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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, un ∈ 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, n ≥ n0 .
ku − u
(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)
ψk+1 (x) ≡ π2
Uk (x), k = 0, 1, . . . ,
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
(1 − t2 )−1/2
w
en =
π
Z1
(1 − x2 )1/2
−1
w
en (x)dx
.
t−x
We have
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
2
wk =
Z1
(1 − x2 )1/2 (w − Pn w)2 dx =
−1
=
Z1
(1 − x2 )1/2 (w −
−1
≤
Z1
n
X
[w, ψk ]ψk )2 ds ≤
k=1
(1 − x2 )1/2 (w − Pn−1 )2 dx ≤
π
kw − Pn−1 k2C ,
2
−1
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) ).
462
A. JISHKARIANI∗ AND G. KHVEDELIDZE
Furthermore,
kKS −1 P (n) vk2 = kKS −1
∞
X
∞
X
[v, ψk ]ψk k2 = kK
[v, ψk ]ϕk k2 =
k=n+1
k=n+1
∞
1
2
X
= 2
[v, ψk ] K(x, t), ϕk (t) ≤
π
k=n+1
∞
n ∞
o1/2 n X
2 o1/2 2
1
X
≤ 2
[v, ψk ]2
K(x, t), ϕk (t)
k ≤
×
π
k=n+1
kvk2
π2
≤
k=n+1
Z1
(1 − x2 )1/2
k=n+1
−1
kvk2
π2
=
Z1
(1 − x2 )1/2
Z1
−1
∞
X
k=n+1
−1
kvk2
= 2
π
∞
X
2
dx =
K(x, t), ϕk (t)
∞
X
K(x, t), Tbk (t) L
2 1/2
(1 − x )
Z1
∞
X
K(x, t), Tbk (t) L
−1
=
Z1
−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 =
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 = 21 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)
, k = 0, 1, . . . ,
ϕk ≡ ck (1 − x)1/2 (1 + x)−1/2 Pk
(−1/2,1/2) −1/2
c0 = π, 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
−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
Then
ku − u
en k = kS −1 (w − w
en k ≤ CkKS −1 P (n) k · kP (n) wk.
en )k = kw − w
466
A. JISHKARIANI∗ AND G. KHVEDELIDZE
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 ϕ k .
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
(n)
wn + Πn−1 KS −1 wn = Πn−1 f1 , 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 Erd¨os–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
(1 − x2 )1/2 w
en (x)dx
.
t−x
Then
e n k = kS −1 (w − w
en )k = kw − w
en k = kΦ − Φ
en k ≤
ku − u
−1 (n−1)
−1 (n−1)
≤ C kKS Π
Kk + kKS Π
f1 k ,
(23)
e n + pπ −1 (1 − x2 )−1/2 .
en ≡ Φ
where u
It is known [12] that
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
−1 (n−1)
2 1/2
S Π
(1 − τ ) (1 − τ 2 )−1/2 (K(x, τ ), v(τ )) =
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,
t),
(x,
τ
)SK
(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])
c′ 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 c′m ≡ 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)]
=
U
δ
f
SK
(x,
t),
[P
≤
1
1
π 2
2n
1 π 1/2 1 (n−1) (n)
b
δ
f
SK
(x,
t)
U
(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 ≤
′
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
≤
(1 − t2 )1/2 U
w(t)k = n
≤ n kU
n (t)δ
2
2
ku − un k ≤ CkΠ(n−1) wk = C
−1
470
A. JISHKARIANI∗ AND G. KHVEDELIDZE
n
C1
≤ n kδ (n) wkC
2
Z1
−1
bn2 (t)dt
(1 − t2 )1/2 U
o1/2
=
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 ϕ k .
k=1
We compose the algebraic system by the condition
Πn (Sun + Kun − f ) = 0,
which results in
a0 ψ 0 +
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 − x2 )−1/2
−1
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
(n)
wn + Πn KS −1 wn = Πn f, 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
472
A. JISHKARIANI∗ AND G. KHVEDELIDZE
we find
u
en = S
−1
(1 + t)1/2 (1 − t)−1/2
w
en =
π
Z1
(1 − x)1/2 (1 + x)−1/2
−1
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 Erd¨os–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.
e5 (x) are calComputations are carried out to within 10−7 . u5 (x) and u
culated.
