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 {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.

458 A. JISHKARIANI^{∗} AND G. KHVEDELIDZE

Assume that we have found an approximate solution u_{n} of equation (2). 457 Take one iteration (see [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.

1072-947X/96/0900-0457$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

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

ũ_{n} − T P_{n}ũ_{n} = f.  (4)

From (1) and (4) we have

(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})^{−1}. Therefore

∥u − ũ_{n}∥ ≤ ∥(I − T P_{n})^{−1}∥ · ∥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 ≡ \frac{1}{π} ∫_{−1}^{1} \frac{u(t)dt}{t−x}, −1 < x < 1, is a singular integral operator, and Ku ≡ \frac{1}{π} ∫_{−1}^{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, a^{2} + b^{2} > 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 L_{2,ρ}[−1, 1], where the weight ρ = ρ_{1} = (1 − x^{2})^{1/2}. The scalar product [u, v] = ∫_{−1}^{1} ρ_{1}uv dx. For the index κ = 1 we have the additional condition

∫_{−1}^{1} u(t)dt = 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})^{−1/2}.

In the space L_{2,ρ} the following two systems of functions are orthonormalized and complete:

(1) φ_{k}(x) ≡ (1 − x^{2})^{−1/2}T̂_{k}(x), k = 0, 1, …,

T̂_{0} = (\frac{1}{π} )^{1/2}T_{0}, T̂_{k+1} = (\frac{2}{π} )^{1/2}T_{k+1}, k = 0, 1, …,

where T_{k}, k = 0, 1, …, are the Chebyshev polynomials of first kind, and

(2) ψ_{k+1}(x) ≡ (\frac{2}{π} )^{1/2}U_{k}(x), k = 0, 1, …, where U_{k}, k = 0, 1, …, are the Chebyshev polynomials of second kind.

Denote Φ ≡ u − pπ^{−1}(1 − x^{2})^{−1/2}. Then problem (6)–(7) can be written in the form (see [5])

SΦ + KΦ = f_{1}, Φ ∈ L_{2,ρ}^{(2)}, f_{1} ∈ L_{2,ρ},  (8)
∫_{−1}^{1} Φ(t)dt = 0,  (9)

where f_{1} ≡ f − pπ^{−1}K(1 − t^{2})^{−1/2}, L_{2,ρ} = L_{2,ρ}^{(1)} ⊕ L_{2,ρ}^{(2)} is the orthogonal decomposition, L_{2,ρ}^{(1)} is the linear span of the function φ_{0} = (1−x^{2})^{−1/2}, and L_{2,ρ}^{(2)} is its orthogonal complement. In the sequel, under S we shall mean its restriction on L_{2,ρ}^{(2)}. Then S(L_{2,ρ}^{(2)}) = L_{2,ρ} and S^{−1}(L_{2,ρ}) = L_{2,ρ}^{(2)}.

The relations

Sφ_{k} = ψ_{k}, k = 1, 2, …,  (10)

(see [6]) are valid. An approximate solution of equation (8) is sought in the form

Φ_{n} = ∑_{k=1}^{n} a_{k}φ_{k}.

Owing to (10), the algebraic system composed of the conditions

[SΦ_{n} + KΦ_{n} − f_{1}, ψ_{i}] = 0, i = 1, 2, …, n,

yields

a_{i} + ∑_{k=1}^{n} 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^{−1})^{−1} 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} + pπ^{−1}(1 − x^{2})^{−1/2}

converges to the exact solution u in the metric of the space L_{2,ρ}. Similar results are valid for κ = −1, 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^{−1}w_{n} = P_{n}f_{1}, w_{n} ≡ SΦ_{n} = ∑_{k=1}^{n} a_{k}ψ_{k}.  (12)

From the initial equation (8) we have

w + KS^{−1}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

w̃_{n} = −KS^{−1}w_{n} + f_{1} = −KΦ_{n} + f_{1}.  (14)

where w̃_{n} satisfies the equation

w̃_{n} = −KS^{−1}P_{n}w̃_{n} + f_{1}.

