Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces

Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

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Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces Iurie Caraus a,∗ , Feras M. Al Faqih b a

Moldova State University, Mateevici 60 str., Chisinau, Moldova, MD-2009, Republic of Moldova

b

King Faisal University, Department of Mathematics and Statistics, Al Hafouf 1982, P.O. Box 5909, Saudi Arabia

article

info

Article history: Received 9 January 2011 Received in revised form 14 January 2012 Keywords: Collocation method Lebesgue spaces Systems of singular integro-differential equations Fejér points

abstract Computational schemes for the collocation method and the mechanical quadrature method for the approximate solution of systems of singular integro-differential equations with a Cauchy kernel are elaborated. The case where the systems of equations are defined on an arbitrary smooth closed contour of a complex plane is examined. The methods researched are based on Fejér points. Estimates of the rate of convergence in Lebesgue space are obtained. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Singular integral equations (SIE) and singular integro-differential equations with Cauchy kernels (SIDE) and their systems arise in several problems of elasticity theory, aerodynamics, mechanics, thermoelasticity, and queuing analysis (see [1–5] and the literature cited therein). The general theory of SIE and SIDE has been widely investigated in the last few decades [6–10]. It is known that the exact solution for systems of SIDE can be found in some particular cases. That is why there is a necessity to elaborate the approximative methods for solving systems of SIDE and proving the convergence. In this article we study the collocation method and the mechanical quadrature method for the approximative solution of systems of SIDE. Usually the collocation method and the mechanical quadrature method for such systems of SIDE have been investigated for two cases: when the contour Γ was a unit circle and when the contour Γ was an interval on the real line [11–14]. However, the case where the contour Γ is assumed to be a smooth closed Jordan boundary of a simply connected domain around the origin has not been studied enough. It should be noted that conformal mapping from the arbitrary smooth closed contour to the unit circle does not solve the problem. Moreover, it makes more difficulties. The transition to another contour, different from the standard one, implies many difficulties. (1) The coefficients, the kernels and the right part of the transformed equation can be more complicated and can no longer be smooth. (2) The convergence analysis can be more complicated due to the transformation of the contour. We note that for the collocation method, the reduction method and the mechanical quadrature method for SIDE and their systems the convergence in Hölder spaces has been obtained in [15–21]. The equations have been defined on an arbitrary smooth closed contour.

∗

Corresponding author. E-mail addresses: [email protected] (I. Caraus), [email protected] (F.M. Al Faqih).

0377-0427/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2012.01.026

I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

3797

2. The main definitions and notation Let Γ be an arbitrary smooth closed contour bounding a simply connected region F + of the complex plane, and let t = 0 ∈ F + and F − = C \ {F + ∪ Γ } where C is the complex plane. Let z = ψ(w) be a function, mapping comfortably and unambiguously the outside of the unit circle Γ0 = {|w| = 1} on the domain F − such that

ψ (′) (∞) = 1.

ψ(∞) = ∞,

(1)

We assume that the function z = ψ(w) has a second derivative, satisfying on Γ0 the Hölder condition with some parameter ν (0 < ν < 1); the class of such contours is denoted by C (2; ν) [15,16]. Let [Lp (Γ )]m (1 < p < ∞) be the complex spaces of the vector functions g (t ) = (g1 (t ), . . . , gm (t )), gj (t ) ∈ Lp (Γ ), j = 1, . . . , m, with the norm

∥g ∥ =

m 

∥gk ∥p ,

  ∥g k ∥p =

k=1

1 l

Γ

|gk |p |dτ |

 1p

,

(2)

where l is the length of Γ . Let Un be the Lagrange interpolating polynomial

(Un g )(t ) =

2n 

g (ts ) · ls (t ),

(3)

s=0

where g ∈ [Lp (Γ )]m and t ∈ Γ , lj (t ) =

 n 2n n   t − tk tj (j) Λk t k , ≡ t − t t j k k=−n k=0,k̸=j

t ∈ Γ , j = 0, . . . , 2n,

where ts ∈ Γ , s = 0, . . . , 2n, are distinct points of Γ . We denote by [Hβ (Γ )]m , 0 < β ≤ 1, the Banach space of m-dimensional vector functions, satisfying on Γ the Hölder condition with degree β . The norm is defined as

