2 The integral equation for the derivative of the displacement

A numerical approach for a special crack problem

Peter Junghanns^a·Robert Kaiser^a

Communicated by G. Milovanovi´c and D. Occorsio

Abstract

A collocation-quadrature method is proposed and studied for the numerical solution of a singular integral equation concerned with the two-dimensional elasticity problem of a crack at a circular cavity surface.

These investigations are based on anC^∗-algebra approach presented in a recent paper[5]on the numerical solution of an integral equation for the notched half-plane problem.

1 Introduction

In[11, (37.5)], the integral equation 1 π

Z1+2L 1

• 1

s−t+k₀(t,s)˜

v₀(s)ds=f₀(t) +C, 1<t<1+2L, (1) is given for studying the crack problem, which considers a circular hole of radius 1 and radial cut of length 2Lat the surface of this hole in an elastic plane, which is subjected at infinity to tensile forcesPperpendicular to the cut. Here,

k0(t,s) =(t−s)(t²−1) ts(ts−1)³ − 1

s³(ts−1)− 1

ts² and f0(t) =P

 1 4t³+ 1

4t− t 2

‹

(2)

P

P 1

0

t 1+2L

The unknown functionv0(t)of equation (1) measures the normal displacement of the face of the cut and has to satisfy the condition

v₀(1+2L) =0 . (3)

Also the constantC∈Ris unknown. In case ofL=0.5 , equation (1) takes the form (cf. also[2, (14.7)]) 1

π Z1

0

• 1

y−x +k₀(1+x, 1+y)˜

v(y)d y=f₀(1+x) +C, 0<x<1 , (4) together with the condition

v(1) =0 , (5)

aChemnitz University of Technology, Faculty of Mathematics, Reichenhainer Str. 39, D-09107 Chemnitz, Germany, [email protected], [email protected]

wherev(x) =v₀(1+x)and

k₀(1+x, 1+y) = (x−y)x(x+2)

(x+1)(y+1)(y+x+x y)³− 1

(y+1)³(y+x+x y)− 1 (x+1)(y+1)², f0(1+x) =P

• 1

4(1+x)³+ 1

4(1+x)−1+x 2

˜ . For the general caseL>0 , we get

1 π

Z1

−1

• 1

y−x +k₀^L(x,y)˜

v(y)d y=f₀^L(x) +C, −1<x<1 , (6) together with (5),v(x) =v0(1+L(1+x)), and

k₀^L(x,y) =L k₀(1+L(1+x), 1+L(1+y)), f₀^L(x) =f₀(1+L(1+x)).

It turns out that the unknown constantCin (1) leads to problems in handling this operator equation as well analytically as numerically (cf. also the discussion in Section4). For that reason, in Section2we transform equation (1) into an integral equation the unknown function of which is the derivative of the normal displacement function. In Section3we propose a collocation-quadrature method for the numerical solution of this integral equation and study the stability of this method, basing on aC^∗-algebra approach for an integral equation of the notched half-plane problem presented in[5]. Section4contains a discussion of the numerical results obtained with the method of the present paper in comparison with results available from the literature. In Section5, we give the technical proof of Lemma2.2.

Note, that we do not loose information on the solution of (1) by transforming this equation into an equivalent one for the derivative of the normal displacement functionv₀(t). Of course,v₀(t)can be recovered from its derivative by integration.

Moreover, the important stress intensity factor att=1+2Lcan also be computed directly fromv₀⁰(t)(cf. (34)).

2 The integral equation for the derivative of the displacement

By elementary calculations one can see that, fort,s>1 and for ek₀(t,s) = t

ts−1−

 t+1

t − 2 t³

‹ 1 (ts−1)²+

t−2 t + 1

t³

‹ 1 (ts−1)³−

 1+ 1

t²

‹ 1 s+1

t

‹

, (7)

we have

∂k₀(t,s)

∂t =∂ek₀(t,s)

∂s and ek₀(t, 1) = 1

1−t. (8)

We assume that the homogeneous equation (6) (i.e., f₀^L(x) +C≡0) has only the trivial solution in the space \

1<p<∞

L^p(−1, 1). Then, the following lemma holds ([2, Section 14, 2^oand Theorem 14.1]).

Lemma 2.1. Equation(6)has a unique solution v∈ \

1<p<∞

L^p(−1, 1).This solution satisfies(5)and possesses a generalized derivative v⁰∈ [

1<p<∞

L^p(−1, 1),where both v(x)and v⁰(x)are bounded in a neighbourhood of x=−1and belong toC^∞(−1, 1).Moreover, p1−x v⁰(x)is locally Hölder continuous in each point of(−1, 1]with Hölder exponent¹₂.

