GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, 457-474 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 {En }, En ⊂ E, and of the corresponding projectors {Pn } be given so that D(Pn ) ⊂ E, En ⊂ D(Pn ), Pn (D(Pn )) = En , T E ⊂ D(Pn ), f ∈ D(Pn ), n = 1, 2, . . . , where D(Pn ) denotes the domain of definition of Pn . Applying the Galerkin method to equation (1), we obtain an approximate equation [1] un − Pn T un = Pn f, u n ∈ En . (2) It is known [1] that if the operator I − T is continuously invertible, and kP (n) T k → 0 for n → ∞, where P (n) ≡ I − Pn , then for sufficiently large n the approximating equation (2) has a unique solution un , and the estimate ku − un k = O(kP (n) uk) is valid. Assume that we have found an approximate solution un of equation (2). 1991 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. 457 c 1997 Plenum Publishing Corporation 1072-947X/96/0900-0457$12.50/0 458 A. JISHKARIANI∗ AND G. KHVEDELIDZE Take one iteration (see [2]) u en = T un + f. (3) u en − T Pn u en = f. (4) The element u en ∈ E, being the approximate solution of equation (1) by the Galerkin method, satisfies the equation From (1) and (4) we have en ) = T P (n) u. (I − T Pn )(u − u If the operator I −T is continuously invertible, and kT P (n) k → 0 for n → ∞, then for sufficiently large n there exists the inverse bounded operator (I − T Pn )−1 . Therefore Since en k ≤ k(I − T Pn )−1 k · kT P (n) uk, ku − u n ≥ n0 . (5) kT P (n) uk ≤ kT P (n) k kP (n) uk, en k compared to ku − un k can be increased by the rate of convergence ku − u means of a good estimate kT P (n) k. In the present paper we consider in the weighted space a singular integral equation of the form Su + Ku = f, (6) R1 where Su ≡ π1 −1 u(t)dt t−x , −1 < x < 1, is a singular integral operator, and R 1 1 Ku ≡ π −1 K(x, t)u(t)dt is an integral operator of the Fredholm type (see [3], [4]). For the singular integral equation (6) we may have three index values: κ = −1, 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 + K)u = f, where a and b are real numbers, a2 + b2 > 0. APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 459 § 1. The Bubnov–Galerkin Method with One Iteration 1.1. Index κ = 1. We take a weighted space L2,ρ [−1, 1], where the weight R1 ρ = ρ1 = (1 − x2 )1/2 . The scalar product [u, v] = −1 ρ1 uv dx. For the index κ = 1 we have the additional condition Z1 u(t)dt = p, (7) −1 where p is a given real number. The operator S is bounded in L2,ρ (see [4]). We require of the kernel K(x, t) that the operator K be completely continuous in L2,ρ . The homogeneous equation Su = 0 in the space L2,ρ has a nontrivial solution u = (1 − x2 )−1/2 . In the space L2,ρ the following two systems of functions are orthonormalized and complete: (1) ϕk (x) ≡ (1 − x2 )−1/2 Tbk (x), k = 0, 1, . . . , Tb0 = 1 1/2 π T0 , Tbk+1 = 2 1/2 π Tk+1 , k = 0, 1, . . . , where Tk , k = 0, 1, . . . , are the Chebyshev polynomials of