∀g (t ) = {g1 (t ), . . . , gm (t )},

∥g ∥β =

m  (∥gk ∥C + H (gk , β)), k =1

∥g ∥C = max |g (t )|, t ∈Γ

H (g ; β) = sup {|t ′ − t ′′ |−β · |g (t ′ ) − g (t ′′ )|} t ′ ̸=t ′′

t ′ , t ′′ ∈ Γ .

(q)

We denote by [Hβ (Γ )]m the space of functions g (t ) containing the derivatives of degree q inclusive, and g (q) (t ) ∈ [Hβ (Γ )]m .

m

Let G(t ) = Gk,l (t ) k,l=1 be a non-singular m × m matrix function with Gk,l (t ) ∈ Hβ (Γ ).



Definition 1. A factorization of a non-singular matrix function G(t ) relative to the contour Γ is a representation of G in the form G(t ) = G+ (t )1(t )G− (t ), where G± (t ) are matrix functions analytic and non-singular in D± , satisfying det G± (t ) ̸= 0, respectively, 1(t ) = diag {t κ1 , t κ2 , . . . , t κm }, and κ1 , κ2 , . . . , κm are integers. The numbers κ1 , κ2 , . . . , κm are called left partial indexes [22]. 3. Numerical schemes for the collocation method and the mechanical quadrature method We consider in the complex space [Lp (Γ )]m (1 < p < ∞) the system of singular integro-differential equations

(Mx ≡)

q  

A˜r (t )x(r ) (t ) + B˜ r (t )

r =0

1

πi

x(r ) (τ )

 Γ

τ −t

dτ +

1 2π i

 Γ

Kr (t , τ ) · x(r ) (τ )dτ



= f (t ),

t ∈ Γ,

(4)

where A˜ r (t ), B˜ r (t ) and Kr (t , τ ) (r = 0, . . . , q) are known m × m matrix functions (MF); f (t ) is a known vector function (VF), dr x(t ) x(0) (t ) = x(t ) is an unknown VF, x(r ) (t ) = dt r (r = 1, . . . , q), and q is a positive integer. We search for the solution of Eq. (4) in the class of VF satisfying the conditions 1 2π i

 Γ

x(τ )τ −k−1 dτ = 0,

k = 0, . . . , q − 1.

(5)

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Using the Riesz operators P = 12 (I + S ), Q = I − P (where I is an identical operator, and S is a singular operator (with Cauchy kernel)), we rewrite (4) in the following form, convenient for consideration:

(Mx ≡)

q  

Ar (t )(Px(r ) )(t ) + Br (t )(Qx(r ) )(t ) +

r =0

1



2π i

Γ

Kr (t , τ )x(r ) (τ )dτ



= f (t ),

t ∈ Γ,

(6)

where Ar (t ) = A˜ r (t ) + B˜ r (t ), Br (t ) = A˜ r (t ) − B˜ r (t ), r = 0, . . . , q. Eq. (4) with conditions (5) is denoted as ‘‘problem (4)–(5)’’. We search for the approximate solution of problem (4)–(5) in the form n 

xn (t ) =

−1 

ξk(n) t k+q +

ξk(n) t k ,

t ∈ Γ,

(7)

k=−n

k=0

(n)

with unknown m-dimensional complex vector numbers ξk = ξk (k = −n, . . . , n). We note that VF xn (t ), constructed using formula (7), satisfies the conditions (5). Let Rn (t ) = Mxn (t ) − f (t ) be the residual of systems of SIDE. The collocation method consists in setting it equal to zero at some distinct points tj , j = 0, . . . , 2n, on Γ . Thus we obtain the system of linear algebraic equations (SLAE) for the unknown m-dimensional complex vector numbers ξk (k = −n, . . . , n) which can be determined by solving Rn (tj ) = 0,

j = 0, . . . , 2n.