The proof of the following lemma is given in the appendix Section5.

Lemma 2.2. For x>0 ,y≥0 ,and the functionek₀(x,y)in(7), the repesentation ek₀(1+x, 1+y) = 1

y+x − 6x

(y+x)²+ 4x²

(y+x)³−h(x,y) (9)

holds with a bounded and continuously differentiable function h:(0,∞)×[0,∞)−→R. Taking into account Lemma2.1and the formula ([12, Chapter II, Lemma 6.1])

d d t

Z1+2L 1

v₀(s)ds s−t = v₀(1)

1−t − v₀(1+2L) 1+2L−t +

Z1+2L 1

v₀⁰(s)ds s−t

(3)= v₀(1) 1−t +

Z1+2L 1

v₀⁰(s)ds s−t , we get

d d t

Z1+2L 1

• 1

s−t+k0(t,s)

˜

v0(s)ds ⁽⁸⁾= d d t

Z1+2L 1

v₀(s)ds s−t +

Z1+2L 1

∂ek₀(t,s)

∂s v0(s)ds

= v₀(1)

1−t −ek0(t, 1)v0(1) + Z1+2L

1

• 1

s−t−ek0(t,s)

˜ v₀⁰(s)ds

(8)= Z1+2L

1

• 1

s−t −ek0(t,s)

˜ v₀⁰(s)ds.

Consequently, instead of (6) we can consider the integral equation 1

π Z1

−1

• 1

y−x+ek₀^L(x,y)˜

u₀(y)d y=g₀^L(x), −1<x<1 , (10) whereg₀^L(x) =f₀⁰(1+L(1+x)),

ek₀^L(x,y) =−Lek0(1+L(1+x), 1+L(1+y)), and u0(x) =v₀⁰(1+L(1+x)). Exploring (9), equation (10) takes the form

1 π

Z1

−1

• 1

y−x − 1

2+y+x+ 6(1+x)

(2+y+x)²− 4(1+x)²

(2+y+x)³+h₀^L(x,y)

˜

u0(y)d y=g₀^L(x), (11)

−1<x<1 , where

h₀^L(x,y) =L h(L(1+x),L(1+y)). (12) We write (11) as

(A₀+H₀)u₀=g₀^L, (13)

where

(A₀u₀) (x) = 1 π

Z1

−1

• 1 y−x +h0

1+x 1+y

‹ 1 1+y

˜

u₀(y)d y

withh0(t) =− 1

1+t+ 6t

(1+t)²− 4t² (1+t)³ and

(H₀u₀) (x) = 1 π

Z1

−1

h₀^L(x,y)u₀(y)d y. By using[2, Theorem 9.1], it was already mentioned in[1, Corollary 2.3]that the operator

A₀:L²_ϕ→L²_ϕ (14)

is a bounded and invertible one, whereϕ(x) =p

1−x². Here, for a Jacobi weightρ(x) =v^α,β(x):= (1−x)^α(1+x)^β, byL²_ρ there is denoted the Hilbert space of all (classes of) functions which are square integrable w.r.t. the weightρ(x), equipped with the inner product and the respective norm

〈f,g〉_ρ:=

Z1

−1

f(x)g(x)ρ(x)d x and kfk_ρ:=q

〈f,f〉_ρ.

Let us denote the associated normalized orthogonal polynomial (with positive leading coefficient) of degreenbyp^ρ_n(x). Due to Lemma2.1, the solution of (11) can be written in the form

u₀(x) = eu0(x) p1−x

with a bounded and locally Hölder continuous functioneu₀:(−1, 1]−→C, which is infinitely differentiable on(−1, 1). For that reason we are interested in approximate solutions of (11) of the form

p_n(x)

p1−x, (15)

wherep_n(x)is an algebraic polynomial of degree less thann. The results on the stability of polynomial collocation methods [8,6](cf. also[9,10,4]) and collocation-quadrature methods[7]for Cauchy singular integral equations with additional fixed singularities of Mellin-type like in (11) do not cover the case (15), since in all these mentioned papers the approximate solution is of the form(1−x)^γ(1+x)^δp_n(x)withγ6=0 andδ6=0 . Hence, we follow the approach described in[1, Section 2]and[5, Section 1]and use the isometrical isomorphismJ₀:L²_ϕ−→L²_µ,f 7→p