For n ≥ n_{0} we obtain

∥w − w̃_{n}∥ ≤ C∥KS^{−1}P^{(n)}∥ · ∥P^{(n)}w∥.

Let Φ̃_{n} ≡ S^{−1}w̃_{n}. To find Φ̃_{n}, it is necessary to calculate the integral

S^{−1}w̃_{n} = \frac{(1 − t^{2})^{−1/2}}{π} ∫_{−1}^{1}(1 − x^{2})^{1/2} \frac{w̃_{n}(x)dx}{t − x}.

We have

∥u − ũ_{n}∥=∥Φ − Φ̃_{n}∥=∥S^{−1}(w − w̃_{n})∥=∥w − w̃_{n}∥≤
≤C∥KS^{−1}P^{(n)}∥ · ∥P^{(n)}w∥,

where ũ_{n} = Φ̃_{n} + pπ^{−1}(1 − x^{2})^{−1/2}.

Theorem 1. If there exists the inverse operator (I + KS^{−1})^{−1} 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, 1], are fulfilled for the derivatives, then the estimate

∥u − ũ_{n}∥ = O(n^{−(m+α)−(l+α_{1})})

is valid.

Proof. We have

P_{n}w = ∑_{k=1}^{n}[w, ψ_{k}]ψ_{k}.

∥P^{(n)}w∥^{2} = ∫_{−1}^{1}(1 − x^{2})^{1/2}(w − P_{n}w)^{2}dx =

= ∫_{−1}^{1}(1 − x^{2})^{1/2}(w − ∑_{k=1}^{n}[w, ψ_{k}]ψ_{k})^{2}ds ≤

≤ ∫_{−1}^{1}(1 − x^{2})^{1/2}(w − P_{n−1})^{2}dx ≤ \frac{π}{2} ∥w − P_{n−1}∥_{C}^{2},

where P_{n−1} is the polynomial of the best uniform approximation. By Jackson’s theorem [7] we have

∥w − P_{n−1}∥_{C} ≤ \frac{C(w)}{(n − 1)^{m+α}} , n > 1,

with a constant C(w) depending on w and its derivatives, i.e., ∥P^{(n)}w∥ = O(n^{−(m+α)}).

Furthermore,

∥KS^{−1}P^{(n)}v∥^{2} = ∥KS^{−1} ∑_{k=n+1}^{∞}[v, ψ_{k}]ψ_{k}∥^{2} = ∥K ∑_{k=n+1}^{∞}[v, ψ_{k}]φ_{k}∥^{2} =

= \frac{1}{π^{2}} ∥ ∑_{k=n+1}^{∞}[v, ψ_{k}](K(x, t), φ_{k}(t))∥^{2} ≤

≤ \frac{1}{π^{2}} ∥\{ ∑_{k=n+1}^{∞}[v, ψ_{k}]^{2}\}^{1/2} × \{ ∑_{k=n+1}^{∞} (K(x, t), φ_{k}(t))^{2}\}^{1/2}∥^{2} ≤

≤ \frac{∥v∥^{2}}{π^{2}} ∫_{−1}^{1}(1 − x^{2})^{1/2}( ∑_{k=n+1}^{∞} (K(x, t), φ_{k}(t))^{2})dx =

= \frac{∥v∥^{2}}{π^{2}} ∫_{−1}^{1}(1 − x^{2})^{1/2}( ∑_{k=n+1}^{∞} (K(x, t), (1 − t^{2})^{−1/2}T̂_{k}(t))^{2})dx =

= \frac{∥v∥^{2}}{π^{2}} ∫_{−1}^{1}(1 − x^{2})^{1/2}∥ ∑_{k=n+1}^{∞} [K(x, t), T̂_{k}(t)]_{L_{r,ρ^{−1}}} T̂_{k}(t)∥_{L_{2,ρ^{−1}}}^{2} dx,

∥ ∑_{k=n+1}^{∞} [K(x, t), T̂_{k}(t)]_{L_{r,ρ^{−1}}} T̂_{k}(t)∥_{L_{2,ρ^{−1}}}^{2} =