first kind, and 1/2 (2) Uk (x), k = 0, 1, . . . , ψk+1 (x) ≡ π2 where Uk , k = 0, 1, . . . , are the Chebyshev polynomials of second kind. Denote Φ ≡ u − pπ −1 (1 − x2 )−1/2 . Then problem (6)–(7) can be written in the form (see [5]) (2) (8) Φ(t)dt = 0, (9) SΦ + KΦ = f1 , Φ ∈ L2,ρ , f1 ∈ L2,ρ , Z1 −1 (1) (2) where f1 ≡ f − pπ −1 K(1 − t2 )−1/2 , L2,ρ = L2,ρ ⊕ L2,ρ is the orthogonal (1) decomposition, L2,ρ is the linear span of the function ϕ0 = (1−x2 )−1/2 , and (2) L2,ρ is its orthogonal complement. In the sequel, under S we shall mean its (2) (2) (2) restriction on L2,ρ . Then S(L2,ρ ) = L2,ρ and S −1 (L2,ρ ) = L2,ρ . The relations Sϕk = ψk , k = 1, 2, . . . , (10) 460 A. JISHKARIANI∗ AND G. KHVEDELIDZE (see [6]) are valid. An approximate solution of equation (8) is sought in the form n X Φn = a k ϕk . k=1 Owing to (10), the algebraic system composed of the conditions [SΦn + KΦn − f1 , ψi ] = 0, i = 1, 2, . . . , n, yields ai + n X ak [Kϕk , ψi ] = [f1 , ψi ], i = 1, 2, . . . , n. (11) k=1 It is known [5] that if there exists the inverse operator (I + KS −1 )−1 mapping L2,ρ onto itself, then for sufficiently large n the algebraic system (11) has a unique solution (a1 , a2 , . . . , an ), and the sequence of approximate solutions un = Φn + pπ −1 (1 − x2 )−1/2 converges to the exact solution u in the metric of the space L2,ρ . Similar results are valid for κ = −1, 0. With the help of the orthoprojector Pn which maps L2,ρ onto the linear span of the functions ψ1 , . . . , ψn we can rewrite the algebraic system (11) as wn + Pn KS −1 wn = Pn f1 , wn ≡ SΦn = n X ak ψ k . (12) k=1 From the initial equation (8) we have w + KS −1 w = f1 , w ∈ L2,ρ , f1 ∈ L2,ρ , w ≡ SΦ. (13) Equation (12) is the Bubnov–Galerkin approximation for (13). As in [2], let us introduce the iteration w en = −KS −1 wn + f1 = −KΦn + f1 . where w en satisfies the equation For n ≥ n0 we obtain en + f1 . w en = −KS −1 Pn w kw − w en k ≤ CkKS −1 P (n) k · kP (n) wk. (14) APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 461 e n ≡ S −1 w e n , it is necessary to calculate the integral Let Φ en . To find Φ S −1 We have (1 − t2 )−1/2 w en = π Z1 −1 (1 − x2 )1/2 w en (x)dx . t−x e n k = kS −1 (w − w en )k = kw − w en k ≤ en k = kΦ − Φ ku − u ≤ CkKS −1 P (n) k · kP (n) wk, e n + pπ −1 (1 − x2 )−1/2 . en = Φ where u Theorem 1. If there exists the inverse operator (I + KS −1 )−1 mapping L2,ρ onto itself, and the conditions w(n) ∈ LipM α, 0 < α ≤ 1, and K (l) (x, t) ∈ LipM α1 , 0 < α1 ≤ 1, ∀x ∈ [−1, 1], are fulfilled for the derivatives, then the estimate en k = O(n−(m+α)−(l+α1 ) ) ku − u is valid. Proof. We have Pn w = n X [w, ψk ]ψk . k=1 (n) kP = ≤ Z1 −1 wk = Z1 −1 2 Z1 −1 (1 − x2 )1/2 (w − Pn w)2 dx = (1 − x2 )1/2 (w − n X k=1 [w, ψk ]ψk )2 ds ≤ (1 − x2 )1/2 (w − Pn−1 )2 dx ≤ π kw − Pn−1 k2C , 2 where Pn−1 is the polynomial of the best uniform approximation. By Jackson’s theorem [7] we have kw − Pn−1 kC ≤ C(w) , n > 1, (n − 1)m+α with a constant C(w) depending on w and its derivatives, i.e., kP (n) wk = O(n−(m+α) ). A. JISHKARIANI∗ AND G. KHVEDELIDZE 462 Furthermore, kKS −1 P (n) vk2 = kKS −1 ∞ X [v, ψk ]ψk k2 = kK k=n+1 ∞ X [v, ψk ]ϕk k2 = k=n+1 ∞ 1 2 X = 2 [v, ψk ] K(x, t), ϕk (t) ≤ π k=n+1 ∞ o1/2 n X n ∞ 2 o1/2 2 1 X ≤ 2 [v, ψk ]2 K(x, t), ϕk (t) k ≤ × π k=n+1 kvk2 π2 ≤ kvk2 π2 = Z1 −1 kvk2 = 2 π Z1 −1 k=n+1 Z1 −1 (1 − x2 )1/2 (1 − x2 )1/2 k=n+1 ∞ X k=n+1 = Z1 −1 ≤ Z1 ∞ X K(x, t), Tbk (t) L −1 (1 − t2 )1/2 2 bk (t) T −1 ∞ X K(x, t), Tbk (t) L 2,ρ Z1 −1 dx, = 2 bk (t) dt = T −1 n X K(x, t), Tbk (t) L (1 − t2 )1/2 K(x, t) − k=0 L2,ρ−1 L2,ρ−1 r,ρ k=n+1 2 bk (t) T −1 r,ρ k=n+1 k=n+1 = 2 K(x, t), (1 − t2 )−1/2 Tbk (t) dx = ∞ X K(x, t), Tbk (t) L 2 1/2 (1 − x ) ∞ X 2 dx = K(x, t), ϕk (t) 2,ρ Tbk (t) −1 2 dt ≤ 2 2 (1 − t2 )1/2 K(x, t) − Pn (x, t) dt ≤ π Ent K(x, t) , where x is a parameter, Pn (x, t) is the polynomial of the best uniform approximation with respect to t, and Ent (K(x, t)) the corresponding deviation K(x, t) − Pn (x, t) ≤ Ent K(x, t) , −1 < x, t < 1. (l) If Kt (x, t) ∈ LipM1 α1 , 0 < α1 ≤ 1 ∀x ∈ [−1, 1], and is continuous with respect to x in [−1, 1], then (see [8, Ch.XIV, §4]) Ent K(x, t) = O n−(l+α1 ) . APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 463 Furthermore, 2 KS −1 P (n) v 2 ≤ kvk 2 π Z1 −1 2 2 (1 − x2 )1/2 π Ent (K(x, t)) dx = 2 kvk π t π En (K(x, t)) . 2 π 2 −(l+α ) (n) −1 1 Thus KS P = O n . Finally, we get = ku − u en k ≤ kKS −1 P (n) wk ≤ ≤ kKS −1 P (n) k · kP (n) wk = O(n−(m+α)−(l+α1 ) ). For the approximate solution un we have ku − un k ≤ CkP (n) wk = O(n−(m+α) ), en performed over un when l = m, α1 = α, we obtain while for one iteration u the estimate ku − u en k = O(n−2(m+α) ). 1.2. Index κ = −1. The operator S is bounded in the weighted space L2,ρ [−1, 1], where ρ = ρ2 = (1 − x2 )−1/2 (see [4]). We require of the kernel K(x, t) that the operator K be completely continuous in L2,ρ . The equation Su = 0 in L2,ρ has the zero solution only, while the equation S ∗ u = 0 has the nonzero solution u = 1. If in the weighted space L2,ρ the equation Su + Ku = f has a solution u, then [Ku − f, 1] = 0. This condition will be fulfilled if K(LL2,ρ ) ⊥ 1 and [f, 1] = 0, which can be achieved by specific transform [9]. In the space L2,ρ the following two systems of functions are complete and orthonormal: 2 1/2 ϕk+1 (x) ≡ (1) Uk (x), k = 0, 1, . . . , π where Uk , k = 0, 1, . . . are Chebyshev polynomials of second kind, and 2 ψk+1 (x) ≡ − (2) Tk+1 (x), k = 0, 1, . . . , π where Tk+1 , k = 0, 1, . . . are Chebyshev polynomials of first kind. Relations Sϕk = ψk , k = 1, 2, . . . (see [6]) are valid. An approximate solution is again sought in the form un = n X k=1 a k ϕk . 