The Numerical solution of systems of Singular Integral ...

The Numerical solution of systems of Singular Integral Equations by reduction methods in generalized Holder spaces. Feras Al Faqih Al-Hussein Bin Talal University Department of Mathematics Jordan [email protected] [email protected]

Iurie Caraus University of Quebec of Montreal, Department of Computer Science Canada [email protected]

Nikos E. Mastorakis WSEAS A.I. Theologou 17-23 15773 Zographou, Athens Greece [email protected]

Abstract: Approximation of functions of a complex variable by various finite-dimensional aggregates is an important problem not only in constructive function theory and approximation but also in the theoretical background of direct approximate methods for functional equations. This problem has been well studied for the case of functions defined on standard contours (a straight line segment, the unit circle, and so on). In the case of an arbitrary closed smooth contour in the complex plane, the problem is not studied enough. We have obtained the numerical schemes of the reduction method over the system of FaberLaurent polynomials for the approximate solution of systems of singular integral equations defined on smooth closed contours in the complex plane. The convergence has been obtained in Generalized Holder spaces. Key–Words: Faber- Laurent polynomials, Singular Integral Equations,Generalized H¨older spaces

1 Introduction

many difculties. It should be noted that conformal mapping from the arbitrary smooth closed

Singular integral equations with Cauchy kernels

contour to the unit circle does not solve the prob-

(SIE) and their systems are used to model many

lem. Moreover, it makes more difcult.

problems in elasticity theory, aerodynamics, mechanics, thermoelasticity, queuing system analysis, etc.[1]-[6]. The problem of numerical solu-

- The coefcients, kernel and right part of transformed equation lose their smoothness;

tion of sigular integral equations (SIE) has been

- The numerical schemes of researched meth-

studied in scientific bibliography [7]-[11]. The

ods become more difcult. The singularity

general theory of SIE and SIDE has been widely

appears in new kernel.

investigated in the last decades. In the same time the reduction method applied to the numerical

In this paper the SIE has been studied in gener-

solution of SIE has been studied enough. The

alized Holder spaces. In case when the SIE is

SIE are defined on an arbitrary smooth closed

defined on the unit circle the reduction method

contour(different from unit circle. We should

has been studied in [15]-[17](in Lebesgue spaces

mention only [12]-[14]. Transition to another

and classical Holder spaces). The SIE is defined

contour, different from the standard one, implies

on the unit circle. The case of Singular integro-

differential equations has been studied in [20]-

solving equation (1) based on the Faber- Laurent

[25]. In the present work the numerical schemes

polynomials[12],[13],[19].

of the reduction method using the Faber-Laurent polynomials is elaborated. The theoretical back-

We are looking for the approximate solution of system of SIE (1) in polynomial form:

ground has been obtained in generalized Holder

n X

xn (t) =

spaces.

αk tk , t ∈ Γ, n ∈ N,

(2)

k=−n

In Section 2 we formulate the main problem. We present the numerical scheme of reduction method. We introduce the main definitions and notations. in Section 3. In Section 4 we formulate auxiliary results. We formulate the convergence theorem in Section 5. We study the non elliptical case in Section 6.

with unknown numerical vectors αk , k

=

0, ±1, ±2, . . . , ±n. We determine these coefficients from the condition that the partial sum for n order polynomial of the Faber- Laurent series for the function Mxn (t) − f (t) is zero: Sn [Mxn − f ] ≡ 0

(3)

2 Problem Formulation

where we denote the Faber-Laurent reduction op-

Let us consider the system of Singular Integral

erator by Sn

Equation

(Sn g)(t) =

Mϕ ≡ c(t)x(t) + 1 2πi

Z

d(t) πi

Z Γ

ak Φk (t) +

k=0

x(τ ) dτ + τ −t

k(t, τ )x(τ ) = f (t), t ∈ Γ,

n X

n X

k=1

Fk

1 , t

 

(4)

ak (k = 0, 1, 2, . . . , n) and bk (k = 1, 2, . . . , n)

(1)

Γ

where Γ is an arbitrary smooth closed contour, limiting the domain D + of the complex plane (we

are Faber-Laurent coefficients of g(t) function, 1 but Φk (t) k = 0, 1, 2, . . . , n and F ( , k = t 1, 2, . . . are Faber- Laurent polynomials generated by the contour Γ. For the simplicity we introduce the notations   1 t

, k = 1, 2, . . . , a−k = bk , k =

assume that z = 0 belongs to the D + domain),

Φ−k (t) = Fk

c(t), d(t), f (t) and k(t, τ ) are known m × m

1, 2, . . . . We obtain the following formula

matrix-functions (m.f) defined on Γ and x(t) is an

(Sn g)(t) =

unknown vector-function (v.f). It is well known that the system of SIE (1) can be solved exactly only in some particular cases. Even in these cases the solution can be expressed by complicated formula which is very difficult and the practical ap-

n X

ak Φk (t), t ∈ Γ.