= ∫_{−1}^{1}(1 − t^{2})^{1/2}( ∑_{k=n+1}^{∞} [K(x, t), T̂_{k}(t)]_{L_{2,ρ^{−1}}} T̂_{k}(t))^{2}dt =

= ∫_{−1}^{1}(1 − t^{2})^{1/2}(K(x, t) − ∑_{k=0}^{n} [K(x, t), T̂_{k}(t)]_{L_{2,ρ^{−1}}} T̂_{k}(t))^{2}dt ≤

≤ ∫_{−1}^{1}(1 − t^{2})^{1/2}(K(x, t) − P_{n}(x, t))^{2}dt ≤ π(E_{n}^{t}(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 E_{n}^{t}(K(x, t)) the corresponding deviation

|K(x, t) − P_{n}(x, t)| ≤ E_{n}^{t}(K(x, t)), −1 < x, t < 1.

If K_{t}^{(l)}(x, t) ∈ Lip_{M_{1}} α_{1}, 0 < α_{1} ≤ 1 ∀x ∈ [−1, 1], and is continuous with respect to x in [−1, 1], then (see [8, Ch.XIV, §4])

E_{n}^{t}(K(x, t)) = O(n^{−(l+α_{1})}).

Furthermore,

∥KS^{−1}P^{(n)}v∥^{2} ≤ \frac{∥v∥^{2}}{π^{2}} ∫_{−1}^{1}(1 − x^{2})^{1/2}π(E_{n}^{t}(K(x, t)))^{2}dx =

= \frac{∥v∥^{2}}{π^{2}} π \frac{π}{2} (E_{n}^{t}(K(x, t)))^{2}.

Thus ∥KS^{−1}P^{(n)}∥ = O(n^{−(l+α_{1})}).

Finally, we get

∥u − ũ_{n}∥ ≤ ∥KS^{−1}P^{(n)}w∥ ≤
≤ ∥KS^{−1}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)}).

1.2. Index κ = −1. The operator S is bounded in the weighted space L_{2,ρ}[−1, 1], where ρ = ρ_{2} = (1 − x^{2})^{−1/2} (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_{L_{2,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:

(1) φ_{k+1}(x) ≡ (\frac{2}{π} )^{1/2}U_{k}(x), k = 0, 1, …,

where U_{k}, k = 0, 1, … are Chebyshev polynomials of second kind, and

(2) ψ_{k+1}(x) ≡ −(\frac{2}{π} )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} = ∑_{k=1}^{n} a_{k}φ_{k}.

The Bubnov–Galerkin method results in the algebraic system

a_{i} + ∑_{k=1}^{n} a_{k}[Kφ_{k}, ψ_{i}] = [f, ψ_{i}], i = 1, 2, …, n.  (15)

Denote w_{n} ≡ Su_{n} = ∑_{k=1}^{n} a_{k}ψ_{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

w_{n} + P_{n}KS^{−1}w_{n} = P_{n}f.  (16)

Let the approximate solution w_{n} be found.

Taking one iteration

w̃_{n} = −KS^{−1}w_{n} + f = −Ku_{n} + f,

we find that ũ_{n} = S^{−1}w̃_{n} = \frac{(1 − t^{2})^{1/2}}{π} ∫_{−1}^{1}(1 − x^{2})^{−1/2} \frac{w̃_{n}(x)dx}{t − x}.

Theorem 2. If there exists the inverse operator (I + KS^{−1})^{−1} mapping L_{2,ρ}^{(2)} onto itself, and the conditions w^{(m)} ∈ Lip_{M} α, 0 < α ≤ 1, K_{t}^{(l)}(x, t) ∈ Lip_{M} α_{1}, 0 < α_{1} ≤ 1, ∀x ∈ [−1, 1], are fulfilled for the derivatives, then the estimate

∥u − ũ_{n}∥ = O(n^{−(m+α)−(l+α_{1})})

