This article was downloaded by: [Stanford University Libraries]On: 06 October 2012, At: 09:42Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable Analysis: An InternationalJournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gapa20

Numerical solution of systems of singularintegral equations withP. S. Theocaris, G. Tsamasphyros & M. Z. Nashed

Version of record first published: 10 May 2007.

To cite this article: P. S. Theocaris, G. Tsamasphyros & M. Z. Nashed (1979): Numerical solution ofsystems of singular integral equations with, Applicable Analysis: An International Journal, 9:1, 37-52

To link to this article: http://dx.doi.org/10.1080/00036817908839250

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly or indirectlyin connection with or arising out of the use of this material.

Appirarhiv Anoi,al\. 1979. Vol 9, 37-52 0003.681 1 79.0901-0037 104.50.0 0 Gordon and Breach Sclenre Publishers, I n c . 1979 Prmted ~n Great Britam

Numerical Solution of Systems of Singular Integral Equations with Variable Coefficients

P. S. THEOCARIS and G. TSAMASPHYROS

Department of Theoretical and Applied Mechanics, The National Technical University, Athens (625), Greece

(Receiced October 3, 1977)

Communicated by M. Z. Nashed

Simple numerical methods for solving singular integral equations of the first and second kind with variable coefficients are described. The methods are based on a generalized Gauss- Jacobi quadrature formula for singular integrals. A further extension of the methods is presented for generalized Cauchy kernels.

1. INTRODUCTION The solution of a large class of mixed boundary value problems in physics and engineering can be reduced to a system of one-dimensional singular integral equations. It is further supposed that this system is transformed to a system of integral equations with uncoupled dominant parts of the form:

bi(t) 1' a dx + kij& t ) u j ( x ) dx =A( t ) ai( t )o i ( t ) +-- 7T - I x - t

In relation (1.1), the real functions a,, b, and J;. are given in the interval ( - 1, I ) , and they satisfy a Holder condition in the same interval; the kernels kij are also known and satisfy a Holder condition in each of the variables x and t. Finally, it is supposed that the unknown functions w, are required to satisfy a Holder condition. Moreover, at the end points t = * I the functions mi, or their first derivatives, have an integrable singularity.

37

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

38 P. S. THEOCARIS A N D G. TSAMASPHYROS

The singular behaviour of the functions oj around t = + 1 may be obtained from the dominant part of the integral equations (1.1) by applying a method given in [I] and [2]. It can readily be shown that the fundamental functions, which characterize the singular behaviour of oj, are given by:

where Rj(x) is a non-vanishing in ( - 1 , l ) Holder-continuous function, pJ and p; (j = 1,2, .. . , M) are integers, chosen in such a way that the behaviour of the fundamental functions X j at + 1 is compatible with the expected singular behaviour of the unknown functions wj(x) (i.e. either bounded if O<Re x J , Re f l j < 1, or infinite, but integrable, if - 1 <Re r j , Re Bj < 0). The constant xj, expressed by:

x . = - (p: + p'.') J J (I .6]

is known as the index of Xj(x). If any of the indices x, ( I = q l , q ,,..., q,) are greater than zero, then, in

addition to satisfying the system (1.1), oi must also satisfy N additional conditions, which are invariably of the following form:

where A, is a constant. In the above solutions aj and pj are taken as real quantities. The case where ccj or j j are complex may be examined in the same way.

The general theory of the system of integral equations (1.1) is given in [2], where a standard technique of reducing it to a system of Fredholm integral equations is discussed. Besides the difficulty encountered in obtaining the fundamental functions, the direct method of reducing the system of singular integral equations to a system of Fredholm equations (for which no analytic solution is possible) is rather cumbersome. For

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS 3 9

these reasons the development of an approximate method, preserving the correct nature of singularities of the unknown functions, seems to be desirable.