464 A. JISHKARIANI∗ AND G. KHVEDELIDZE The Bubnov–Galerkin method results in the algebraic system ai + n X ak [Kϕk , ψi ] = [f, ψi ], i = 1, 2, . . . , n. (15) k=1 Pn Denote wn ≡ Sun = k=1 ak ψk . Then using the orthoprojector Pn mapping L2,ρ onto the linear span of the functions ψ1 , ψ2 , . . . , ψn the algebraic system (15) can be rewritten as wn + Pn KS −1 wn = Pn f. (16) Let the approximate solution wn be found. Taking one iteration en = −KS −1 wn + f = −Kun + f, w en = we find that u en = S −1 w (1 − t2 )1/2 R 1 w en (x)dx (1 − x2 )−1/2 . −1 π t−x Theorem 2. If there exists the inverse operator (I + KS −1 )−1 mapping (l) onto itself, and the conditions w(m) ∈ LipM α, 0 < α ≤ 1, Kt (x, t) ∈ LipM α1 , 0 < α1 ≤ 1, ∀x ∈ [−1, 1], are fulfilled for the derivatives, then the estimate ku − u en k = O(n−(m+α)−(l+α1 ) ) (2) L2,ρ is valid. This theorem as well as Theorem 3 which will be formulated in the next subsection can be proved similarly to Theorem 1. 1.3. Index κ = 0. Here we may have two cases: (1) α = − 21 , β = 12 and (2) α = 21 , β = − 12 . Let us consider the first case. The second one is considered analogously. The operator S is bounded in the weighted space L2,ρ [−1, 1] with the weight ρ = ρ3 = (1 − x)1/2 (1 + x)−1/2 [4]. We require of the kernel K(x, t) that the operator K be completely continuous in L2,ρ [−1, 1]. In the space L2,ρ the equations Su = 0 and S ∗ u = 0 have only trivial solution u = 0, S(L2,ρ ) = L2,ρ , where S is the unitary operator. We have the equation Su + Ku = f, u ∈ L2,ρ , f ∈ L2,ρ . (17) In L2,ρ we take two complete and orthonormal systems of functions (see [10]): (1/2,−1/2) (1) ϕk ≡ ck (1 − x)1/2 (1 + x)−1/2 Pk , k = 0, 1, . . . , c0 = π, (−1/2,1/2) −1/2 ck = (hk ) , k = 1, 2, . . . , APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS (−1/2,1/2) hk (1/2,−1/2) where Pk (1/2,−1/2) = hk 465 2Γ(k + 1/2)Γ(k + 3/2) , (2k + 1)(k!)2 = , k = 0, 1, . . . , are the Jacobi polynomials; (−1/2,1/2) (2) ψk ≡ −ck Pk The relations . Sϕk = ψk , k = 0, 1, . . . (18) (see [6]) are valid. We seek an approximate solution of equation (17) in the form un = n X ak ϕk . k=1 With regard to (18) the Bubnov–Galerkin method [Sun + Kun − f, ψi ] = 0, i = 0, 1, . . . , n, yields the algebraic system ai + n X ak [Kϕk , ψi ] = [f, ψi ] i = 0, 1, . . . , n, (19) k=0 which, by means of the orthoprojector Pn mapping L2,ρ onto the linear span of the functions ψ0 , ψ1 , . . . , ψn , can be written in the form wn + Pn KS −1 wn = Pn f, wn ≡ Sun = n X ak ψk . (20) k=0 Let the approximate solution wn be found. Taking one iteration w en = −KS −1 wn + f = −Kun + f, we find u en = S Then −1 (1 + t)1/2 (1 − t)−1/2 w en = π Z1 −1 (1 − x)1/2 (1 + x)−1/2 w en (x)dx . t−x ku − u en k = kS −1 (w − w en k ≤ CkKS −1 P (n) k · kP (n) wk. en )k = kw − w A. JISHKARIANI∗ AND G. KHVEDELIDZE 466 Theorem 3. If there exists the inverse operator (I + KS −1 )−1 mapping (l) L2,ρ onto itself, and the conditions w(m) ∈ LipM α, 0 < α ≤ 1, Kt (x, t) ∈ LipM α1 , 0 < α1 ≤ 1, ∀x ∈ [−1, 1], are fulfilled for the derivatives, then the estimate en k = O(n−(m+α)−(l+α1 ) ) ku − u 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. 