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 = −\frac{1}{2}, β = \frac{1}{2} and (2) α = \frac{1}{2}, β = −\frac{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, 1] with the

weight ρ = ρ_{3} = (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 L_{2,ρ}[−1, 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)^{−1/2}P_{k}^{(1/2,−1/2)}, k = 0, 1, …,

c_{0} = π, c_{k} = (h_{k}^{(−1/2,1/2)})^{−1/2}, k = 1, 2, …,

h_{k}^{(−1/2,1/2)} = h_{k}^{(1/2,−1/2)} = \frac{2Γ(k + 1/2)Γ(k + 3/2)}{(2k + 1)(k!)^{2}},

where P_{k}^{(1/2,−1/2)}, k = 0, 1, …, are the Jacobi polynomials; (2) ψ_{k} ≡ −c_{k}P_{k}^{(−1/2,1/2)}. 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} = ∑_{k=1}^{n} a_{k}φ_{k}.

With regard to (18) the Bubnov–Galerkin method

[Su_{n} + Ku_{n} − f, ψ_{i}] = 0, i = 0, 1, …, n,

yields the algebraic system

a_{i} + ∑_{k=0}^{n} a_{k}[Kφ_{k}, ψ_{i}] = [f, ψ_{i}] i = 0, 1, …, n,  (19)

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^{−1}w_{n} = P_{n}f, w_{n} ≡ Su_{n} = ∑_{k=0}^{n} a_{k}ψ_{k}.  (20)

Let the approximate solution w_{n} be found.

Taking one iteration

w̃_{n} = −KS^{−1}w_{n} + f = −Ku_{n} + f,

we find

ũ_{n} = S^{−1}w̃_{n} = \frac{(1 + t)^{1/2}(1 − t)^{−1/2}}{π} ∫_{−1}^{1}(1 − x)^{1/2}(1 + x)^{−1/2} \frac{w̃_{n}(x)dx}{t − x}.

Then

∥u − ũ_{n}∥ = ∥S^{−1}(w − w̃_{n})∥ = ∥w − w̃_{n}∥ ≤ C∥KS^{−1}P^{(n)}∥ · ∥P^{(n)}w∥.

Theorem 3. If there exists the inverse operator (I + KS^{−1})^{−1} mapping L_{2,ρ} onto itself, and the conditions w^{(m)} ∈ Lip_{M} α, 0 < α ≤ 1, K_{t}^{(l)}(x, t) ∈ Lip_{M} α_{1}, 0 < α_{1} ≤ 1, ∀x ∈ [−1, 1], are fulfilled for the derivatives, then the estimate

∥u − ũ_{n}∥ = O(n^{−(m+α)−(l+α_{1})})

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

Φ_{n} = ∑_{k=1}^{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}]_{x_{j}} = 0, j = 1, 2, …, n.

This, owing to (10), results in the algebraic system

∑_{k=1}^{n} a_{k}ψ_{k}(x_{j}) + ∑_{k=1}^{n} 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^{−1})^{−1} 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 = −1, 0.

Let us take the space of continuous functions C[−1, 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−1}KS^{−1}w_{n} = Π_{n−1}f_{1}, w_{n} ∈ L_{2,ρ}^{(n)},

where L_{2,ρ}^{(n)} 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

w̃_{n} = −KS^{−1}w_{n} + f_{1} = −KΦ_{n} + f_{1}.

where w̃_{n} satisfies the equation

w̃_{n} + KS^{−1}Π_{n−1}w̃_{n} = f_{1}.  (22)

By means of (13) and (22) we obtain

w − w̃_{n} + KS^{−1}w − KS^{−1}Π_{n−1}w̃_{n} + KS^{−1}w̃_{n} − KS^{−1}w̃_{n} = 0,

(I + KS^{−1})(w − w̃_{n}) = −KS^{−1}Π^{(n−1)}w̃_{n}, Π^{(n−1)} ≡ I − Π_{n−1},

w − w̃_{n} = −(I + KS^{−1})^{−1}KS^{−1}Π^{(n−1)}(−Kϕ_{n} + f_{1}),
∥w − w̃_{n}∥ ≤ C(∥KS^{−1}Π^{(n−1)}K∥ · ∥Φ_{n}∥ + ∥KS^{−1}Π^{(n−1)}f_{1}∥).