(8)

Using the formulae in [15,20]

(Px)(r ) (t ) = (Px(r ) )(t ),

(Qx)(r ) (t ) = (Qx(r ) )(t ),

(9)

and the relations

(k + q)! k+q−r t , (k + q − r )!

(t k+q )(r ) =

(t −k )(r ) = (−1)r

k = 0, . . . , n,

(k + r − 1)! −k−r t , (k − 1)!

k = 1, . . . , n,

(10)

from (8) we obtain the SLAE for the collocation method:

 q 

n  (k + q)! k+q−r (k + r − 1)! −k−r · ξ−k tj ξk + Br (tj ) (−1)r t ( k + q − r )! (k − 1)! j k=0 r =0 k=1  n n  1  (k + q)! (k + r − 1)! + Kr (tj , τ )τ k+q−r dτ · ξk + (−1)r 2π i k=0 (k + q − r )! Γ (k − 1)! k=1  

Ar (tj )

×

n 

1

2π i

Γ

Kr (tj , τ )τ −k−r dτ · ξ−k

= f (tj ),

j = 0, . . . , 2n,

(11)

where tj (j = 0, . . . , 2n) is a set of distinct points on Γ . We approximate integrals in SLAE (11) using the quadrature rule 1



2π i

Γ

g (τ )τ l+k dτ ∼ =

1 2π i

 Γ

Un (τ l+1 · g (τ ))τ k−1 dτ ,

(where k = 0, . . . , n for l = 0, 1, 2, . . . and k = −1, . . . , −n, for l = −1, −2, . . .). Un is the Lagrange interpolating polynomial (3). Thus, we obtain the following SLAE for the mechanical quadrature method:

 q 

n 

r =0

k=0

Ar (tj )

+

n  k=0

×

2n  s =0

n  (k + q)! k+q−r (k + r − 1)! −k−r tj ξk + Br (tj ) (−1)r t · ξk (k + q − r )! (k − 1)! j k=1

2n n  (k + q)!  (k + r − 1)! (s) Kr (tj , ts )ts1+k−r Λ−k + (−1)r ξk (k + q − r )! s=0 (k − 1)! k=1 

(s)

Kr (tj , ts )ts−1−r Λk ξ−k

= f (tj ),

j = 0, . . . , 2n.

(12)

I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

(q)

3799

(q)

˚ p ]m = {g ; ∃g (r ) ∈ C (Γ ), r = 1, . . . , q − 1, g (q) ∈ [Lp (Γ )]m } and ∀g ∈ [W ˚ p ]m the condition (5) is satisfied and Let [W ˚ p(q) ]m is determined by the equality the norm in [W

∥g ∥p,q = ∥g (q) ∥[Lp ]m . We denote by [Lp,q ]m the image of space [Lp ]m with mapping P + t −q Q with the same norm as in [Lp ]m . We formulate Lemmas 1 and 2 from [23]. We use these lemmas for proving of convergence theorems. ˚ p(q) ]m → [Lp,q ]m , (Dq g )(t ) = g (q) (t ) is continuously invertible and its inverse Lemma 1. The differential operator Dq : [W ( q ) −q ˚ p ]m is determined by the equality operator D : [Lp,q ]m → [W

(D−q g )(t ) = (N + g )(t ) + (N − g )(t ),   (−1)q q −1 + (Pg )(τ )(τ − t ) log 1 − (N g )(t ) = 2π i(q − 1)! Γ   (−1)q−1 (N − g )(t ) = (Qg )(τ )(τ − t )q−1 log 1 − 2π i(q − 1)! Γ

t



τ τ t

dτ , dτ .

