2,ρ = 1 (K(x, t, S Π (n f 1 = = 1 ( 1/2 [SK1 (x, t, Ûn(t 2 2 n δ(n f 1 = = 1 ( 1/2 1 [δ (n 2 2 n f 1 SK 1 (x, t, P (n Û n (t] = = 1 ( 1/2 1 [P (n 2 2 n δ (n f 1 SK 1 (x, t, Ûn(t] 1 ( 1/2 1 P (n 2 2 n δ (n f 1 SK 1 (x, t Ûn (t 1 ( 1/2 1 P (n 2 2 n δ (n f 1 SK 1 (x, t. Under the conditions of the theorem ( δ (n f 1 SK 1 (x, t (m LipM α, 0 < α 1, x [, 1]. Therefore we have P (n δ (n f 1 SK 1 (x, t C m 2m+α M (n m+α, ( KS Π (n 1 f 1 = O 2 n 1 (n 1 m+α. (25 With the help of the obtained estimates (24 and (25, from (23 we finally get ( 1 u ũ n = O 2 n 1 (n 1 m+α. Remark 1. Under the conditions of the theorem we obtain for the approximate solution u n that ( 1/2 u u n C Π (n Û n (t w = C 2 2 n δ(n w C 1 2 n Ûn(tδ (n w(t = C { 1/2 1 2 n (1 t 2 1/2 Ûn(t(δ 2 (n w dt} 2

14 470 A. JISHKARIANI AND G. KHVEDELIDZE C 1 2 n δ(n w C { (1 t 2 1/2 Û 2 n(tdt} 1/2 = = C 1 2 n δ(n w C Ûn(t = C 1 2 n δ(n w C = C 1 2 n δ(n (f 1 KΦ C = = C 1 δ (n 2 n (f 1 1 [ K1 (x, t, Φ(t ] C C 1 ( δ (n 2 n f 1 C + 1 [ δ (n K 1 (x, t, Φ(t ] C C 1 ( δ (n 2 n f 1 C + 1 δ (n K 1 (x, t Φ(t C C 1 ( δ (n 2 n f 1 C + 1 δ (n K 1 (x, t C C 2 2 n Index κ =. Introduce the subspace C 0 [, 1] C[, 1]; v C 0 [, 1] if [v, 1] = 0. C (n [, 1] C[, 1] is a linear span of polynomials ψ 0, ψ 1,..., ψ n. The projector can be determined as follows [11]: Π n v = L n v a (n 0 ψ 0, where L n v C (n [, 1] is the Lagrange polynomial and a (n 0 is the coefficient of the Fourier series expansion a (n 0 [L n v, ψ 0 ]. Again, as in Subsection 1.2, we seek an approximate solution in the form u n = a k ϕ k. We compose the algebraic system by the condition which results in a 0 ψ 0 + a k ψ k (x j + Π n (Su n + Ku n f = 0, a k (Kϕ k (x j = f(x j, j = 0, 1,..., n. Using the projector, we can rewrite this algebraic system as w n + Π n KS w n = Π n f, w n L (n 2,ρ, where L (n 2,ρ is the linear span of the system of functions ψ 0, ψ 1,..., ψ n. Let the approximate solution w n be found. Taking one iteration w n = KS w n + f = Ku n + f,

15 we find APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 471 ũ n = S w n = (1 t2 1/2 Denote K 1 (x, t (1 t 2 1/2 K(x, t. (1 x 2 /2 w n(xdx t x. Theorem 5. If there exists the inverse operator (I + KS mapping L (r 2,ρ, ρ = ρ 2, onto itself, the roots of the Chebyshev polynomial of the first kind T n+1 are taken as collocation nodes, (SK 1 (x, t m Lip M α, 0<α 1, x [, 1], and the divided differences of all orders of the functions f(x and K 1 (x, t with respect to x are uniformly bounded t [, 1], then the estimate ( 1 u ũ n = O 2 n 1 n m+α is valid. This theorem as well as the next one can be proved similarly to Theorem 4. Remark 2. As in Subsection 2.1, under the conditions of the theorem we obtain ( 1 u u n = O 2 n Index κ = 0. As in Subsection 1.3, an approximate solution is again sought in the form u n = a k ϕ k. Equating the residuals to zero at the points x 1,..., x, we obtain [ Sun + Ku n f ] x j = 0 j = 0, 1,..., n, which yields the algebraic system a k ψ k (x j + a k (Kϕ k (x j = f(x j, j = 0, 1,..., n. k=0 k=0 Just as for the index κ = 1 we can rewrite this system as w n + Π n KS w n = Π n f, w n L (n 2,ρ, where L (n 2,ρ is the linear span of functions ψ 0,..., ψ n. Let the approximate solution w n be found. Taking one iteration w n = KS w n + f = Ku n + f