For sufficiently large n we have [11]

w_{n} = (I + Π_{n−1}KS^{−1})^{−1}Π_{n−1}f_{1},
∥Φ_{n}∥ = ∥S^{−1}w_{n}∥ ≤ C∥Π_{n−1}f_{1}∥,
Π_{n−1}f_{1} → f_{1}, for n → ∞, ∀f_{1} ∈ C[−1, 1],

i.e., ∥Φ_{n}∥ are uniformly bounded owing to the Erdös–Turan [10] and Banach–Steinhaus [8] theorems. Therefore

∥w − w̃_{n}∥ ≤ C(∥KS^{−1}Π^{(n−1)}K∥ + ∥KS^{−1}Π^{(n−1)}f_{1}∥).

We find that

Φ̃_{n} ≡ S^{−1}w̃_{n} = \frac{1}{π} (1 − t^{2})^{−1/2} ∫_{−1}^{1} \frac{(1 − x^{2})^{1/2}w̃_{n}(x)dx}{t − x}.

Then

∥u − ũ_{n}∥ = ∥Φ − Φ̃_{n}∥ = ∥S^{−1}(w − w̃_{n})∥ = ∥w − w̃_{n}∥ ≤
≤ C(∥KS^{−1}Π^{(n−1)}K∥ + ∥KS^{−1}Π^{(n−1)}f_{1}∥),  (23)

where ũ_{n} ≡ Φ̃_{n} + pπ^{−1}(1 − x^{2})^{−1/2}.

It is known [12] that

Π^{(n−1)}v(t) = ω(t)δ^{(n)}v(t), ∀v ∈ C[−1, 1],

where ω(t) ≡ ∏_{i=1}^{n}(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])

Π^{(n−1)}v(t) = (\frac{π}{2} )^{1/2} \frac{Û_{n}(t)}{2^{n}}δ^{(n)}v(t) (Û_{n}(t) = (\frac{2}{π} )^{1/2}U_{n}(t)).

Denote K_{1}(x, t) ≡ (1 − t^{2})^{−1/2}K(x, t).

Theorem 4. If there exists the inverse operator (I + KS^{−1})^{−1} 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_{P_{M}} α, 0 < α ≤ 1, ∀x ∈ [−1, 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, 1], then the estimate

∥u − ũ_{n}∥ = O(\frac{1}{2^{n}} · \frac{1}{(n − 1)^{m+α}} )

is valid.

Proof. Let us estimate the norms on the right-hand side of inequality (23). We have

∥KS^{−1}Π^{(n−1)}Kv∥_{L_{2,ρ}} = ∥\frac{1}{π} (K(x, t),

S^{−1}Π^{(n−1)}(K(x, τ), v(τ)))∥ = \frac{1}{π^{2}} ∥((1 − t^{2})^{1/2}(1 − t^{2})^{−1/2}K(x, t),

S^{−1}Π^{(n−1)}(1 − τ^{2})^{1/2}(1 − τ^{2})^{−1/2}(K(x, τ), v(τ)))∥ =

= \frac{1}{π^{2}} ∥[SK_{1}(x, t), Π^{(n−1)}[K_{1}(x, τ), v(τ)]]∥ =

= \frac{1}{π^{2}} ∥[SK_{1}(x, t), [Π^{(n−1)}K_{1}(x, τ), v(τ)]]∥ =

= \frac{1}{π^{2}} ∥[[SK_{1}(x, t), Π^{(n−1)}K_{1}(x, τ)], v(τ)]]∥ =

= \frac{1}{π^{2}} (\frac{π}{2} )^{1/2}∥[[SK_{1}(x, t), \frac{Û_{n}(t)}{2^{n}}δ^{(n)}K_{1}(x, τ), v(τ)]∥ =

= \frac{1}{π^{2}} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥[δ^{(n)}K_{1}(x, τ)SK_{1}(x, t), P^{(n−1)}Û_{n}(t)]v(τ)]∥ =