From Lemma 1 follows: (q)

˚ p ]m → [Lp ]m , B = (P + t q Q )Dq is invertible and Lemma 2. The operator B : [W B−1 = D−q (P + t −q Q ). The proofs of Lemmas 1 and 2 can be found in [23]. The convergence of the collocation method and the mechanical quadrature method are given in the following theorems. Theorem 3. Let the following conditions be satisfied:

Γ ∈ C (2, ν), 0 < ν < 1; MF Ar (t ) and Br (t ) belong to the space [Hα (Γ )]m×m , 0 < α < 1, r = 0, . . . , q; det Aq (t ) · det Bq (t ) ̸= 0, t ∈ Γ ; 1 all left partial indexes of MF t q B− q (t )Aq (t ) are equal to zero; MF Kr (t , τ ) (r = 0, . . . , q) ∈ Hβ [(Γ × Γ )]m×m , 0 < β ≤ 1, and VF f (t ) ∈ [C (Γ )]m ; ˚ p(q) ]m → [Lp (Γ )]m is linear and invertible; (6) the operator M : [W (7) the points tj (j = 0, . . . , 2n) form a system of Fejér knots on Γ (see [24,25]): (1) (2) (3) (4) (5)



tj = ψ exp

2π i



2n + 1

(j − n)



,

j = 0, . . . , 2n, i2 = −1.

Then, there exists N1 such that, for n ≥ N1 , SLAE (11) of the collocation method has the unique solution ξk (k = −n, . . . , n). The ˚ p(q) ]m to the exact approximate solutions xn (t ), constructed using formula (7), converge when n → ∞ in the norm of space [W solution x(t ) of the problem (4)–(5) and the following estimate for the convergence speed holds:

 ∥x − xn ∥p,q = O

1 nα



      1 1 def +O ω f; + O ω t h; = δn . n

n

(13)

MF h(t , τ ) is a continuous MF relative to t and τ on Γ . The obvious form of h(t , τ ) is large and can be found in [26]. Theorem 4. Let all conditions (1)–(7) of Theorem 3 be satisfied. Then, there exists N2 (≥N1 ) such that, for n ≥ N2 , SLAE (12) has ˚ p(q) ]m to the exact a unique solution ξk , k = −n, . . . , n. The approximate solutions (7) converge when n → ∞ in the norm [W solution x(t ) of the problem (4)–(5) and the following estimation for the convergence is true:

   1 ∥x − xn ∥p,q = δn + O ωτ h; . n

(14)

4. Auxiliary results We formulate one result from [26], establishing the equivalence (in the sense of solvability) of problem (4)–(5) and the dq (Px)(t ) dq (Qx)(t ) system of SIE. We use this result for proving Theorems 3 and 4. VF dt q and dt q can be represented by integrals of

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I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

Cauchy type with the same density v(t ): dq (Px)(t )

 v(τ ) +  d τ , t ∈ F  dt q 2π i Γ τ − t . q −q  d (Qx)(t ) t v(τ ) −   = d τ , t ∈ F dt q 2π i Γ τ − t 

1

=

(15)

Using the integral representation (15) we reduce the problem (4)–(5) to a form equivalent (in terms of solvability) to the system of SIE

(Υ v ≡)C (t )v(t ) +

D(t )



πi

Γ

for unknowns VF v(t ) where C (t ) =

1

v(τ ) 1 dτ + τ −t 2π i

[Aq (t ) + t −q Bq (t )],

D(t ) =

 Γ

h(t , τ )v(τ )dτ = f (t ),

t ∈ Γ,

(16)

1

[Aq (t ) − t −q Bq (t )]. (17) 2 2 From the condition (5) of Theorem 3, h(t , τ ) is a MF from [C (Γ × Γ )]m×m , by both variables. The obvious form of h(t , τ ) is large and is given in [26]. Lemma 5. The system SIE (16) and problem (4)–(5) are equivalent in terms of solvability. That is, for each solution v(t ) of system SIE (16), there is a solution of problem (4)–(5) determined by the formulae



(−1)q (Px)(t ) = 2π i(q − 1)!