2.1. Index κ = 1. As in Subsection 1.1 we seek an approximate solution of problem (8)–(9) in the form Φn = n X ak ϕk . k=1 By the collocation method the residual SΦn +KΦn −f1 (f1 is introduced above by (8)) at discrete points will be equated to zero, [SΦn + KΦn − f1 ]xj = 0, j = 1, 2, . . . , n. This, owing to (10), results in the algebraic system n X k=1 ak ψk (xj ) + n X ak (Kϕk )(xj ) = f1 (xj ), j = 1, 2, . . . , n. (21) k=1 As is known [11], if there exists the operator (I + KS −1 )−1 mapping L2,ρ onto itself, and as the collocation nodes are taken the roots of the Chebyshev polynomials of the second kind Un , then for sufficiently large n the algebraic system (21) has a unique solution, and the process converges in the space L2,ρ . Analogous results are valid for the index κ = −1, 0. Let us take the space of continuous functions C[−1, 1]. Let Πn be the projector defined by the Lagrange interpolation polynomial Πn v = Ln v. With the help of this projector the algebraic system (21) can be rewritten in the form wn + Πn−1 KS −1 wn = Πn−1 f1 , (n) wn ∈ L2,ρ , (n) where L2,ρ is the linear span of functions ψ1 , ψ2 , . . . , ψn (ψk is a polynomial of degree k − 1). Let the approximate solution wn be found. As in the Bubnov–Galerkin method we take one iteration w en = −KS −1 wn + f1 = −KΦn + f1 . APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS where w en satisfies the equation en = f1 . w en + KS −1 Πn−1 w 467 (22) By means of (13) and (22) we obtain w−w en + KS −1 w − KS −1 Πn−1 w en + KS −1 w en − KS −1 w en = 0, en ) = −KS −1 Π(n−1) w en , (I + KS −1 )(w − w Π(n−1) ≡ I − Πn−1 , w−w en = −(I + KS −1 )−1 KS −1 Π(n−1) (−Kφn + f1 ), kw − w en k ≤ C kKS −1 Π(n−1) Kk · kΦn k + kKS −1 Π(n−1) f1 k . For sufficiently large n we have [11] wn = (I + Πn−1 KS −1 )−1 Πn−1 f1 , kΦn k = kS −1 wn k ≤ CkΠn−1 f1 k, ∀f1 ∈ C[−1, 1], Πn−1 f1 → f1 , for n → ∞, i.e., kΦn k are uniformly bounded owing to the Erdös–Turan [10] and Banach–Steinhaus [8] theorems. Therefore kw − w en k ≤ C kKS −1 Π(n−1) Kk + kKS −1 Π(n−1) f1 k . We find that 1 e n ≡ S −1 w Φ en = (1 − t2 )−1/2 π Z1 −1 Then (1 − x2 )1/2 w en (x)dx . t−x e n k = kS −1 (w − w en k = kΦ − Φ en )k = kw − w ku − u en k ≤ −1 (n−1) −1 (n−1) ≤ C kKS Π Kk + kKS Π f1 k , e n + pπ −1 (1 − x2 )−1/2 . en ≡ Φ where u It is known [12] that (23) Π(n−1) v(t) = ω(t)δ (n) v(t), ∀v ∈ C[−1, 1], Qn where ω(t) ≡ i=1 (t − ti ) 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 Un are taken as interpolation nodes, then (see [13]) Π(n−1) v(t) = π 1/2 U bn (t) 2 2n δ (n) v(t) Denote K1 (x, t) ≡ (1 − t2 )−1/2 K(x, t). bn (t) = U 2 1/2 π Un (t) . 