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. Szeg¨o, 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
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, un ∈ 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, n ≥ n0 .
ku − u
(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)
ψk+1 (x) ≡ π2
Uk (x), k = 0, 1, . . . ,
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
(1 − t2 )−1/2
w
en =
π
Z1
(1 − x2 )1/2
−1
w
en (x)dx
.
t−x
We have
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
2
wk =
Z1
(1 − x2 )1/2 (w − Pn w)2 dx =
−1
=
Z1
(1 − x2 )1/2 (w −
−1
≤
Z1
n
X
[w, ψk ]ψk )2 ds ≤
k=1
(1 − x2 )1/2 (w − Pn−1 )2 dx ≤
π
kw − Pn−1 k2C ,
2
−1
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) ).
462
A. JISHKARIANI∗ AND G. KHVEDELIDZE
Furthermore,
kKS −1 P (n) vk2 = kKS −1
∞
X
∞
X
[v, ψk ]ψk k2 = kK
[v, ψk ]ϕk k2 =
k=n+1
k=n+1
∞
1
2
X
= 2
[v, ψk ] K(x, t), ϕk (t) ≤
π
k=n+1
∞
n ∞
o1/2 n X
2 o1/2 2
1
X
≤ 2
[v, ψk ]2
K(x, t), ϕk (t)
k ≤
×
π
k=n+1
kvk2
π2
≤
k=n+1
Z1
(1 − x2 )1/2
k=n+1
−1
kvk2
π2
=
Z1
(1 − x2 )1/2
Z1
−1
∞
X
k=n+1
−1
kvk2
= 2
π
∞
X
2
dx =
K(x, t), ϕk (t)
∞
X
K(x, t), Tbk (t) L
2 1/2
(1 − x )
Z1
∞
X
K(x, t), Tbk (t) L
−1
=
Z1
−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 =
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 = 21 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)
, k = 0, 1, . . . ,
ϕk ≡ ck (1 − x)1/2 (1 + x)−1/2 Pk
(−1/2,1/2) −1/2
c0 = π, 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
−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
Then
ku − u
en k = kS −1 (w − w
en k ≤ CkKS −1 P (n) k · kP (n) wk.
en )k = kw − w
466
A. JISHKARIANI∗ AND G. KHVEDELIDZE
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 ϕ k .
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
(n)
wn + Πn−1 KS −1 wn = Πn−1 f1 , 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 Erd¨os–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
(1 − x2 )1/2 w
en (x)dx
.
t−x
Then
e n k = kS −1 (w − w
en )k = kw − w
en k = kΦ − Φ
en k ≤
ku − u
−1 (n−1)
−1 (n−1)
≤ C kKS Π
Kk + kKS Π
f1 k ,
(23)
e n + pπ −1 (1 − x2 )−1/2 .
en ≡ Φ
where u
It is known [12] that
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
−1 (n−1)
2 1/2
S Π
(1 − τ ) (1 − τ 2 )−1/2 (K(x, τ ), v(τ )) =
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,
t),
(x,
τ
)SK
(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])
c′ 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 c′m ≡ 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)]
=
U
δ
f
SK
(x,
t),
[P
≤
1
1
π 2
2n
1 π 1/2 1 (n−1) (n)
b
δ
f
SK
(x,
t)
U
(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 ≤
′
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
≤
(1 − t2 )1/2 U
w(t)k = n
≤ n kU
n (t)δ
2
2
ku − un k ≤ CkΠ(n−1) wk = C
−1
470
A. JISHKARIANI∗ AND G. KHVEDELIDZE
n
C1
≤ n kδ (n) wkC
2
Z1
−1
bn2 (t)dt
(1 − t2 )1/2 U
o1/2
=
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 ϕ k .
k=1
We compose the algebraic system by the condition
Πn (Sun + Kun − f ) = 0,
which results in
a0 ψ 0 +
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 − x2 )−1/2
−1
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
(n)
wn + Πn KS −1 wn = Πn f, 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
472
A. JISHKARIANI∗ AND G. KHVEDELIDZE
we find
u
en = S
−1
(1 + t)1/2 (1 − t)−1/2
w
en =
π
Z1
(1 − x)1/2 (1 + x)−1/2
−1
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 Erd¨os–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.
e5 (x) are calComputations are carried out to within 10−7 . u5 (x) and u
culated.
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.
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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