(−1)q (Qx)(t ) = 2π i(q − 1)!



v(τ ) (τ − t )

Γ

q−1

 log 1 −

 Γ

v(τ )τ

−q

(τ − t )

q−1



t



τ

log 1 −

+

q−1 

 αk τ

q−k−1 k

t

dτ ,

k=1

τ t

+

q −2 

 βk τ

q−k−1 k

t

dτ ,

(18)

k=1

(αk , k = 1, . . . , q − 1, and βk , k = 1, . . . , q − 2, are real numbers). On the other hand, for each solution x(t ) of the problem (4)–(5), there is a solution v(t )

v(t ) =

dq (Px)(t ) dt q

+ tq

dq (Qx)(t ) dt q

,

to the system of SIE (16). Furthermore, for linearly independent solutions of (16), there are corresponding linearly independent solutions of the problem (4)–(5) from (18) and vice versa. In the formulae of (18), log(1 − t /τ ) and log(1 − τ /t ) (for given τ ), there are branches that vanish at the points t = 0 and t = ∞ respectively. 5. Proofs of theorems In this section we prove Theorems 3 and 4. Proof of the Theorem 3. Using the conditions (8) Rn (tj ) = 0,

j = 0, . . . , 2n,

(19)

we obtain that (11) is equivalent to the operator equation Un MUn xn = Un f ,

(20)

where M is an operator defined in (4). We should n (≥N1 ) is  large enough, the operator Un MUn is invertible.  show that if n The operator acts from the subspace [X˚ n ]m = t q αk t k + −1 αk t k (the norm as in [W˚ p(q) ]m ) to the subspace k=0

k=−n

n

[Xn ]m =



rk t k ,

t ∈ Γ,

k=−n

(the norm as in [Lp (Γ )]m , and rk are arbitrary complex vector numbers). dq (Pxn )(t ) dq (Qxn )(t ) In similar way, using the formulae (10) the VF and are given by Cauchy type integrals with the same dt q dt q density vn (t ): dq (Pxn )(t )

 vn (τ ) +  d τ , t ∈ F  dt q 2π i Γ τ − t  . dq (Qxn )(t ) vn (τ ) t −q −   = d τ , t ∈ F dt q 2π i Γ τ − t =

1



(21)

I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

3801

Using formulae (9) and relations (10) we obtain from (21)

vn (t ) =

n  (k + q)!

k!

k=0

t k ξk + (−1)q

n  (k + q − 1)!

(k − 1)!

k=1

t −k ξ−k

and so vn (t ) ∈ [Xn ]m , t ∈ Γ . Using (20), (21) as well as the problem (4)–(5) can be reduced to an equivalent equation (in the sense of solvability): U n Υ U n vn = U n f ,

(22)

considered as an equation in the subspace [Xn ]m . Obviously, (22) is the equation of the collocation method for the system SIE (16). The collocation method was considered in [15–17], where sufficient conditions for the solvability and convergence of this method were obtained. From (21) and vn (t ) ∈ [Xn ]m we conclude that if VF vn (t ) is the solution of Eq. (22) then VF is the discrete solution yn (t ) of the system Un MUn xn = Un f and vice versa. We can determine the VF yn (t ) from the relations (18):



(−1)q (Pyn )(t ) = 2π i(q − 1)!



(−1)q (Qyn )(t ) = 2π i(q − 1)!



vn (τ ) (τ − t )

q −1

Γ

 log 1 −

 Γ

t

 +

τ

q −1 

 αk τ

q−k−1 k

t

dτ ;

k=1



vn (τ )τ −q (τ − t )q−1 log 1 −

τ t

+

q −1 

 βk τ q−k−1 t k dτ ;

(23)

k=1

As was mentioned above, the VF yn (t ) is determined through vn (t ) from (23) uniquely. It follows that if (22) has a unique solution vn (t ) in subspace [Xn ]m , then the following relation is true: yn (t ) = xn (t ).