1+.f, where againϕ(x) =p

1−x²andµ(x) = v t1−x

1+x, to get the following equation equivalent to (13),

(A+H)u=g (16)

withA=J₀A₀J₀⁻¹:L²_µ−→L²_µ,H=J₀H₀J₀⁻¹:L²_µ−→L²_µ,g=J₀g₀^L∈L²_µ, andu=J₀u₀. It follows (Au) (x) = 1

π Z1

−1

• 1 y−x +h

1+x 1+y

‹ 1 1+y

˜ u(y)d y

whereh(t) =p

th0(t)− 1 1+p

t, and (Hu) (x) = 1

π Z1

−1

h^L(x,y)u(y)d y, h^L(x,y) = v t1+x

1+yh^L₀(x,y), g(x) =p

1+x g₀^L(x). (17)

In the following lemma we collect some mapping properties of the operators involved in equation (16). For this, as usual by C[−1, 1]we denote the Banach space of all continuous functionsf:[−1, 1]−→Cequipped with the supremum normkfk_∞. Note that the operatorAis the operator of an integral equation for the notched half-plane problem, for which a collocation-quadrature method (which we introduce in the following section) is studied in[5]. This enables us to use the results from[5]to study the properties of this numerical method applied to equation (16).

Lemma 2.3. With the above notations the following holds.

(a) The operatorA:L²_µ−→L²_µis bounded and invertible.

(b) The operatorsH:L²_µ−→C[−1, 1]andH:L²_µ−→L²_µare compact.

Proof. Assertion (a) is a consequence of the already mentioned boundedness and invertibility of the operator (14). Since the function[−1, 1]²−→R,(x,y)7→p

1+x h₀^L(x,y)is continuous (see Lemma2.2), the estimates

|(Hu)(x)|=

Z1

−1

p1+x h₀^L(x,y)u(y)d y p1+y

≤

p1+x h₀^L(x, .) ∞

v u t

Z1

−1

d y p1−y² kuk_µ and

|(Hu)(x₁)−(Hu)(x₂)| =

Z1

−1

p1+x₁h₀^L(x₁,y)−p

1+x₂h₀^L(x₂,y)u(y)d y p1+y

≤

p1+x₁h₀^L(x₁, .)−p

1+x₂h^L₀(x₂, .) ∞

v u t

Z1

−1

d y p1−y² kukµ

show that the set¦

Hu:u∈L²_µ,kuk_µ≤1©

is a uniformly bounded and equicontinuous set of functions. Hence,H:L²_µ−→C[−1, 1] is compact together withH:L²_µ−→L²_µin virtue of the continuous imbeddingC[−1, 1]⊂L²_µ. tu

3 A collocation-quadrature method

Here, we describe a collocation-quadrature method for the operator equation (16). Letx^σ_kn=cos2k−1

2n π,k=1, . . . ,n,n∈N denote the Chebyshev nodes of first kind and

L^σ_nf (x) =

n

X

k=1

f(x^σ_kn)`^σ_kn(x)

the respective interpolation polynomial of a function f :(−1, 1)−→C, where`^σ_kn(x)are the usual fundamental Lagrange interpolation polynomials w.r.t. these nodes. Moreover, letM^σ_n=νL^σ_nµIbe the weighted interpolation operator given by

M^σ_nf

(x) =ν(x) L^σ_nµf (x)

withν(x) = v t1+x

1−x the Chebyshev weight of third kind. Finally, letL_n:L²_µ−→L²_µdenote the orthogonal projection L_nf=

n−1

X

j=0

f,ep_j

µep_j,

whereep_n=νp^ν_n,n=0, 1, . . . forms a complete orthonomal system inL²_µ. In its image space imL_nwe look for an approximate solutionu_nby solving

(A_n+H_n)u_n=M^σ_ng, (18)

where the operatorsA_n=M^σ_n S+B⁰_n

L_nandH_n=M^σ_nH⁰_nL_nare defined by (Su_n) (x) = 1

π Z1

−1

u_n(y)d y y−x , B⁰_nu_n

(x) = 1 π

Z1

−1

•

L^σ_nϕh1+x 1+.

‹ u_n 1+.

˜

(y)σ(y)d y,

H⁰_nu_n

(x) = 1 π

Z1

−1

L^σ_nϕh^L(x, .)u_n

(y)σ(y)d y,

andσ(x) = 1

p1−x². Note that, for the quadrature operators we have, for example, H⁰_nu_n

(x) =1 n

n

X

k=1

ϕ(x^σ_kn)h^L(x,x^σ_kn)u_n(x^σ_kn).