(5)

k=−n

We get the following relation for unknown function: xn (t) =

n X

xk Φk (t), t ∈ Γ.

(6)

k=−n

plication will be very difficult. That is why the

It is obvious that the coefficients αk from (2) are

problem for finding the numerical solution for

expressed by xk , k = 0, ±1, ±2, . . . , by non-

systems of SIE (1) is very important. In this

degenerated matrix (A). In the similar way the

paragraph we present the reduction method for

coefficients xk by αk by inverse matrix A−1 . It is

simple to find the matrix A. In this article we will

monograph [18]. Also we present the properties

not present the obvious form of matrix A. Thus

of the singular operators, studied in [Hω ]m . Let Γ be an arbitrary smooth closed contour

the approximate solution of equation (1) can be

limiting a monoconex domain D + of complex

considered in the form (6). Developing the operational equation (3) us-

plane C; D − = C \ {D +

S

Γ} . We consider that

ing the orthogonality property of the system

the point t = ∞ ∈ ∞. By d = diamD + =

{Φk (t)}∞ k=−∞ and identifying the coefficients be-

max |t − t |, t , t ∈ Γ.

′

sides Φk (t) k = 0, ±1, ±2, . . . , ±n. n X

(n) aj−k xk

k=0 n X

+

−1 X

′′

′

′′

Further, we assume that ω(δ) is the module of continuity, and

(n) bj−k xk

′

ω(δ) =

k=−n

′′

sup |f (t − f (t |, δ ∈ [0; d] |t′ −t′′ |≤δ

(n)

Ajk xk = fj , j = 0, ±1, ±2, . . . , ±n. (7)

k=0

aj , bj and Ajk k, j = 0, ±1, ±2, . . . , ±n are

is the continuity modulus of the function f (t), t ∈ Γ.

=

By [Hω ]m = [Hω (Γ)]m we will denote the

c(t) Z+ d(t), b(t) = c(t) − d(t) and Ak (t) = 1 k(t, τ )Φk (τ )dτ, k = 0, ±1, ±2, . . . , ±n. 2πi Γ We should note that the Faber-Laurent

Banach space of m- dimensional v.f. g(t) contin-

a(t)

Faber-Laurent coefficients of m.f.

uous on Γ (g(t) ∈ C(Γ)) that satisfy the condition: H(g; f ) ≡ sup

cooefficients[19] of the v.f. g(t) are determinate

ω(g, δ) < ∞, δ ∈ (0; d). ω(δ)

by formulae: 1 Z gk = 2πi

|w|=1

1 Z yk = 2πi

|w|=1

g(ψ(w)) dw, k = 0, 1, 2, . . . ; w k+1

The norm in [Hω ]m is defined by the equality: ∀g(t) = {g1 (t), . . . , gm (t)},

g(ϕ(w)) dw, k = −1, −2, −3, . . . ; w k+1

||g||ω,m =

m X

(||g||C + H(gk ; ω)),

k=1

||g||c = max |g(t)|.

The functions ψ(w) and ϕ(w) are Riemann func-

t∈Γ

(8)

Thus [Hω ]m is nonseparable Banach space

tions of the contour Γ that apply of exterior of Γ0 in the exterior of D + , ψ(∞), ψ (∞) > 0,

[11],[19].

similarly the exterior of Γ0 in D + , ϕ(∞) = 0,

that the continuity modulus ω satisfies the Bari-

′

′

ϕ (∞) > 0.

In what follows we will consider

Stechkin condition [18]: Zh

3 Definitions of Function Spaces and Notation In this section we introduce the Holder generalized spaces [Hω ]m determined the module of continuity. These spaces have been introduced in the

0

Zδ 0

ω(ξ) dξ+ ξ

Zh δ

ω(ξ) dξ < ∞, ξ

(9)

ω(ξ) dξ = O(ω(δ)) → 0, δ → +0. ξ2 (10)

In this case, according to [19], the singular inte-

closed contour of complex plane C by partial

gral operator:

sums of the Faber-Laurent series in the general-

(Sg)(t) =

1 πi

Z Γ

g(τ ) dτ, t ∈ Γ, τ −t

ized Holder spaces have been obtained. Besides, (11)

the proper interest of the approximation theory of the complex variable functions, these results have

is bounded in [Hω ]m . We are in the presence of

an essential application in the theory of the ap-

the following theorem, essentially used further.

proximating methods of solution of equations. In

Theorem 1 Assume that the continuity modulus

special to obtain the numerical solution for sys-

ω1 (δ) satisfies (9), and continuity modulus ω2 (δ)

tem of SIE by reduction methods.

satisfies the conditions (9) and (10) simultane-

As it is known that the [Hω ]m spaces is not

ously. Then for every v.f. g(t) ∈ [Hω2 ]m the op-

separable. That is why the approximation of the

erator Sg − gS is bounded as an operator acting

whole [Hω ]m space in the norm (8) with finite

from [Hω1 ]m to [Hω2 ]m and

dimensional generating sets is impossible. But some subsets from [Hω ]m this problem can be

||Sg − gS||ω1→ω2 ≤ const||g||ω2 .