= \frac{1}{π^{2}} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥[[P^{(n−1)}δ^{(n)}K_{1}(t, τ)SK_{1}(x, t), Û_{n}(t)]v(τ)]∥ ≤

≤ \frac{1}{π^{2}} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥ ∥[P^{(n−1)}δ^{(n)}K_{1}(t, τ)SK_{1}(x, t), Û_{n}(t)]∥ × ∥v(τ)∥ ∥ ≤

≤ \frac{∥v∥}{π^{2}}(\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥ ∥ ∥P^{(n−1)}δ^{(n)}K_{1}(t, τ)SK_{1}(x, t)∥ × ∥Û_{n}(t)∥ ∥ ∥ ≤

≤ \frac{∥v∥}{π^{2}}(\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥ ∥ ∥P^{(n−1)}δ^{(n)}K_{1}(t, τ)SK_{1}(x, t)∥∥ ∥.

Under the conditions of the theorem

(δ^{(n)}K_{1}(t, τ)SK_{1}(x, t))^{(m)} ∈ Lip_{M} α, 0 < α ≤ 1, ∀x, τ ∈ [−1, 1].

By Jackson’s theorem (see [7], [8])

∥P^{(n−1)}δ^{(n)}K_{1}(t, τ)SK_{1}(x, t)∥ ≤ \frac{c′_{m}2^{m+α}M}{(n − 1)^{m+α}} ,

where c′_{m} ≡ 12\frac{6^{m}m^{m}}{m!} (\frac{m+1}{2})^{α}.

Therefore we obtain

∥KS^{−1}Π^{(n−1)}K∥ = O(\frac{1}{2^{n}} · \frac{1}{(n − 1)^{m+α}} ).  (24)

Furthermore,

∥KS^{−1}Π^{(n−1)}f_{1}∥_{L_{2,ρ}} = \frac{1}{π} ∥(K(x, t), S^{−1}Π^{(n−1)}f_{1})∥ =

= \frac{1}{π} (\frac{π}{2} )^{1/2}∥[SK_{1}(x, t), \frac{Û_{n}(t)}{2^{n}}δ^{(n)}f_{1}∥ =

= \frac{1}{π} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥[δ^{(n)}f_{1}SK_{1}(x, t), P^{(n−1)}Û_{n}(t)]∥ =

= \frac{1}{π} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥[P^{(n−1)}δ^{(n)}f_{1}SK_{1}(x, t), Û_{n}(t)]∥ ≤

≤ \frac{1}{π} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥ ∥P^{(n−1)}δ^{(n)}f_{1}SK_{1}(x, t)∥ × ∥Û_{n}(t)∥ ∥ ≤

≤ \frac{1}{π} (\frac{π}{2} )^{1/2} \frac{1}{2^{n}} ∥ ∥P^{(n−1)}δ^{(n)}f_{1}SK_{1}(x, t)∥ ∥.

Under the conditions of the theorem

(δ^{(n)}f_{1}SK_{1}(x, t))^{(m)} ∈ Lip_{M} α, 0 < α ≤ 1, ∀x ∈ [−1, 1].

Therefore we have ∥P^{(n−1)}δ^{(n)}f_{1}SK_{1}(x, t)∥ ≤ \frac{C′_{m}2^{m+α}M}{(n−1)^{m+α}},

∥KS^{−1}Π^{(n−1)}f_{1}∥ = O(\frac{1}{2^{n}} · \frac{1}{(n − 1)^{m+α}} ).  (25)

With the help of the obtained estimates (24) and (25), from (23) we finally get

∥u − ũ_{n}∥ = O(\frac{1}{2^{n}} · \frac{1}{(n − 1)^{m+α}} ).