It is well known that, in the investigation of numerical methods for operator equations, the stability of the respective operator sequence plays an essential role. The sequence(A_n+H_n)in (18) is called stable (inL²_µ) if, for all sufficiently largen, the operatorsA_n+H_n: imL_n−→imL_nare invertible and if the norms

(A_n−H_n)⁻¹L_n

L(L²µ)are uniformly bounded. Note that, if the method is stable and if(A_n+H_n)L_nconverges strongly toA+H∈L(L²_µ), then the operatorA+His injective. If additionally the image ofA+HequalsL²_µ, then

g−M^σ_ng

_µ−→0 implies theL²_µ-convergence of the solutionu_nof (18) to the (unique) solutionu∈L²_µof (16).

To investigate the stability of the operator sequence in (18) we follow theC^∗-algebra approach already used in, for example, [4,6,7](cf. also[8,9,10]). Moreover, in[5]a collocation-quadrature method for the numerical solution of an integral equation for the notched half plane problem was studied with the help of this approach. We will consider the operator sequence under consideration as an element of aC^∗-algebra, which we describe in the following. For this, by`²we denote the Hilbert space of all square summable sequencesξ= ξj

∞

j=0,ξj∈Cwith the inner product〈ξ,η〉= X∞

j=0

ξjηj. Moreover, we define the operators

W_n:L²_µ−→L²_µ, u7→

n−1

X

j=0

u,ep_n−1−j

µep_j, P_n:`<sup>2</sup>−→`², (ξj)_j=0^∞7→(ξ0,· · ·,ξn−1, 0, . . .), and

V_n: imL_n−→imP_n, u_n7→

π n

Æ1−x^σ_1nu_n(x^σ_1n), . . . ,π n

Æ1−x^σ_nnu_n(x^σ_nn), 0, . . . , Ve_n: imL_n−→imP_n, u_n7→π

n

Æ1−x^σ_nnu(x_nn^σ), . . . ,π n

Æ1−x^σ_1nu(x_1n^σ), 0, . . . . LetT={1, 2, 3, 4}, set

X⁽¹⁾=X⁽²⁾=L²_µ, X⁽³⁾=X⁽⁴⁾=`², L⁽¹⁾_n =L⁽²⁾_n =L_n, L⁽³⁾_n =L⁽⁴⁾_n =P_n, and defineE_n^(t): imL_n−→X^(t)_n :=imL^(t)_n fort∈T by

E_n⁽¹⁾=L_n, E_n⁽²⁾=W_n, E_n⁽³⁾=V_n^σ, E_n⁽⁴⁾=Ve_n^σ.

Here and at other places, we use the notionL_n,W_n, . . . instead ofL_n|imL_n,W_n|imL_n, . . . , respectively. All operatorsE_n^(t),t∈T are unitary with the inverses

E_n⁽¹⁾−1

=E_n⁽¹⁾, E⁽²⁾_n −1

=E⁽²⁾_n , E_n⁽³⁾−1

=V_n⁻¹, E_n⁽⁴⁾−1

=Ve_n⁻¹, where, forξ∈imP_n,

V_n⁻¹ξ=

n

X

k=1

nξk−1

πp

1−x^σ_kne`^σkn and Ve_n⁻¹ξ=

n

X

k=1

nξn−k

πp

1−x^σ_kne`^σkn, and where

e`^σ_kn(x) = ν(x)

ν(x^τ_kn)`^σ_kn(x), k=1, . . . ,n,

are the weighted fundamental interpolation polynomials. It is easily seen that, for all indicesr,t∈T withr6=t, the operators E_n^(r) E^(t)_n −1

L^(t)_n (19)

as well as their adjoints converge weakly to zero (cf., for example, the proof of[4, Lemma 2.1]). Now we can introduce the algebra of operator sequences we are interested in. ByFwe denote the set of all sequences(A_n)of linear operatorsA_n: imL_n−→imL_n for which the strong limits

W^t(A_n):= lim

n→∞E_n^(t)A_n E^(t)_n −1

L^(t)_n and

(W^t(A_n))^∗= lim

n→∞

€E_n^(t)A_n E_n^(t)−1

L^(t)_n Š∗

, t∈T,

exist. IfFis provided with the supremum normk(A_n)kF:=sup_n≥1kA_nL_nk_L(L²_ν)and with the operations(A_n) + (B_n):= (A_n+B_n), (A_n)(B_n):= (A_nB_n), and(A_n)^∗:= (A^∗_n), thenF_{becomes a}C^∗-algebra with the identity element(L_n). Furthermore, we introduce the setJ_⊂Fof all sequences of the form

‚ ₄ X

t=1

E_n^(t)−1

L^(t)_n T_tE_n^(t)+C_n

Π,

where the linear operatorsT_t:X^(t)−→X^(t)are compact and where the sequence(C_n)∈Fbelongs to the closed idealGof all sequences fromFtending to zero in norm, i.e., lim

n→∞kC_nL_nk_L(L²_ν)=0 . From[13,14, Theorem 10.33](see also[3, Theorem 6.1]) we infer the following proposition.