468 A. JISHKARIANI∗ AND G. KHVEDELIDZE Theorem 4. If there exists the inverse operator (I + KS −1 )−1 mapping L2,ρ , ρ = ρ1 , onto itself, the roots of the second kind Chebyshev polynomial Un are taken as collocation nodes, (SK1 (x, t))m ∈ LPM α, 0 < α ≤ 1, ∀x ∈ [−1, 1], and the divided differences of all orders of the functions f1 (x) and K1 (x, t) with respect to x are uniformly bounded ∀t ∈ [−1, 1], then the estimate 1 1 en k = O n · ku − u 2 (n − 1)m+α is valid. Proof. Let us estimate the norms on the right-hand side of inequality (23). We have 1 kKS −1 Π(n−1) KvkL2,ρ = K(x, t), π 1 −1 (n−1) (K(x, τ ), v(τ )) = 2 (1 − t2 )1/2 (1 − t2 )−1/2 K(x, t), S Π π 2 1/2 −1 (n−1) (1 − τ ) (1 − τ 2 )−1/2 (K(x, τ ), v(τ )) = S Π 1 = 2 SK1 (x, t), Π(n−1) [K1 (x, τ ), v(τ )] = π 1 = 2 SK1 (x, t), [Π(n−1) K1 (x, τ ), v(τ )] = π 1 = 2 [SK1 (x, t), Π(n−1) K1 (x, τ )], v(τ )] = π bn (t) 1 π 1/2 U = 2 δ (n) K1 (x, τ ), v(τ ) = [SK1 (x, t), n π 2 2 1 π 1/2 1 (n) (n−1) b P U (x, τ )SK (x, t), (t)]v(τ δ K ) = 2 = n 1 1 π 2 2n 1 π 1/2 1 (n−1) (n) bn (t)]v(τ ) δ K1 (t, τ )SK1 (x, t), U = 2 [P ≤ n π 2 2 1 π 1/2 1 bn (t)] × kv(τ )k P (n−1) δ (n) K1 (t, τ )SK1 (x, t), U ≤ 2 ≤ π 2 2n kvk π 1/2 1 bn (t)k ≤ 2 P (n−1) δ (n) K1 (t, τ )SK1 (x, t) × kU ≤ n π 2 2 kvk π 1/2 1 P (n−1) δ (n) K1 (t, τ )SK1 (x, t) ≤ 2 . π 2 2n Under the conditions of the theorem (δ (n) K1 (t, τ )SK1 (x, t))(m) ∈ LipM α, 0 < α ≤ 1, ∀x, τ ∈ [−1, 1]. By Jackson’s theorem (see [7], [8]) c0 2m+α M (n−1) (n) , δ K1 (t, τ )SK1 (x, t) ≤ m P (n − 1)m+α APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS 6m mm m+1 α where c0m ≡ 12 m! Therefore we obtain 2 469 . kKS −1 Π(n−1) Kk = O 1 1 . · 2n (n − 1)m+α (24) Furthermore, 1 k(K(x, t), S −1 Π(n−1) f1 )k = π bn (t) U 1 π 1/2 = [SK1 (x, t), n δ (n) f1 = π 2 2 1 π 1/2 1 (n) (n−1) b = t), f SK (x, P (t)] U = [δ 1 1 n π 2 2n 1 π 1/2 1 (n−1) (n) bn (t)] = δ f SK (x, t), U [P ≤ 1 1 π 2 2n 1 π 1/2 1 (n−1) (n) b δ f SK U (t) (x, t) × ≤ P ≤ 1 1 n π 2 2n 1 π 1/2 1 ≤ P (n−1) δ (n) f1 SK1 (x, t) . n π 2 2 kKS −1 Π(n−1) f1 kL2,ρ = Under the conditions of the theorem (m) δ (n) f1 SK1 (x, t) ∈ LipM α, 0 < α ≤ 1, ∀x ∈ [−1, 1]. Therefore we have kP (n−1) δ (n) f1 SK1 (x, t)k ≤ 0 Cm 2m+α M (n−1)m+α , 1 1 . KS −1 Π(n−1) f1 = O n · 2 (n − 1)m+α (25) With the help of the obtained estimates (24) and (25), from (23) we finally get 1 1 . ku − u en k = O n · 2 (n − 1)m+α Remark 1. Under the conditions of the theorem we obtain for the approximate solution un that π 1/2 U bn (t) (n) n δ w ≤ 2 2 1 Z o1/2 C1 n C1 b (n) bn2 (t)(δ (n) w)2 dt ≤ w(t)k = n (1 − t2 )1/2 U ≤ n kU n (t)δ 2 2 ku − un k ≤ CkΠ(n−1) wk = C −1 A. JISHKARIANI∗ AND G. KHVEDELIDZE 470 n C1 ≤ n kδ (n) wkC 2 Z1 −1 o1/2 bn2 (t)dt (1 − t2 )1/2 U = C1 bn (t)k = C1 kδ (n) wkC = C1 kδ (n) (f1 − KΦ)kC = = n kδ (n) wkC × kU 2 2n 2n C1 1 = n δ (n) (f1 − K1 (x, t), Φ(t) ≤ 2 π C C1 (n) 1 (n) ≤ n kδ f1 kC + δ K1 (x, t), Φ(t) ≤ 2 π C 1 C1 (n) (n) ≤ n kδ f1 kC + δ K1 (x, t) × Φ(t) ≤ 2 π C C2 1 C1 (n) (n) ≤ n kδ f1 kC + δ K1 (x, t)C ≤ n . 