Remark 1. Under the conditions of the theorem we obtain for the approximate solution u_{n} that

∥u − u_{n}∥ ≤ C∥Π^{(n−1)}w∥ = C(\frac{π}{2} )^{1/2}∥\frac{Û_{n}(t)}{2^{n}}δ^{(n)}w∥ ≤

≤ \frac{C_{1}}{2^{n}} ∥Û_{n}(t)δ^{(n)}w(t)∥ = \frac{C_{1}}{2^{n}} \{ ∫_{−1}^{1}(1 − t^{2})^{1/2}Û_{n}^{2}(t)(δ^{(n)}w)^{2}dt\}^{1/2} ≤

≤ \frac{C_{1}}{2^{n}} ∥δ^{(n)}w∥_{C}\{ ∫_{−1}^{1}(1 − t^{2})^{1/2}Û_{n}^{2}(t)dt\}^{1/2} =

= \frac{C_{1}}{2^{n}} ∥δ^{(n)}w∥_{C} × ∥Û_{n}(t)∥ = \frac{C_{1}}{2^{n}} ∥δ^{(n)}w∥_{C} = \frac{C_{1}}{2^{n}} ∥δ^{(n)}(f_{1} − KΦ)∥_{C} =

= \frac{C_{1}}{2^{n}} ∥δ^{(n)}(f_{1} − \frac{1}{π} [K_{1}(x, t), Φ(t)]∥_{C} ≤

≤ \frac{C_{1}}{2^{n}} (∥δ^{(n)}f_{1}∥_{C} + \frac{1}{π} ∥[δ^{(n)}K_{1}(x, t), Φ(t)]∥_{C} ≤

≤ \frac{C_{1}}{2^{n}} (∥δ^{(n)}f_{1}∥_{C} + \frac{1}{π} ∥ ∥δ^{(n)}K_{1}(x, t)∥ × ∥Φ(t)∥ ∥_{C} ≤

≤ \frac{C_{1}}{2^{n}} (∥δ^{(n)}f_{1}∥_{C} + \frac{1}{π} ∥ ∥δ^{(n)}K_{1}(x, t)∥_{C}) ≤ \frac{C_{2}}{2^{n}} .

2.2. Index κ = −1. Introduce the subspace C_{0}[−1, 1] ⊂ C[−1, 1]; v ∈ C_{0}[−1, 1] if [v, 1] = 0. C^{(n)}[−1, 1] ⊂ C[−1, 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_{0}^{(n)}ψ_{0},

where L_{n}v ∈ C^{(n)}[−1, 1] is the Lagrange polynomial and a_{0}^{(n)} is the coefficient of the Fourier series expansion a_{0}^{(n)} ≡ [L_{n}v, ψ_{0}].

Again, as in Subsection 1.2, we seek an approximate solution in the form

u_{n} = ∑_{k=1}^{n} a_{k}φ_{k}.

We compose the algebraic system by the condition

Π_{n}(Su_{n} + Ku_{n} − f) = 0,

which results in

a_{0}ψ_{0} + ∑_{k=1}^{n} a_{k}ψ_{k}(x_{j}) + ∑_{k=1}^{n} 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^{−1}w_{n} = Π_{n}f, w_{n} ∈ L_{2,ρ}^{(n)},

where L_{2,ρ}^{(n)} 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

w̃_{n} = −KS^{−1}w_{n} + f = −Ku_{n} + f,

we find

ũ_{n} = S^{−1}w̃_{n} = \frac{(1 − t^{2})^{1/2}}{π} ∫_{−1}^{1}(1 − x^{2})^{−1/2} \frac{w̃_{n}(x)dx}{t − x}.

Denote K_{1}(x, t) ≡ (1 − t^{2})^{1/2}K(x, t).

Theorem 5. If there exists the inverse operator (I + KS^{−1})^{−1} mapping L_{2,ρ}^{(r)}, 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, 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, 1], then the estimate

∥u − ũ_{n}∥ = O(\frac{1}{2^{n}} · \frac{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

∥u − u_{n}∥ = O(\frac{1}{2^{n}} ).

2.3. Index κ = 0. As in Subsection 1.3, an approximate solution is again sought in the form

u_{n} = ∑_{k=1}^{n} a_{k}φ_{k}.