(24)

From the conditions (3), (4), (6) of Theorem 3, Lemmas 1 and 2, the invertibility of operator Υ : [Lp (Γ )]m → [Lp (Γ )]m follows. We should show that for (22) all conditions of Theorem 1 from [15,16] are satisfied. Theorem 1 from the citation gives the convergence of the collocation method for systems SIE in spaces [Lp (Γ )]m . From condition 3 of Theorem 1 in [16] and from (17) we obtain condition 3 of Theorem 3. From the equality 1 [C (t ) − D(t )]−1 [C (t ) + D(t )] = t q B− q Aq (t ),

we conclude that the partial indices of the MF [C (t )− D(t )]−1 [C (t )+ D(t )] are equal to zero, which coincides with condition 4 of Theorem 3. Other conditions of Theorem 3 coincide with conditions of Theorem 1 of [16]. Conditions (1)–(6) in Theorem 3 provide the validity of all conditions of Theorem 1 in [16]. Therefore beginning with numbers n ≥ n1 , (22) is uniquely solvable. The approximate solutions vn (t ) of (22) converge to the exact solution of the system of (16) in the norm of space [Lp (Γ )]m as n → ∞. Therefore the system Eqs. (20) and the SLAE (11) have unique solutions for (n ≥ N1 ). From Theorem 1 of [16] the following estimation holds:

∥v − vn ∥[Lp ]m ≤ O



1 nα



      1 1 +O ω f; + O ω t h; . n

n

(25)

From (15) and (23) we obtain

(Px)(ν) (t ) = (P v)(t ),

(Qx)(ν) (t ) = t −ν (Q v)(t ).

Therefore we have

(Pxn )(ν) (t ) = (P vn )(t ),

(Qxn )(ν) (t ) = t −ν (Q vn )(t ).

We proceed to get an error estimate:

∥x − xn ∥p,ν = ∥x(ν) − x(ν) n ∥[Lp ]m ≤ ∥P (v − vn )∥[Lp ]m + ∥t −ν Q (v − vn )∥[Lp ]m ≤ ∥P ∥ · ∥v − vn ∥[Lp ]m + ∥t −ν ∥∥Q ∥ · ∥v − vn ∥[Lp ]m ≤ (∥P ∥ + ∥t −ν ∥∥Q ∥)∥v − vn ∥[Lp ]m .

(26)

Using the inequality

∥t −ν ∥Lp =

  1 l

|t −ν |p dt Γ

 1p

  =

1 l

|t −ν p |dt Γ

 1p

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I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

 1p

 1 ≤

1

l min |t |pν

 1p

 =

l

t ∈Γ

1 min |t |pν

 = c1 ,

t ∈Γ

from (25) and (26), we obtain (13). Theorem 3 is proved.



Proof of the Theorem 4. It is easy to verify that SLAE (12) is equivalent to the operational equation Un

  q 

1

r =0

2π i

Ar (t )(Px(nr ) )(t ) + Br (t )(Qx(nr ) )(t ) +

+



1 2π i

Γ

1 (τ ) −r −1 Un [τ K (t , τ )](Qx(nr ) )(τ )dτ

τ

 Γ

1 (τ ) q+1−r Un [τ K (t , τ )](Px(nr ) )(τ )dτ

τ

 = Un f ,

(27)

which after the application of the integral representation (23) is equivalent (in the same sense of solvability) to the operator equation



1

Un C (t )vn (t ) + D(t )(S vn )(t ) +

2π i

 Γ

1 (τ ) Un [τ h(t , τ )] · vn (τ )dτ



τ

= Un f ,

(28)

where the VF C (t ), D(t ) are determined above and MF h(t , τ ) is determined in [26]. Eq. (27) represents an equation for the mechanical quadrature method for system SIE (16). It is easy to verify (as in the proof of Theorem 3) that the conditions of Theorem 4 provide the validity of all conditions of Theorem 2 from [17,15] (for the mechanical quadrature method). It follows that (28) is uniquely solvable for n ≥ N2 . Moreover, the approximate solutions vn (t ) ∈ [Xn ]m of this equation converge to the exact solution v(t ) of system SIE (16) in the norm [Lp (Γ )]m as n → ∞ and the following estimation is true:

∥v − vn ∥[Lp (Γ )]m = O



1



nα

         1 1 1 +O ω f; + O ω τ h; + O ω t h; . n

n

n

(29)

The VF xn (t ) can be expressed via the VF vn (t ) using formula (23). Using the definition of the norm in the space [Lp (Γ )]m , and the relations (18), (24), (29), we obtain (14). Theorem 4 is proved.  Remark 1. The stability of the collocation method was proved in [21]. 6. A numerical experiment Let us consider the SIDE (4) for m = 1. In this example we take the exact solution as x(t ) =

1 t −1

.