2 π 2 2.2. Index κ = −1. Introduce the subspace C0 [−1, 1] ⊂ C[−1, 1]; v ∈ C0 [−1, 1] if [v, 1] = 0. C (n) [−1, 1] ⊂ C[−1, 1] is a linear span of polynomials ψ0 , ψ1 , . . . , ψn . The projector can be determined as follows [11]: (n) Πn v = Ln v − a0 ψ0 , (n) where Ln v ∈ C (n) [−1, 1] is the Lagrange polynomial and a0 is the coeffi(n) cient of the Fourier series expansion a0 ≡ [Ln v, ψ0 ]. Again, as in Subsection 1.2, we seek an approximate solution in the form un = n X ak ϕk . k=1 We compose the algebraic system by the condition Πn (Sun + Kun − f ) = 0, which results in a 0 ψ0 + n X k=1 ak ψk (xj ) + n X ak (Kϕk )(xj ) = f (xj ), j = 0, 1, . . . , n. k=1 Using the projector, we can rewrite this algebraic system as (n) wn + Πn KS −1 wn = Πn f, wn ∈ L2,ρ , (n) where L2,ρ is the linear span of the system of functions ψ0 , ψ1 , . . . , ψn . Let the approximate solution wn be found. Taking one iteration w en = −KS −1 wn + f = −Kun + f, APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS we find u en = S −1 (1 − t2 )1/2 w en = π 2 1/2 Denote K1 (x, t) ≡ (1 − t ) Z1 −1 (1 − x2 )−1/2 471 w en (x)dx . t−x K(x, t). Theorem 5. If there exists the inverse operator (I + KS −1 )−1 mapping ρ = ρ2 , onto itself, the roots of the Chebyshev polynomial of the first kind Tn+1 are taken as collocation nodes, (SK1 (x, t))m ∈ LipM α, 0 < α ≤ 1, ∀x ∈ [−1, 1], and the divided differences of all orders of the functions f (x) and K1 (x, t) with respect to x are uniformly bounded ∀t ∈ [−1, 1], then the estimate 1 1 ku − u en k = O n · m+α 2 n is valid. (r) L2,ρ , 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 ku − un k = O n . 2 2.3. Index κ = 0. As in Subsection 1.3, an approximate solution is again sought in the form n X un = ak ϕk . k=1 Equating the residuals to zero at the points x1 , . . . , x, we obtain Sun + Kun − f x = 0 j = 0, 1, . . . , n, j which yields the algebraic system n X k=0 ak ψk (xj ) + n X ak (Kϕk )(xj ) = f (xj ), j = 0, 1, . . . , n. k=0 Just as for the index κ = 1 we can rewrite this system as wn + Πn KS −1 wn = Πn f, (n) wn ∈ L2,ρ , (n) where L2,ρ is the linear span of functions ψ0 , . . . , ψn . Let the approximate solution wn be found. Taking one iteration w en = −KS −1 wn + f = −Kun + f A. JISHKARIANI∗ AND G. KHVEDELIDZE 472 we find u en = S −1 (1 + t)1/2 (1 − t)−1/2 w en = π Z1 −1 (1 − x)1/2 (1 + x)−1/2 en (x)dx w . t−x Denote K1 (x, t) ≡ (1 + t)1/2 (1 − t)−1/2 K(x, t). Theorem 6. If there exists the inverse operator (I + KS −1 )−1 mapping ( 1 ,− 1 ) 2 2 are taken L2,ρ , ρ = ρ3 , onto itself, the roots of the Jacobi polynomial Pn+1 (m) ∈ L : pM α, 0 < α ≤ 1, ∀x ∈ [−1, 1], as collocation nodes, (SK1 (x, t)) and the divided differences of all orders of the functions f (x) and K1 (x, t) with respect to x are uniformly bounded ∀t ∈ [−1, 1], then the estimate 1 1 ku − u en k = O n · m+α 2 n is valid. Remark 3. Under the conditions of the theorem 1 ku − un k = O n . 