Proposition 3.1. The setJforms a two-sided closed ideal in the C^∗-algebraF. Moreover, a sequence(A_n)∈Fis stable if and only if the operatorsW^t(A_n):X^(t)−→X^(t), t∈T and the coset(A_n) +J_∈F_/Jare invertible.

LetA₀denote the smallestC^∗-subalgebra ofFcontaining all sequences fromJand all sequences(A_n)with A_n=M^σ_n(aI+bS+B⁰_n)L_n,

wherea,bare piecewise continuous functions on[−1, 1], whereI:L²_µ−→L²_µis the identity operator, and whereSas well asB⁰_n are defined after (18). Then, the following is proved in[5, Theorem 3.12].

Proposition 3.2. A sequence(A_n)∈A₀is stable inL²_µif and only if all limit operatorsW^t(A_n):X^(t)−→X^(t),t=1, 2, 3, 4are invertible.

The following statement we infer from[5, Section 4].

Lemma 3.3. LetA_nbe the operators in(18), i.e.,A_n=M^σ_n(S+B_n⁰)L_n.Then, the limit operatorsW¹(A_n),W²(A_n):L²_µ−→L²_µ andW³(A_n):`<sup>2</sup>−→`²are invertible, and the fourth limit operatorW⁴(A_n):`<sup>2</sup>−→`²is Fredholm with index0 .

Now, we are turning to study the remaining part of the operator sequence involved in equation (18), namely(H_n). First, we remark that (see[4, Corollary 3.3])

nlim→∞

f −L^σ_nf

_ν=0 for all f∈C_γ,δ, (20)

where 0≤γ < ¹4, 0≤δ < ³4, where againν=µ⁻¹, and whereC_γ,δdenotes the Banach space consisting of all continuous functions f :(−1, 1)−→Cfor whichv^γ,δf :[−1, 1]−→Cis continuous with v^γ,δf

(1) =0 ifγ >0 and v^γ,δf

(−1) =0 if δ >0 . The norm inC_γ,δis defined by

kfk_γ,δ,∞:=

v^γ,δf ∞. Furthermore (see[5, Corollary 2.12]),

n→∞lim

f−M^σ_nf

_µ=0 for all f ∈C_α,β, (21)

where 0≤α <³₄and 0≤β <¹₄.

Lemma 3.4. Assume that the function[−1, 1]²−→C,(x,y)7→r(x,y)v^α,β(y)is continuous, where0≤α <¹₄ and0≤β < ³₄. Then,

n→∞limsup

L^σ_nr(x, .)−r(x, .)

_ν:−1≤x≤1 =0 .

Proof. Fixγ,δ∈Rsuch thatα < γ <¹4andβ < δ <³4. By assumptionr^x∈C_γ,δ, wherer^x(y) =r(x,y), and

xlim→x₀kr^x−r^x0k_γ,δ,∞=0 for all x₀∈[−1, 1].

Suppose that the assertion of the lemma is not true. Then, there is an" >0 and a sequencen₁<n₂<. . . of natural numbers satisfying

sup n

L^σ_n

kr^x−r^x

_ν:−1≤x≤1 o

≥2" for all k∈N. Hence, for everyk∈N, there is anx_k∈[−1, 1]such that

L^σ_n

kr^xk−r^xk

_ν≥", and we can assume thatx_k−→x^∗ifk−→ ∞. By (20),M:=supn

L^σ_n

C_γ,δ→L²_ν:n∈N

o<∞and

M₀:=

v u t

Z1

−1

(1−x)^−12−2γ(1+x)^12−2δd x<∞. Moreover, there is ak₀∈Nsuch that

L^σ_n

kr^x∗−r^x∗ ν<"

3 and

r^xk−r^x∗

γ,δ,∞< "

3 min{M0,M} for all k≥k₀. It follows, fork≥k0,

"≤ L^σ_n

kr^xk−r^xk _ν ≤

L^σ_n

k(r^xk−r^x∗) _ν+

L^σ_n

kr^x∗−r^x∗ _ν+

r^x∗−r^xk _ν

≤ M

r^xk−r^x∗ _γ,δ,

∞+"

3+M₀

r^xk−r^x∗ _γ,δ,

∞< ",

which is a contradiction. tu

Lemma 3.5. The sequence(H_n)belongs to the idealJof the algebraF.In particular,W¹(H_n) =HandW^t(H_n) =0for t=2, 3, 4 .