To obtain a solution by numerical means, the following expressions are used :

As yet only the case of constants a,, b, is considered [3, 41. In [3] a series expansion of qi(x) in Jacobi polynomials is used. With the aid of well- known orthogonality properties of the Jacobi polynomials an infinite system of equations is obtained. References [5], [6] and [7] describe methods based on the extension of standard rules for numerical in- tegration of regular integrals to the case of Cauchy-type principal value integrals with arbitrary integration intervals and weight functions. Furthermore, in a series of publications [5, 61 and [13 to 151 the numerical solution of Cauchy-type singular integral equations with or without generalized kernels, of the first and the second kind, with both constant and variable coefficients, was reduced to systems of linear algebraic equations by using numerical integration rules for Cauchy-type principal value integrals [8] and the existence of a sufficient number of collocation points was proved in almost all cases. These developments were quite general and concerned both finite and infinite integration intervals, but their attention was mainly focused on the interval [ - l,1] with a weight function W(x) given by equation (1.3). Although any selection of the abscissae used in the numerical integration rules was generally possible, yet primarily Gaussian integration rules of Gauss (open), Radau (semi-closed) and Lobatto (closed) type have been used. Moreover, the case of complex singularities in the weight function has been also considered for Cauchy type singular integral equations with a regular Fredholm kernel [15] where complex collocation points lying outside the integration interval were introduced.

The purpose of this paper is to systematize and extend the results noted above by developing a generalized interpolation formula from which we derive a generalized Gaussian quadrature scheme to include an arbitrary number of preassigned real nodes, for the Cauchy type integrals. If the preassigned node (s) is at the one limit of the interval of integration or at both limits, then the integration formulas often go by the names of Radau and Lobatto respectively. Based on this integration formula we propose two methods in paragraphs 3 and 4 respectively for the numerical solution of the system of singular integral equations (1.1) with variable coefficients.

Particular attention was given to the more accurate method described

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

40 P. S. THEOCARIS AND G. TSAlMASPHYROS

in paragraph 3. But, in order to be in measure to apply it, it is necessary to prove the existence of a sufficient number of roots for a complicated transcendental equation. The two theorems, which are given in the text, permit to find a lower limit of the number of roots. Much more practical interest is presented by the two corollaries which, under certain re- strictions, give the exact number of roots and their position.

The disadvantage of this method is that the zeros depend not only on the numerical integration rule, but also on the ratio ai(t)/bi(t). Thus we are obliged to find in each case the zeros of the transcendental equation which is not a simple matter. But the most serious limitation of the method is the nonexistence of a sufficient number of zeros. Evidently the other method is not restrained by these limitations and it is the alternative for these cases.

Special attention must be made also to the serious advantage offered from the proposed method of numerical integration, to include an arbitrary number of preassigned nodes. In fact, in many physical problems we are not interested for all the values of distribution coi(x), but for the values of specific points not appertaining into ( - 1, 1). This is particularly true, for example. if the equations (1.1) prescribe the boundary values of a cracked medium and of course our interest is limited to the evaluation of the stress intensity factor, i.e. the value of cp,(x) at the crack tip. In this case, a Lobatto method is recommended.

2. A QUADRATURE FORMULA

Let the singular integral I(q) be defined by:

We derive a generalized Gaussian quadrature scheme for the singular integral (2.1) to include an arbitrary number m of real preassigned nodes yk ( k = l ,2 , ..., m) lying outside the interval ( - 1, 1) and let:

where Q(x) is one signed for - 1 < x < 1, and s is equal either to 1, or to - 1, in order that Q(x) are positive in the interval - 1 < x < 1 and:

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS

be a set of orthogonal polynomials such that:

These polynomials satisfy the three-term recurrence formula:

where:

with:

We also have the Christoffel-Darboux formula

" .k(x).k(y) Zn + 1 (x).n(~) -nn(x).n+ 1 (Y ) (2.10) k = 0 x-y

For x = y we have:

We define now the associated function $,(z) as:

j1 w x x d x , z $ [ - 1 , l J $n(z) = - 2 - 1 z - x

With the usual convention that for t E (- 1 , l )

$,(t) = lim $[$,(t + ie) + $,(t - ic)] & + O

it follows that:

1 f1 W(x).n(x) dx, $,(t) = - - l < t < l .