16 472 A. JISHKARIANI AND G. KHVEDELIDZE we find ũ n = S w n = (1 + t1/2 (1 t /2 Denote K 1 (x, t (1 + t 1/2 (1 t /2 K(x, t. (1 x 1/2 (1 + x /2 w n(xdx t x. Theorem 6. If there exists the inverse operator (I + KS mapping L 2,ρ, ρ = ρ 3, onto itself, the roots of the Jacobi polynomial P ( 1 2, 1 2 n+1 are taken as collocation nodes, (SK 1 (x, t (m L : p M α, 0 < α 1, x [, 1], and the divided differences of all orders of the functions f(x and K 1 (x, t with respect to x are uniformly bounded t [, 1], then the estimate ( 1 u ũ n = O 2 n 1 n m+α is valid. Remark 3. Under the conditions of the theorem ( 1 u u n = O 2 n. Remark 4. If we require only that w (m Lip M α, 0 < α 1, then for the collocation method for all values of the index κ = 1, 0, we obtain the same order of convergence ( ln n u u n = O n m+α for an approximate solution u n in the respective weighted spaces. Indeed, for any v C[, 1] we have Π (n v = Π (n P (n v, where Π (n I Π n, P (n I P n, where Π n is the Lagrange interpolation operator, and P n is the orthoprojector with respect to polynomials T 0, T 1,..., T n. If the nodes in the interpolation Lagrange polynomial are taken with respect to the weight, then L n : C L 2,ρ are bounded by the Erdös Turan theorem. Therefore (see [13] u u n L2,ρ C Π (n u L2,ρ C 1 P (n u C = ( ln n = O n m+α (Π n = L n. As an example of the application of the above methods in the case of the index κ =, let us consider the equation 1 u(tdt t x + 1 (x 8 t 8 + x 7 t 7 u(tdt =

17 APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 473 ( 2 1/2 ( 3 = 128 x7 32x x 4 18x with the exact solution ( 2 1/2(32x u(x = ϕ 6 (x = 5 32x 3 + 6x(1 x 2 1/2, where {ϕ k (x}, k = 0, 1,..., is the orthonormal system of functions in L 2,ρ2. We find the fifth approximation u 5 (x = 5 ( 2 1/2(1 a k ϕ k (x, ϕ k (x = x 2 1/2 U k (x, where U k (x, k = 1, 2,..., are the Chebyshev polynomials of the second kind. Computations are carried out to within u 5 (x and ũ 5 (x are calculated. In the case of the Bubnov Galerkin method we have u 5 = 1, , ũ 5 = 0, , u 5 u 100, 01%, ũ 5 u 1, 52% for an absolute and a relative error, respectively, while in the case of the collocation method we obtain u 5 = 1, , ũ 5 = 0, , u 5 u 100, 03%, ũ 5 u 1, 60%. The result is the expected one for u 5 (x, since in our example the function ϕ 6 (x is the exact solution. References 1. M. A. Krasnoselsky, G. M. Vainikko, P. P. Zabreiko, Ja. B. Rutitskii, and V. Ja. Stetsenko, Approximate solution of operator equations. (Russian Nauka, Moscow, F. Chatelin and R. Lebbar, Superconvergence results for the iterative projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem. J. Integral Equations 6(1984, N. I. Muskhelishvili, Singular integral equations. (Translation from the Russian P. Nordhoff (Ltd., Groningen, 1953.

18 474 A. JISHKARIANI AND G. KHVEDELIDZE 4. B. V. Khvedelidze, Linear discontinuous boundary value problems of function theory, singular integral equations and some of their applications. (Russian Trudy Tbilis. Mat. Inst. Razmadze 23( A. V. Dzhishkariani, To the solution of singular integral equations by approximate projective methods. (Russian J. Vychislit. Matem. i Matem. Fiz. 19(1979, No. 5, F. G. Tricomi, On the finite Hilbert transformation. Quart. J. Math., 2(1951, No. 7, I. P. Natanson, Constructive theory of functions. (Rusian Gosizdat. Techniko-Theoret. Literatury, Moscow Leningrad, L. V. Kantorovich and G. P. Akilov, Functional analysis. (Russian Nauka, Moscow, A. V. Dzhishkariani, To the problem of an approximate solution of one class of singular integral equations. (Russian Trudy Tbilis. Mat. Inst. Razmadze 81(1986, G. Szegö, Orthogonal polynomials. American Mathematical Society Colloquium Publications, New York XXIII( A. V. Dzhishkariani, To the solution of singular integral equations by collocation methods. (Russian J. Vychislit. Matem. i Matem. Fiz. 21(1981, No. 2, I. S. Berezin and N. P. Zhitkov, Computation methods, vol. 1. (Russian Fizmatgiz, Moscow, P. K. Suetin, Classical orthogonal polynomials. (Russian Nauka, Moscow, (Received Authors address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi Republic of Georgia