Proof. Due toL_n−→Istrongly inL²_µ, due to (21), and due to Lemma2.3,(b), the sequence L_nHL_n−M^σ_nHL_n

belongs to the idealG. Since, foru_n=νp_n∈imL_n,ϕu_n= (1+.)p_nis a polynomial of degree at mostn, we have

H⁰_nu_n (x) = 1

π Z1

−1

L^σ_nh^L(x, .)

(y)p_n(y)ν(y)d y

and consequently, again using (21), M^σ_nHL_nu_n−H_nu_n

_µ ≤ M^σ_n

C→L²_µ

Hu_n−H⁰_nu_n ∞

≤ csup

¨Z1

−1

h^L(x,y)−

L^σ_nh^L(x, .) (y)

|pn(y)|ν(y)d y:−1≤x≤1

«

≤ csup

h^L(x, .)−L^σ_nh^L(x, .)

_ν:−1≤x≤1 kpnk_ν ≤ c_nkunk_µ, where lim

n→∞c_n=0 in virtue of the continuity ofh^L(x,y)p

1+yon[−1, 1]²(cf. Lemma2.2, (12), and (17)) and Lemma3.4.

Hence, also M^σ_nHL_n−H_n

belongs to the idealG. Now, the assertion follows by using the compactness ofH, the strong convergence ofL_n, and the weak convergences of the operators (19) and their adjoints implying(L_nHL_n)∈Jtogether with

W¹(H_n) =HandW^t(H_n) =0 fort=2, 3, 4 . tu

Combining Proposition3.2, Lemma3.3, and Lemma3.5, we get the following stability result.

Proposition 3.6. The sequence(A_n+H_n)of the operators in the collocation-quadrature method(18)is stable inL²_µ,if and only if the homogeneous equation(A+H)u=0has only the trivial solution inL²_µand if also the operator

−S+H:`<sup>2</sup>−→`² (22)

has a trivial null space, where

S=• 1−(−1)^j−k

j−k −1−(−1)^j+k+1 j+k+1

˜∞ j,k=0

and H=

h

(j+¹2)² (k+¹2)²

2 k+¹2

∞ j,k=0

.

Proof. Due to Proposition3.2, the stability of the operator sequence(A_n+H_n)is equivalent to the invertibility of all limit operatorsW^t(A_n+H_n). Since the first operator equalsW¹(A_n+H_n) =A+H(see Lemma3.5and[5, Prop. 2.17]) and since Lemma2.3is in force, the invertibility of this operator is equivalent to the triviality of its null space. Moreover, by Lemma 3.3and Lemma3.5, the second and third limit operators are invertible. The fourth limit operator equals (see[5, Prop. 2.17]) π⁻¹(−S+H):`<sup>2</sup>−→`²and is Fredholm with index zero (see[5, Section 4]), and the proposition is proved. tu Remark1. Here, we focus on the use of the Chebyshev nodes of first kind as collocation nodes, since it is obvious from the results of[5, Section 4]that the collocation-quadrature method based on the Chebyshev nodes of third kind is unstable. For Chebyshev nodes of second and fourth kind, we are not able to prove that the respective operator sequences belong to the corresponding algebraF_.

4 Computational aspects and numerical results

Before presenting numerical results obtained by applying the method introduced in Section3, let us discuss the method and results given in[11]for equation (1) resp. (6). KALANDIYA[11, Section 37]applies directly a collocation-quadrature method to equation (6) after multiplying the unknown function byp

1+yand the equation byp

1+x. Hence, the mentioned method is applied to (cf.[11, (37.8)])

1 π

Z1

−1

• 1

y−x +k_∗(x,y)

˜

v_∗(y)d y=f_∗(x) +Cp

1+x, −1<x<1 , (23)

where

k_∗(x,y) = 1 p1+y

p

1+x k₀^L(x,y)− 1 p1+y+p

1+x

, f_∗(x) =p

1+x f₀^L(x), andv_∗(x) =p

1+x v(x)withv(x)and f₀^L(x)from (6). An approximate solutionv_n(x)forv_∗(x)is searched for in the form (cf.

