Reduction Methods for Approximate Solution of the Singular Integro ...

RECENT ADVANCES IN APPLIED MATHEMATICS AND COMPUTATIONAL AND INFORMATION SCIENCES - Volume I

Reduction Methods for Approximate Solution of the Singular Integro-Differential Equations in Lebesgue Spaces Iurie Caraus Moldova State University Faculty of Mathematics and Informatics Mateevici 60, str., Chisinau, MD-2009 Republic of Moldova [email protected]

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

Abstract: We have elaborated the numerical schemes of reduction method by Faber- Laurent polynomials for the approximate solution of system of singular integro- differential equations. The equations are defined on the arbitrary smooth closed contour. The theoretical foundation has been obtained in Lebesgue spaces. Key–Words: singular integro- differential equations, reduction method, Lebesgue spaces

1

Introduction

However the case when the contour of integration can be an arbitrary closed smooth curve (not unit circle) has not been studied enough. Transition to another contour different from standard one implies many difficulties. It should be noted that the conformal mapping from an arbitrary smooth closed contour to the unit circle using the Riemann function does not solve the problem. Moreover it makes more difficult. We note that theoretical background of reduction methods for SIDE in Generalized Holder spaces has been obtained in [12]. The equations have been defined on an arbitrary smooth closed contours.

Singular Integro- differential equations (SIDE) with Cauchy kernels (SIDE) are used to model many problems in elasticity theory, aerodynamics, mechanics, thermoelasticity, queueing system analysis, etc. [13]-[15]. The general theory of SIE and SIDE has been widely investigated in the last decades. It is known that the exact solution for SIDE is possible in some particularly cases. That is why there is a necessity to elaborate the numerical methods for solving SIDE with corresponding theoretical background. The theoretical background means the following a) the establishment of convergence of methods;

2

The results of approxima-

b) the examination of the rate of convergence;

tion functions by Faber-

c) the effective estimation of error.

Laurent polynomials

In this article we study the reduction method for approximate solution of SIDE. The problem for approximate solution of SIDE by reduction methods has been studied in []. The equations have been defined on the unit circle. ISSN: 1790-5117

Let Γ is an arbitrary smooth closed contour limiting the one-spanned area of complex plane D+ , the point z = 0 ∈ D+ and D− = C \ {D+ ∪ Γ}, C is the full complex plane. Let z = ψ(w) and z = ϕ(w) are the func50

ISBN: 978-960-474-071-0

RECENT ADVANCES IN APPLIED MATHEMATICS AND COMPUTATIONAL AND INFORMATION SCIENCES - Volume I

tions mapping conformably and unambiguously the exterior of unit circle Γ0 on D−0 and interior on D+ so that ψ(∞) = ∞, ψ (∞) > 0 0 and ϕ(∞) = 0, ϕ (∞) > 0. We denote by w = Φ(z) and w = F (z) the reversible functions for z = ψ(w) and z = ϕ(w). We suppose 1 Φ(z) = 1 z→∞ z lim

and

ν < 1); the class of such contours is denoted [7] by C(2; ν). We denote dk , k = 1, 2, . . . , the constants which are not depend of n. We will not be interested their values. Theorem 1 Let Γ ∈ C(2; µ), g(t) ∈ Hα (Γ), r = 0, 1, 2, . . . , then for all continuous function g(t) the following inequality takes place

lim zF (z) = 1; (1)

z→∞

(see [1]). We obtain from (1) that the functions w = Φ(z) and w = F (z) admit in the vicinity of points z = ∞ and z = 0 the following decompositions accordingly Φ(z) = z+

∞ X

rk z −k

||g − Sn g||p ≤ (1 + m1 (p))d1

k=0

3

We denote by Φn (z), n = 0, 1, 2, . . . , the set of members by nonnegative degrees z for decomposing Φn (z) and by Fn (1/z) , n = 1, 2, . . . the set of members by negative degrees z for decomposing F n (z). The polynomials Φn (z), n = 0, 1, 2, . . . and Fn (1/z) , n = 1, 2, . . . , are FaberLaurent polynomials for Γ (see [1], [2]). We denote by Sn the operator of reduction method by Faber- Laurent polynomials : (Sn g)(t) =

n X

ak Φk (t) +

k=0

bk Fk

k=1

The numerical schemes of reduction method

We introduce the spaces and class of functions where we study these equations. In space Lp (Γ), 0 < β < 1 we study the S.I.D.E.