2 Remark 4. If we require only that w(m) ∈ LipM α, 0 < α ≤ 1, then for the collocation method for all values of the index κ = 1, 0, −1 we obtain the same order of convergence ln n ku − un k = O m+α n for an approximate solution un in the respective weighted spaces. Indeed, for any v ∈ C[−1, 1] we have Π(n) v = Π(n) P (n) v, where Π(n) ≡ I −Πn , P (n) ≡ I −Pn , where Πn is the Lagrange interpolation operator, and Pn is the orthoprojector with respect to polynomials Tb0 , Tb1 , . . . , Tbn . If the nodes in the interpolation Lagrange polynomial are taken with respect to the weight, then Ln : C → L2,ρ are bounded by the Erdös–Turan theorem. Therefore (see [13]) ku − un kL2,ρ ≤ CkΠ(n) ukL2,ρ ≤ C1 kP (n) ukC = ln n = O m+α (Πn = Ln ). n As an example of the application of the above methods in the case of the index κ = −1, let us consider the equation 1 π Z1 −1 1 u(t)dt + t−x π Z1 −1 (x8 t8 + x7 t7 )u(t)dt = APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS = 473 2 1/2 3 x7 − 32x6 + 48x4 − 18x2 + 1 π 128 with the exact solution u(x) = ϕ6 (x) = 2 1/2 π (32x5 − 32x3 + 6x)(1 − x2 )1/2 , where {ϕk (x)}, k = 0, 1, . . . , is the orthonormal system of functions in L2,ρ2 . We find the fifth approximation u5 (x) = 5 X ak ϕk (x), ϕk (x) = k=1 2 1/2 π (1 − x2 )1/2 Uk−1 (x), where Uk−1 (x), k = 1, 2, . . . , are the Chebyshev polynomials of the second kind. Computations are carried out to within 10−7 . u5 (x) and u e5 (x) are calculated. In the case of the Bubnov–Galerkin method we have k∆u5 k = 1, 0001151, k∆u5 k ≈ 100, 01%, kuk k∆e u5 k = 0, 0151855, k∆e u5 k ≈ 1, 52% kuk for an absolute and a relative error, respectively, while in the case of the collocation method we obtain k∆u5 k = 1, 0002931, k∆u5 k ≈ 100, 03%, kuk k∆e u5 k = 0, 0159781, k∆e u5 k ≈ 1, 60%. kuk The result is the expected one for u5 (x), since in our example the function ϕ6 (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, 1969. 2. 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), 71–91. 3. N. I. Muskhelishvili, Singular integral equations. (Translation from the Russian) P. Nordhoff (Ltd.), Groningen, 1953. 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(1957). 5. 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, 1149–1161. 6. F. G. Tricomi, On the finite Hilbert transformation. Quart. J. Math., 2(1951), No. 7, 199–211. 7. I. P. Natanson, Constructive theory of functions. (Rusian) Gosizdat. Techniko-Theoret. Literatury, Moscow–Leningrad, 1949. 8. L. V. Kantorovich and G. P. Akilov, Functional analysis. (Russian) Nauka, Moscow, 1977. 9. 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), 41–49. 10. G. Szegö, Orthogonal polynomials. American Mathematical Society Colloquium Publications, New York XXIII(1959). 11. A. V. Dzhishkariani, To the solution of singular integral equations by collocation methods. (Russian) J. Vychislit. Matem. i Matem. Fiz. 21(1981), No. 2, 355–362. 12. I. S. Berezin and N. P. Zhitkov, Computation methods, vol. 1. (Russian) Fizmatgiz, Moscow, 1959. 13. P. K. Suetin, Classical orthogonal polynomials. (Russian) Nauka, Moscow, 1976. (Received 30.12.1993) Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi 380093 Republic of Georgia