[11, (37.9),(37.10)])

v_n(x) =

n

X

k=1

ξkne`^ϕkn(x), (24)

wheree`<sup>ϕ</sup><sub>kn</sub>(x) = ϕ(x)`^ϕ_kn(x)

ϕ(x^ϕ_kn) and where`^ϕ_kn(x) = U_n(x)

(x−x^ϕ_kn)U_n⁰(x^ϕ_kn) withU_n(cosθ) = sinnθ

sins are the fundamental Lagrange interpolation polynomials w.r.t. the Chebyshev nodesx^ϕ_kn=cos kπ

n+1,k=1, . . . ,nof second kind. The integral operator with the

kernel functionk_∗(x,y)is approximated with the help of the Gaussian rule w.r.t. thex_kn^ϕ’s. After substitutingv_n(x)into (23) and collocating atx^ϕ_jn,j=1, . . . ,none obtains (by using relation (27) below) a system of linear equations

n

X

k=1

αjkξkn−Ç

1+x^ϕ_jnC=f_∗(x^ϕ_jn), j=1, . . . ,n, (25) where (cf.[11, (37.11)]and also[9, (3.10)])

αjk=

–1−(−1)^j+k

x^ϕ_kn−x^ϕ_jn +k_∗(x^ϕ_jn,x^ϕ_kn)

™ϕ(x^ϕ_kn)

n+1 j,k=1, . . . ,n. (26)

The first addend in the brackets equals 0 in case ofj=k. If one considers equation (23) in the spaceL²_σwithσ(x) = (1−x²)⁻¹² again being the Chebyshev weight of first kind, then from the well-known relation

1 π

Z1

−1

ϕ(y)U_n(y)

y−x d y=−T_n+1(x), −1<x<1 , n=0, 1, 2, . . . , (27) where the Chebyshev polynomials of second kindU_n(x) =

sπ

2 p^ϕ_n(x)are already mentioned after equation (24) and whereT_n(x) withT_n(cosθ) =

sπ

2p^σ_n(cosθ) =cosnθ,n≥1 , are the Chebyshev polynomials of first kind, one can conclude the following: If (v_∗,C)∈L²_σ×Ris a solution of (23) then

Z1

−1

–

f_∗(x) +Cp

1+x−1 π

Z1

−1

k_∗(x,y)v_∗(y)d y

™

σ(x)d x=0 . (28)

K^ALANDIYAuses this condition (cf.[11, (37.129,(37.13)]) to get a additional equation to the system (25) by discretizing (28) with the help of the Gaussian rule w.r.t. the Chebyshev nodes of first kind and with the help of the already mentioned discretization of the integral operator with the kernel functionk_∗(x,y). Finally, KALANDIYAends up with a system

Anξn=ηn (29)

of linear equations, whereξn= ξkn

n+1

k=1is the vector of the function valuesξkn=v_n(x^ϕ_kn),k=1, . . . ,n, and the approximate valueξn+1,nofC, whereηn=

ηjn

n+1

j=1withηjn=f_∗(x^ϕ_jn),j=1, . . . ,n, andηn+1,n= 1 n

n

X

k=1

f_∗(x_kn^σ), and where the system matrixAn=

αjk

n+1

j,k=1is given by (26) and by αn+1,k= 1

n(n+1)

n

X

j=1

k_∗(x^σ_jn,x^ϕ_kn)ϕ(x_kn^ϕ), αk,n+1=−q

1+x_kn^ϕ , k=1, . . . ,n,

as well asαn+1,n+1=−2p 2

π . But, condition (28) is an artifical one, since one can only say that this condition is satisfied if (v_∗,C)∈L²_σ×Ris a solution of (23). One cannot use it as a solvability condition for equation (23). Moreover, one should note that, due to the considerations by DUDUCHAVA[2, Section 14], the operator defined by the left hand side of (6) has Fredholm index 0 in the spaceL²_ν, whereν(x) =v^−12,12(x), which means that the operator given by the left hand side of (23) has Fredholm index 0 inL²_σ. These problems are also confirmed by the numerical results, which we present in Table 2 below. In particular, one is interested in the computation of a normalized stress intensity factor, which is independent ofP(cf. (2)) and is given in case of P=1 by the formula (cf.[11, (36.35)])

δ= 1 L lim

x→1−0

v_∗(x)

p1−x. (30)

Using (24),δis approximated by (cf.[11, (37.16),(36.35)]) δn=

p2 L

n

X

k=1

ξkn`^ϕkn(1) ϕ(x^ϕ_kn) =

p2 L

n

X

k=1

(−1)^k+1ξkn

v u t1+x^ϕ_kn

1−x^ϕ_kn= p2

L

n

X

k=1

(−1)^k+1ξkncot kπ

2(n+1). (31)

KALANDIYA[11, p. 257]presents the following results:

L 10.0 5.0 1.0 0.2 0.04 0.01

δ 1.1338 1.2006 1.6281 2.9460 4.3970 4.9063 Table 1:[11, p. 257]

From Table 2 we observe that the results of Table 1 are obviously obtained forn=40 and that the computed approximate values forδstrongly depend onn, which indicates instability of the method (cf. also Table 3).