µ ¶

n X

(3)

The scheme of proof for this theorem is similar as in [11].

∞ 1 X F (z) = + vk z k z k=0

and

H(g; α) . nα

Mϕ ≡

1 , t ∈ Γ, t

q X

 cr (t)ϕ(r) (t) + dr (t)

r=0

1 Z ϕ(r) (τ ) dτ + πi τ −t Γ



1 Z + hr (t, τ )ϕ(r) (τ )dτ  = f (t), 2πi

where ak bk are the Faber- Laurent coefficients for function g(t) :

t ∈ Γ,

Γ

1 Z g(ψ(w)) , ak = 2πi wk+1

(4) where cr (t), dr (t), hr (t, τ ), r = 0, q are given functions and f (t) is given function ϕ(t) is unknown function. We search for the solution ϕ(t) of system of S.I.D.E. (4) satisfying the conditions on Γ

k = 0, 1, 2, . . . ,

Γ0

1 Z g(ϕ(w)) , bk = 2πi wk+1

k = 1, 2, . . . ,

1 Z ϕ(τ )τ −k−1 dτ = 0, k = 0, q − 1. 2πi

Γ0

Let Lp (Γ)(1 < p < ∞) be complex space of the functions g(t) ∈ Lp (Γ) with the norm 

1 ||g|| =  l

◦ (ν)

1

Z

Let W p

p

p

|g| |dτ | ,

(2)

(Γ) = {g; ∃ g (r) ∈ C(Γ), r = ◦ (ν)

0, ν − 1, g (ν) ∈ Lp (Γ)}. For any ∀g ∈W p

Γ

(Γ) ◦ (ν)

satisfies the condition (5) and the norm in W p (Γ) is defined by

where l is the length of Γ. We shall assume that the function z = ψ(w) has second derivative , satisfying on Γ0 the H¨older condition with some parameter ν (0 < ISSN: 1790-5117

(5)

Γ

||g||p,ν = ||g (ν) ||Lp . 51

ISBN: 978-960-474-071-0

RECENT ADVANCES IN APPLIED MATHEMATICS AND COMPUTATIONAL AND INFORMATION SCIENCES - Volume I

Using the Riesz operators (P = 1/2 · (I + S), Q = I − P both I and S are the identity operator and singular operator with Cauchy kernel) the S.I.D.E. (4) can be written as follows:

2. aq (t)· bq (t) 6= 0, t ∈ Γ; 3. the left and right indexes of function aq (t) are equal zero, the left and right indexes of function bq (t) are equal q; for any t ∈ Γ;

q h X

ar (t)P ϕ(r) (t) + br (t)Qϕ(r) (t) +

◦ (ν)

4. the operator M :W p (Γ) → Lp (Γ), where 0 < β < α is linearly invertible;

r=0



1 Z hr (t, τ )ϕ(r) (τ )dτ  = f (t), + 2πi

4. Γ ∈ C(2; µ);

t ∈ Γ,

5. hr (t, τ ) belongs to C(Γ) by both variables.

Γ

(6) where ar (t) = cr (t)+dr (t), br (t) = cr (t)−dr (t), r = 0, q. We name the ”problem (6)-(5)” the of S.I.D.E. (5) and conditions (6) Since the analytic solution for the ”problem (6)-(5)” is rarely available ( see [4]), we look for the approximate solution for the problem ”(6)- (5)”. Most of early approximative methods for SIDE are designed for the case where boundary is a unit circle (see [6,7]). The problem(6)-(5)” was solved by quadrature- interpolation and collocation methods when Γ is an arbitrary smooth closed contour. (see [3,8,9]). We look for the approximate solution of problem ”(6)- (5)” in the form n X

n X

Then beginning with numbers n, the equation of reduction method (9) has the unique solution. The approximate solutions ϕn (t), constructed by formula (7), converge in the norm of ◦ (ν)

space W p (Γ), as n → ∞ to the exact solution ϕ∗ (t) of the ”problem (6) - (5)” for ∀ right part f (t) ∈ C(Γ). Furthermore, the following error estimate holds: µ

∗

||ϕ − ϕn ||p,ν µ

(n)