2 1 t - x

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

42 P. S. THEOCARIS AND G. TSAMASPHYROS

The associated function $,(z) satisfies the same recurrence formula as nn(x) , for n > 1. But for 11 = 0, the recurrence formula is of the form:

Let P,+,(x) be the polynomial of degree n+ m such that:

Then, by using the Lagrangian interpolation formula. we have:

+:- (X - t ) n ( ~ ) ~ , ( ~ ) -- x cp(~,) +

A = I ( x - Y ~ ( J L - ~ ) Q ' ( Y ~ ~ ( Y A ) Q x n n x i t ) . (2.17) Q(t)n,(t)

In (2.16) R,+,+ , is the remainder of the interpolation formula which, for q ( z ) an analytic function, is given by:

where C is a closed contour containing the interval [ - 1, I ] in its interior as well as a11 points y,. We multiply now (2.16) by w ( x ) / ( x - t ) and integrate it from - 1 to 1 to obtain:

f w(x)cp(x) "+" dx-,) - 1 x - t d-X= C iJ.n-- +q,(t)cp(t)+E,+,+1

J = l XI-t

with:

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS

From (2.12) it is clear that for / z I-, x, $,(z)=O(/ z I - " ' ) and hence:

Thus, with the help of (2.23), it follows from (2.24) that

En+,+ = 0 for every q(x)~.P, ,+, ,

where .Y2, + are the class of polynomials of degree 5 2n + rn. We remark that A,,, are the modified standard weights for integration of bounded functions [9]. By using the Christoffel-Darboux formula, we obtain:

Another formula of 3.j,n may be obtained by using (2.11):

If, in particular, the preassigned nodes are chosen to coincide either with the one limit ( m = l ) or with both limits (nz=2) of the interval of integration ( - 1, I), then the integration formulas often go by the names of Radau and Lobatto respectively. If a preassigned node (nz=O) does not exist, the integration formula is simply the modified for singular integrals Gauss-Jacobi formula. In these particular cases we have:

W(x) = (1 - x ) q l + x)' (2.27)

with: if y k # l ( k = l , 2 )

+ if y, or y 2 = 1

and thus the corresponding n,(x) is the Jacobi polyne~~tial P(@'")(x) and so the weights 3.j,n may be readily calculated [6].

It has been also proved [lo] that the prescribed Gauss-Jacobi quadra- ture rule is convergent for Holder-continuous functions.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

44 P. S. THEOCARIS AND G. TSAMASPHYROS

3. SOLUTION OF A SINGULAR INTEGRAL EQUATION

By reason of simplicity we treat only the case of a singular integral equation. Evidently, the case of a system of singular integral equations may be treated in a similar way. The following integral equation is considered :

with:

Using (1.7) and (1.3) the quadrature formula for Cauchy-type integrals (2.19) and the quadrature formula for bounded functions [9], the above equation may be reduced to:

where x,, i,,,, q,(t) are defined by (2.20), (2.21) and (2.22) respectively and p,(t) is the remainder.

In practice, by selecting 11 sufficiently large p,(t) can be made as small as required, and hence, may be neglected. If the index ;1 is greater than zero, then a supplementary equation must be added to (3.1), which results from the additional condition (1.7), that is:

Let us apply now the functional equation (3.1) at the real roots jt,}f=,, t , ~ [ - I , l ] of F(t) . We obtain by this process a linear system of p equations with (n + m ) unknowns. To these equations may be added also the N ( N = O or 1) equations (3.3). In order that the system of these ( p f N ) equations is not under-determined, it is necessary that, ( p + N ) 2 (n + m). Therefore, F ( t ) must have p 2 (n + m - N) real roots in the interval [- 1, 11. The two theorems which follow prove under what conditions F ( t ) has the aforementioned sufficient number of real roots. It is also necessary to note that, if we take the analytic continuation of (3.1) in the whole complex plane, it is possible to apply (3.1) even to real or complex collocation

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS 4 5

points {tk]P,=l not appertaining to the interval [ - 1, 1) of the definition of the integral equation.