Equating the residuals to zero at the points x_{1}, …, x, we obtain

[Su_{n} + Ku_{n} − f]_{x_{j}} = 0 j = 0, 1, …, n,

which yields the algebraic system

∑_{k=0}^{n} a_{k}ψ_{k}(x_{j}) + ∑_{k=0}^{n} a_{k}(Kφ_{k})(x_{j}) = f(x_{j}), j = 0, 1, …, n.

Just as for the index κ = 1 we can rewrite this system as

w_{n} + Π_{n}KS^{−1}w_{n} = Π_{n}f, w_{n} ∈ L_{2,ρ}^{(n)},

where L_{2,ρ}^{(n)} is the linear span of functions ψ_{0}, …, ψ_{n}.

Let the approximate solution w_{n} be found.

Taking one iteration

w̃_{n} = −KS^{−1}w_{n} + f = −Ku_{n} + f

we find

ũ_{n} = S^{−1}w̃_{n} = \frac{(1 + t)^{1/2}(1 − t)^{−1/2}}{π} ∫_{−1}^{1}(1 − x)^{1/2}(1 + x)^{−1/2} \frac{w̃_{n}(x)dx}{t − x}.

Denote K_{1}(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 L_{2,ρ}, ρ = ρ_{3}, onto itself, the roots of the Jacobi polynomial P_{n+1}^{( \frac{1}{2} ,− \frac{1}{2} )} are taken as collocation nodes, (SK_{1}(x, t))^{(m)} ∈ L : p_{M}α, 0 < α ≤ 1, ∀x ∈ [−1, 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, 1], then the estimate

∥u − ũ_{n}∥ = O(\frac{1}{2^{n}} · \frac{1}{n^{m+α}} )

is valid.

Remark 3. Under the conditions of the theorem

∥u − u_{n}∥ = O(\frac{1}{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, −1 we obtain the same order of convergence

∥u − u_{n}∥ = O(\frac{ln n}{n^{m+α}} )

for an approximate solution u_{n} 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 −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 T̂_{0}, T̂_{1}, …, T̂_{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}∥_{L_{2,ρ}} ≤ C∥Π^{(n)}u∥_{L_{2,ρ}} ≤ C_{1}∥P^{(n)}u∥_{C} =

= O(\frac{ln n}{n^{m+α}} ) (Π_{n} = L_{n}).

As an example of the application of the above methods in the case of the index κ = −1, let us consider the equation

\frac{1}{π} ∫_{−1}^{1} \frac{u(t)dt}{t − x} + \frac{1}{π} ∫_{−1}^{1}(x^{8}t^{8} + x^{7}t^{7})u(t)dt =

= (\frac{2}{π} )^{1/2}(\frac{3}{128}x^{7} − 32x^{6} + 48x^{4} − 18x^{2} + 1)

with the exact solution

u(x) = φ_{6}(x) = (\frac{2}{π} )^{1/2}(32x^{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,ρ_{2}}. We find the fifth approximation

u_{5}(x) = ∑_{k=1}^{5} a_{k}φ_{k}(x), φ_{k}(x) = (\frac{2}{π} )^{1/2}(1 − x^{2})^{1/2}U_{k−1}(x),

where U_{k−1}(x), k = 1, 2, …, are the Chebyshev polynomials of the second kind.

Computations are carried out to within 10^{−7}. u_{5}(x) and ũ_{5}(x) are calculated.

In the case of the Bubnov–Galerkin method we have

∥Δu_{5}∥ = 1, 0001151, ∥Δũ_{5}∥ = 0, 0151855,

\frac{∥Δu_{5}∥}{∥u∥} ≈ 100, 01%, \frac{∥Δũ_{5}∥}{∥u∥} ≈ 1, 52%

for an absolute and a relative error, respectively, while in the case of the collocation method we obtain

∥Δu_{5}∥ = 1, 0002931, ∥Δũ_{5}∥ = 0, 0159781,

\frac{∥Δu_{5}∥}{∥u∥} ≈ 100, 03%, \frac{∥Δũ_{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

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(Received 30.12.1993)

Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi 380093 Republic of Georgia