The coefficients are chosen as follows:

 +1 , 2 2 t t    1 1 1 1 B˜ r (t ) = t+ − −1 ,

A˜ r (t ) =

1



t+

2

1

2

−

1

t



1

t

(t + r + 1) Kr ( t , τ ) = , τ r = 0, . . . , 1. The right part is calculated automatically in the program. The contour Γ is an ellipse R1 ∗ cos(ϕ)+ i ∗ R2 ∗ sin(ϕ). In this example R1 = 4, R2 = 11. The program is written in Matlab. 1 It is easy to verify the conditions of Theorem 3. To calculate the index of the function t q B− q (t )Aq (t ) we use the numerical algorithm from [7]. To construct the Fejér points we build the conformal mapping function (1) using the numerical algorithm from [27]. In our test, the non-collocation points have been obtained from the formula

 t˜j = R1 ∗ cos

2π ∗ (j − 1) k

+

π 16



 + i ∗ R2 ∗ sin

2π ∗ (j − 1) k

+

π 16



,

where k is an integer and j = 1, . . . , k + 1. We observe that we should take enough collocation points to guarantee the convergence (Table 1). Remark 2. Similar numerical results have been obtained when the contour Γ is:

I. Caraus, F.M. Al Faqih / Journal of Computational and Applied Mathematics 236 (2012) 3796–3804

3803

Table 1 The error between the exact solution x(t˜j ) and the approximate solution xn (t˜j ) is given at selected points t˜j , j = 1, . . . , k + 1, for different numbers n of collocation points. The error is the largest error in the magnitude of all selected points. No. of collocation points

Error

9 15 19 21

0.0015 5.4350e−04 4.4102e−05 4.6371e−07

  • an epitrochoid: x = (R + r ) ∗ cos(ϕ) − d ∗ cos (R+r r ) ϕ ,   (R + r ) y = (R + r ) ∗ sin(ϕ) − d ∗ sin ϕ , R = 3, d = 0.5, r = 1; r

• a limaçon: x =

a 2

+ b cos(ϕ) + cos(2 ∗ ϕ),

y = b sin(ϕ) +

a 2

a 2

sin(2 ∗ ϕ),

a = 2.3, b = 5.

7. Conclusion In this paper we have proposed numerical schemes for the collocation method and the mechanical quadrature method for solving systems of SIDE. The equations are defined on an arbitrary smooth closed contour. The convergence of these methods in Lebesgue spaces was proved. A numerical example illustrates the performance of the collocation method and the mechanical quadrature method. References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