(12)

Theorem 2 Let the continuity modulus ω1 and ω2 satisfy both conditions (9) and (10). If g(t) ∈ [Hω2 ]m , then the operator Sg − gs mapping the

solved positively. The theorem given below establishes the possibility of function approximation with Faber-Laurent polynomials in generalized Holder spaces.

space [Hω1 ]m to [Hω2 ]m is completely continuous.

Theorem 3 Let ω1 (δ) and ω2 (δ) δ ∈ (0; d] be

The prove of theorems is realized analogously to

the two continuity modulus that satisfy the rela-

the similar outcome get in [18] about the bound[Hβ (Γ)]m with the condition g(t) ∈ [Hα (Γ)]m ,

tions (9) and (10). We suppose that the function ω2 (δ) Φ(δ) = is nondecreasing on (0; d]. Then ω1 (δ) for every function g(t) from [Hω2 ]m the following

and do not here for the deduction.

relation takes place:

ness of the operator T = Sg−gS in Holder space

We note that in case when ω1 (δ) = δ β and α

ω2 (δ) = δ ,

||g−Sn g||ω1,m ≤ d1 +d2 ||Sn ||C )Φ

0 < β ≤ α ≤ 1 the spaces [Hω1 ]m and Hω2 coincide with the classical Holder spaces Hβ and Hα .

4 The approximation of functions in generalized Holder spaces

1 H(g; ω2). n (13)

 

By dk , ck k = 1, 2, . . . we denote the constant values that do not depend from n. We would like to mention in the virtue of imposed condition to the function Φ(δ), the space [Hω2 ]m is included in [Hω1 ]m and || · ||ω1 ≤ c1 || · ||ω2

(14)

In this item the results of approximation functions of complex variables defined on the smooth

Proof. According to the definition (8) of the

norm in the space [Hω1 ]m space

Using the Jackson theorem we get

||g−Sn g||ω1,m = ||g−Sn g||C +H(g−Sn g; ω1) = N1 +N2

N3 ≤

2d3 (1 + ||Sn ||C )

(15) The quantity N1 increases simply. Indeed, let Rn∗ (t) g(t) = En (g).

1 H(g; ω2). n Set out to the quantity evaluation N4 . In the virtue ≤ 2d3 (1 + ||Sn ||C )Φ

(16)

As Sn Rn∗ ≡ Rn∗

=

ω2 ( n1 ) ω(g; n1 ) ≤ ω1 ( n1 ) ω2 ( n1 )

the best uniform approximation for the function ||g −

1 n

2d3 (1 + ||Sn ||C ) =

be the polynomial having the form (2) of

Rn∗ ||C,m

 

ω

 

of the equality (17) we will obtain for the numer-

(17)

ator from the expression of N4 :

and by Jackson theorem ′

′′

′

′′

|g(t ) − g(t )| + |(Sn g)(t ) − (Sn g)(t )|

1 , En (g) ≤ d3 ω g; n 



′

′′

′

′′

≤ |g(t ) − g(t )| + |Rn∗ (t ) − Rn∗ (t )|+

We get

′

′′

|Sn (g −Rn∗ )(t ) −Sn (g −Rn∗ )(t )| = I1 + I2 + I3 .

N1 = ||g − Rn∗ + Sn Rn∗ − Sn g||C ≤ (1 + ||Sn ||C )En (g) ≤   1 d3 (1 + ||Sn ||C )ω g; n 1 We evaluate N2 . Let n < d. Then

For I1 we get ′

′′

′

(18)

′

′′

ω(g; |t − t |) ′ ′′ ω2 (|t − t |) ≤ ′ ′′ ω2 (|t − t |) ′

′

′′

H(g; ω2)ω2 (|t − t |).

N2 = H(g − Sn g; ω1) = ′

′′

I1 = |g(t ) − g(t )| ≤ ω(g; |t − t |) =

′′

For the evaluation of I2 we use the relation be-

′′

g(t ) − (Sn g)(t ) − g(t ) + (Sn g)(t )| sup ′ ′′ ω1 (|t′ − t′′ |) 0≤|t −t |≤d

tween the continuity modulus of the function g(t)

and monograph [15]: ′ ′ ′′ ′′ |g(t ) − (Sn g)(t )| + |g(t ) + (Sn g)(t )| + ≤ sup 1 ω1 (|t′ − t′′ |) ω(Rn∗ ; δ) ≤ d4 ω(g; δ), ∀δ > 0. ≤|t′ −t′′ |≤d n

′

sup ′

′′

1 ≤0