THEOREM 1 Let [a (x ) , b ( r ) ] be Holder-continuous fimc tions. T h e func- tion F ( x ) defined in (3.2) has in the interval I,=(x,, x,+ ,), k = 1 ,2 , . . . , n + m - 1 :

i) An odd nuinber 21+1. (1=0, 1 ,... ), of real zeror, if b ( x ) has 2p, (p=O, 1 ,2 , . . .), roots in I,;

ii) A n even number. 21, (1=0, 1,2, ...), of r e d zeros, i f b ( x ) has an odd number of real roots z,. In particular, it has at least two real zeros, i f there is a z, for which:

Proof We write q, as:

It is evident that the ratio [n,(x) - nn ( t ) ] / ( x - t ) is a polynomial of ( n - 1)- degree. Thus, in order to calculate the first integral it is permitted to use the standard Gauss-quadrature formula and to write:

where $,(t) is the associated function defined by (2.13). It is easy to prove that the only possible singular points of $,(t) are the points i 1. x. Using the last expression, we write (3.2) as:

By taking into consideration (2.26) we see that the modified-Christoffel numbers &,, ( k = l ,2 , ..., n), are positive. By inspection of (3.7) we can see that the expression under brackets tends to + cr;, if x-tx:, and to - x, if x+xk-+ ,, ( k = 1,2, ..., (n - 1)). In this way, .if b ( x ) has one signed value at the limits of I,, i.e. if b ( x ) possesses 2p, (p=O, 1,2, ...), real roots in I,, F ( x ) takes opposite values for x+x: and x+x,,, ( k = 1,2, ...,( n - 1)). Thus, it necessarily possesses an odd number of zeros.

If, in contrast, b ( x ) has an odd number of real roots in I,, F ( x ) tends either to +a, or to -a, if it tends to the limits of the interval I , by

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

46 P. S. THEOCARIS AND G. TSAMASPHYROS

taking values from this interval. Thus, F ( x ) possesses an even number 21, ( 1 =0, 1,2, ...) of real zeros.

It is evident that the inequality (3.4) ensures that F ( x ) has opposite values for x = z , and x+x; (or equivalently, for x+x,,) and thus F ( x ) possesses at least two real zesos. We also note that, by using such elementary arguments, we can calculate a lower bound of the number of roots in each I,.

THEOREM 2 Let the preassigned nodes coincide with the limits i 1 and W ( x ) be defined by (2.27). Then: if p > 0 (or v > 0) , Theorem 1 holds for the interval (x,, 1) (or ( - 1, x,)) . If p<O (or v<0) the function F(x ) has in the intercal (x,, 1) (or ( - 1, x , ) ) :

i) An even number 21, ( I = O , 1,2 ,... ), of real zeros, if b(x ) has 2p, ( p =0, 1,2, ...), roots in the prescribed intercal. In particular, it has at least two real zeros in (x,, 1 ) (or ( - 1 , ~ ~ ) ) . If there is a z, for which it is valid respectively that:

b ( l )a ( z j ) < 0, z j E (x,, 1 ) (7 81

ii) An odd number 21 + 1 , ( I = 0, 1,2, . . .), of real zeros, if b ( x ) has 2p + 1 ip =0, 1,2 ,... ), roots in (x,, 1) (or in ( - 1, x,)) .

Proof We note that (see Tricomi [ I l l ) :

2 p + T ( p ) T ( n + IJ + 1) q,(x) = 71 cot npw(x) -

Tin + p + v + l)Q(x)P?. " ' (x)

1 - .Y n + l , -11-p-V; l a p : - -

2 Or, equivalently:

2 p + " T ( ~ ) T ( n + p + 1) q,(x) = - 71 cot TCvw(x) + ( - 1)"

T (n + p + v + l ) Q ( x ) P ~ , ' ) ( x )

n + l , - n - p - v ; 1-v;---- 2

From (2.27) together with (1.4) and (1.5) we have:

1 a(1)- ib(1)

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS

Consequently, (3.10) and (3.11) can be written as follows:

Taking into consideration (3.14) (or (3.15)) we can calculate the limiting value of F(x) for x+l (or for x+ - 1) :

lirn F(x)= - b(1)2p+"(n!) T(p)T(p + 1)r (n + v + 1)

lirn (2) (3.16) x- I n r ( n + p + v + l ) r ( n + p + l ) R (x )

lim F ( x ) = b(- 1)2p+'(n!) T(v)T(v + 1)T(n + p + 1 )

x-t - I z r ( n t p t ~ + 1 ) ~ ( n + v + 1 ) , + - ~ lim (L) n ( x ) (3.17)

But l / R ( x ) tends to a positive value (+ x, $ or 1, depending on whether there exists a preassigned node, or not) when x tends to 1 or to - 1. Thus, if p>O (or v>0) and, if b(x) has no real roots or has an even number of real roots in (x,, 1) (or in ( - 1, x,)), the function F(x) takes opposite values for x+x; and x+l - (or for x-+ - I t and x+x;). Consequently, F(x) has also in the interval (x,, 1) (or ( - 1 , x , ) ) an odd number of zeros.