αk Φk (t)+

(n)

References: [1] Suetin P. The series by Faber polynomials.M.: Science, 1984. (in Russian)

(8)

[2] Kuprin N. The Faber series and the problem of linear interfacing // Izv. Vuzov Mathematics. 1980, 1. p. 20- 26. (in Russia)

We examine the condition (8) as the operator equation Sn M Sn ϕn = Sn f (9)

[3] Zolotarevskii V. Finite methods of solving the singular integral equations on the closed contours of integration .- Kishinev: ”Stiintsa”, 1991. (in Russian)

for unknown v.f. ϕn (t) in subspace Rn , for functions of the form (7). We note that the equation (9) is equivalent for the system of linear equations with 2n + 1 dimensional unknown v.f. αk , k = −n, n. We do not indicate the obvious form of this system because of difficulty formula.

[4] Muskhelishvili N., Singular integral equations .- M:. Science, 1968. (in Russian) [5] Lifanov I. The method of singular integral equtions and numerical experiment .- M : TOO ”Ianus”, 1995. (in Russian)

Theorem 2 Assume the following conditions are satisfied:

[6] Gabdulahev B. The opimal approximation of solutions for lenear problems .- Kazani: University of Kazani, 1980. (in Russian)

1. m. f. ar (t), br (t) and r = 0, q belong to the space Hα (Γ), 0 < α ≤ 1; ISSN: 1790-5117

¶

The scheme of proof for this theorem is similar as in [12].

µ ¶

α−k Fk

Sn [M ϕn − f ] = 0.

µ

1 1 O ω(f ; ) + O ω t (h; ) . n n

1 , t ∈ Γ, t k=0 k=1 (7) (n) where αk = αk , k = −n, n; are unknown vectors of dimension m. To find the unknowns integers αk , k = −n, n we use the relation ϕn (t) = tq

¶

¶

1 =O α + n

52

ISBN: 978-960-474-071-0

RECENT ADVANCES IN APPLIED MATHEMATICS AND COMPUTATIONAL AND INFORMATION SCIENCES - Volume I

[7] Zolotarevskii V. The direct methods for the solution of singular integro- differential equations on the unit circle // Collection ”Modern problems of applied mathematics”.- Chisinau: Science, 1979. p. 50-64. (in Russian)

Acknowledgements: The research of the first author was supported by SCSTD of ASM grant 07411.08 INDF and MRDA/CRDF Grant CERIM 10006-06

[8] Caraus Iu. The numerical solution of singular integro- differential equations in H¨older spaces. Conference on Scientific Computation. Geneva, 2002. [9] Caraus Iurie, Zhilin Li. A Direct methods and convergence analysis for some system of singular integro- differential equations, May 2003, CRSC-TR03-22, North Caroline, USA. [10] Krikunov Iu. The general Reimann problem and linear singular integro- differential equation // The scientific notes of Kazani university. Kazani.- v. 116(4) -1956. p.3-29. (in Russian) [11] Zolotarevskii V., Tarita I. The application of Faber- Laurent polynomials for the approximate solution to singular integral equations // The collection of Harkiv National University, Nr. 590, Harkiv, 2003. p. 124127. (in Russian) [12] Iurie Caraus, Nikos E. Mastoraskis, Feras M. Al- Faqih, Approximate solution of Singular integro- differential equations by Reduction Methods in Generalized Holder Spaces. WSEAS TRANSACTIONS ON MATHEMATICS, Issue 4, V. 6, 2007, pp. 595-601, ISSN 1109-2769. [13] Beloterkovski S. and Lifanov I., Numerical methods in singular integral equations. Moscow, Nauka: 1985, 256 p. (in Russian) [14] Linkov A. and Boundary Integral Equations in Elasticity. Theory Kluwer Academic: Dordrecht ; Boston, 2002, 268 p. Kalandia A., Mathematical methods of two- dimensional elasticity. Mir Publishers: 1975, 351 p. [15] Bardzokas D. and Filshtinsky M., Investigation of the direct and inverse piezoeffect in the dynamic problem of electroelasticity for an unboundedions medium with a tunnel opening. ACTA MECHANICA 2002, 155(1): pp. 17-25. ISSN: 1790-5117

53

ISBN: 978-960-474-071-0