J. Cohen, O. Boxma, Boundary Value Problems in Queueing System Analysis, North-Holland, Amsterdam, 1983, p. 405. A. Kalandia, Mathematical Methods of Two-Dimensional Elasticity, Mir, Moscow, 1975, p. 351. A. Linkov, Boundary Integral Equations in Elasticity Theory, Kluwer, Dordrecht, 2002, p. 268. N. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion, and Bending, Noordhoff, Leyden, The Netherlands, 1953, p. 704. E. Ladopoulos, Singular Integral Equations: Linear and Non-linear Theory and its Applications in Science and Engineering, Springer, Berlin, Heidelberg, New York, 2000, p. 551. V. Ivanov, The Theory of Approximative Methods and their Application to the Numerical Solution of Singular Integral Equations, Noordhoff, Leyden, The Netherlands, 1976, p. 330. F. Gakhov, in: I.N. Sneddon (Ed.), Boundary Value Problems, Pergamon, Addison-Wesley, Oxford, Reading, MA, 1966, p. 561. N. Muskelishvili, Singular Integral Equations. Boundary Problems of Function Theory and their Application to Mathematical Physics, Noordhoff, Leyden, The Netherlands, 1977 (Edition: Rev. translation from the Russian/edited by Radok, J.R.M.: Reprint of the 1958, p. 447, edition. ISBN: 900160700419). N. Vekua, Systems of Singular Integral Equations, in: Groningen Material, Noordhoff, 1967, p. 216 (Translated from the Russian by Gibbs, A.G. and Simmons, G.M.). I. Gohberg, I. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators, Ştiinţă, Chişinău, 1973 (in Russian), German translation: Birkhauser Verlag, Basel, 1979. S. Prössdorf, B. Silbermann, Numerical Analysis for Integral and Related Operator Equations, Akademie-Verlag, Birkhauser Verlag, Berlin, Basel, 1991, p. 542. S. Mikhlin, S. Prössdorf, Singular Integral Operators, Springer-Verlag, Berlin, 1986, p. 528. S. Prössdord, Some Classes of Singular Equations, Elsevier, North-Holland, 1978, p. 417. B. Gabdulalhaev, The polynomial approximations of solution of singular integral and integro-differential equations by Dzyadik, Izv. Vyssh. Uchebn. Zaved. Mat. N6 (193) (1978) 51–62 (in Russian). V. Zolotarevski, Finite-Dimensional Methods for Solving of Singular Integral Equations on the Closed Contours of Integration, Ştiinţă, Chişinău, 1991, p. 136 (in Russian). V. Zolotaveski, Approximate solution of systems of singular integral equations on some smooth closed contours in Lp spaces, Izv. Vyssh. Uchebn. Zaved. Mat. N2 (1989) 79–82; Translation in Soviet Math. (Iz. VUZ) 33 N2 (1989) 100–105. V. Zolotarevski, Direct methods for solving singular integral equations on closed smooth contours in spaces Lp , Rev. Anal. Numér. Théor. Approx. 25 (1–2) (1996) 257–265. V. Zolotarevski, Zhilin Li, Iu. Caraus, Approximate solution of singular integro-differential equations over Faber–Laurent polynomials, Differ. Equ. 40 (12) (2004) 1764–1769 (Translated from Differ. Uravn. 40 (12) (2004) 1682–1686). Iu. Caraus, The numerical solution for systems of singular integro-differential equations by Faber–Laurent polynomials, in: NAA 2004, in: Lecture notes in Computer Science, vol. 3401, Springer, Berlin, Heidelberg, New York, (ISSN: 0302-9743), 2005, pp. 219–223. Iu. Caraus, Feras M. Al Faqih, Approximate solution of singular integro-differential equations in generalized Hölder spaces, Numer. Algorithms (2007) 205–215. Iu. Caraus, N. Mastorakis, Convergence of collocation methods for singular integro-differential equations, WSEAS Trans. Math. 6 (11) (2007) 859–864. G. Litvinchuk, Solvability Theory of Boundary Value Problem and Singular Integral Equations with Shift, in: Mathematics and its Applications, Kluwer Academic Publishers, 2000. R. Saks, Boundary-value problems for elliptic systems of differential equations, University of Novosibirsk, Novosibirsk, 1975 (in Russian).

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[24] V. Smirnov, N. Lebedev, Functions of a Complex Variable. Constructive Theory, Scripta Technica Ltd., London, 1968, Trans., Iliffe. [25] P. Novati, A polynomial method based on Fejér points for computation of functions of unsymmetric matrices, Appl. Numer. Math. 44 (2003) 201–224. [26] Yu. Krikunov, The general boundary Riemann problem and linear singular integro-differential equation, The Scientific Notes of the Kazani University, Kazani, vol. 116 (4), 1956, pp. 3–29 (in Russian). [27] T.A. Driscoll, Algorithm 756: a MATLAB toolbox for Schwartz–Christoffel mapping, ACM Trans. Math. Software 22 (1996) 168–186.