L

n 40 80 160 320 640 1280

0.01 4.9064 4.8802 4.8571 4.8369 4.8192 4.8036 0.04 4.3970 4.3714 4.3486 4.3287 4.3113 4.2960 0.20 2.9461 2.9237 2.9037 2.8862 2.8709 2.8574 1.00 1.6281 1.6130 1.5994 1.5876 1.5773 1.5683 5.00 1.2006 1.1908 1.1816 1.1737 1.1669 1.1610 10.00 1.1338 1.1253 1.1172 1.1102 1.1042 1.0991

Table 2:δnfrom (31) obtained by KALANDIYA’s method (29) Since theL²_σ-norm ofv_n=ϕp_nis equal to

v u t

Z1

−1

p1−x²|pn(x)|²d x= v u t π

n+1

n

X

k=1

ϕ(x_kn^ϕ)2 p_n(x^ϕ_kn)

2= v u t π

n+1

n

X

k=1

|ξkn|²,

the condition numbers of the maticesAnin (29) should be bounded if the collocation-quadrature method, represented by (29), is stable inL²_σ. Unfortunately, the results shown in Table 3 imply that this is not the case.

L

n 40 80 160 320 640 1280

0.01 2.06E04 1.42E05 1.02E06 7.47E06 5.55E07 4.15E08 0.04 1.94E04 1.34E05 9.62E05 7.05E06 5.23E07 3.91E08 0.20 1.58E04 1.09E05 7.76E05 5.67E06 4.20E07 3.14E08 1.00 1.05E04 7.21E04 5.13E05 3.73E06 2.76E07 2.05E08 5.00 6.36E03 4.36E04 3.09E05 2.24E06 1.64E07 1.22E08 10.00 4.96E03 3.41E04 2.42E05 1.75E06 1.18E07 9.48E07

Table 3: cond(An)forAnfrom (29)

In the Tables 4 and 5 we present the numerical results obtained by applying the collocation-quadrature method described in Section3to equation (16), i.e., to the equation (cf. (10), (16), and (17))

1 π

Z1

−1

• 1

y−x +ek^L(x,y)

˜

u(y)d y=g(x), −1<x<1 , (32) with

ek^L(x,y) = 1 p1+y

p

1+xek₀^L(x,y)− 1 p1+y+p

1+x

, g(x) =p

1+x g₀^L(x), andu(x) =p

1+x u₀(x). The respective system

Bnξn=ηn (33)

of linear equations with the system matrixBn=Sn+Knand the right hand sideηnis given by

Sn =



 v u t1−x^σ_jn

1−x^σ_kn Se`^σ_kn (x^σ_jn)





n

j,k=1

,

Kn =



 v u t1−x^σ_jn

1−x^σ_kn B⁰_n+H⁰_n e`^σkn

(x^σ_jn)





n

j,k=1

=



 1 n

v u t1−x^σ_jn

1−x^σ_knϕ(x^σ_kn)ek^L(x^σ_jn,x^σ_kn)





n

j,k=1

= h 1 n

q1−x^σ_jnÆ

1+x^σ_knek^L(x^σ_jn,x^σ_kn) iⁿ

j,k=1, andηn = Æ

1−x^σ_jng(x^σ_jn) n

j=1. To use, for example, the matrixSn instead of the matrix S⁰n = ” Se`^σ_kn

(x^σ_jn) —n j,k=1 is motivated by the following fact. The matrixSn+Knis equal to the operatorE_n⁽³⁾(A_n+H_n) E_n⁽³⁾−1

: imP_n−→imP_n. Hence, sinceE_n⁽³⁾: imL_n−→imP_nis a unitary operator, the sequence of the matricesSn+Knis stable if and only if the sequence of the operatorsA_n+H_nin (18) is stable inL²_σ. This means, that in case of stability the matrixSn+Knis a preconditioning of the matrixS⁰n+K⁰n.

To computeδin (30) in terms of the solutionu(x)of equation (32) we proceed as follows. By definition ofv_∗(x)we have, withv(x)from (6) andu(x)from (32),

δ= p2

L lim

x→1−0

v(x)

p1−x =−2p 2 L lim

x→1−0v⁰(x)p

1−x=−2 lim

x→1−0u(x)p

1−x, (34)