If, in contrast, b(x) has an odd number of roots, but p >O (or v > O ) , F (x ) takes one-signed values for x+xJ and x + l - (or for x+ - 1' and x 4 x ; ) . Thus, F(x) possesses an even number 21, ( 1 =0,1,2, ...), of real zeros.

If, now, p < 0 (or v < 0), then r ( p ) < 0 (or r ( v ) < 0), and the conclusions concerning the roots of F(x) derived previously are reversed and thus the validity of the second part of the Theorem 2 is demonstrated. The

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

48 P. S. THEOCARIS AND G. TSAMASPHYROS

inequality (3.8) (or (3.9)) ensures that F(x) has opposite values for x = z, and x + l (or x - t l ) and consequently, F(x) possesses at least two real zeros. Now, as an immediate consequence of Theorems 1 and 2, we can arrive at the following corollaries:

COROLLARY 1 If b(x) has no roots in ( - 1, I), then F(x) has at least:

i) (n-1) zeros, f p < O and v<O

ii) n zeros, ij' p . v < 0

iii) (n + 1) zeros, if ,u > 0 and v > 0.

All the zeros alternate with the zeros ofP?svJ(x).

COROLLARY 2 If p + v = k, where k is a negatice (> -2) or positive integer, then:

F(x) = 1

S2(x)P?, "(x)

and in the particular case, where a, b are constarzts, F(x) has ( n + k) roots, which coincide with the roots of Pi;? -"I and for k = - 1,0,1 alternate with the zeros of P?,')(x).

Proof The validity of the Corollary 2 turns out to be evident from relations (3.2) and (3.14), and from the following well-known expression of Jacobi polynomials in terms of hypergeometric functions:

4. ADDITIONAL COMMENTS

We propose now another way to reduce the functional equation (3.1) to a linear system of equations. Thus, if a n-point quadrature formula is used we select at the beginning as collocation points t, the (n+p) zeros of the polynomials nn+,(x) (where p is an integer #0) , subsequently, we use a (n +p)-point quadrature formula and we select as collocation points t, the zeros of n,(x). We obtain by this process from the integral equation a linear system of (2n + p) equations with (2n + p + m) unknowns. To these equations the N (N=O or 1) equations (3.2) may be also added. It is eviderlt that, in order to obtain a number of equations equal to, or greater

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS 49

than, the unknowns, solely for x > 0, one and only one preassigned node is permitted.

It is obvious that the size of the final system of linear equations obtained by the aforementioned process is approximately twice the size of the system obtained by the method described in paragraph 3. The increase of the size of the system is not followed by any increase in the degree of '

accuracy. But, on the other hand, it is simpler to select as collocation points the roots of the polynomials n,(x) and n,+,(x), instead of the roots of F(x). Moreover, the present method has the advantage to be applicable for any functions a(x) and b(x).

Consider now the case where a j or P j are complex:

In this case the polynomials z,(x), orthogonal with respect to W(x), do not have real roots in the interval ( - 1, l ) . An extension of the methods for such polynomials is obviously possible, but, it is perhaps preferable for this case to consider a non-Gaussian quadrature formula. In this way, we are free to select arbitrarily the nodes xj, (j= 1,2, ..., n), where we interpolate ~ ( x ) . We denote now as n,(x) the polynomial:

Then, by using the same procedure as above (relations (2.15) to (2.18), see also [?I), we arrive at the same quadrature formula, as the one defined by (2.19) to (2.23). In this case $,(z) is not an a priori known function and it does not satisfy a recurrence formula, but it is possible to calculate it.

We have also for / z I -+ a, $,(z) = 0() z / - ') and hence:

Thus, with the help of (2.23) it follows from (4.4) that:

Intuitively, we select as n,(x) the polynomial which is orthogonal with respect to the weight function

The methods described in this paper may be applied also to a system of

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

50 P. S. THEOCARIS AND G. TSAMASPHYROS

generalized Cauchy singular integral equations, where r , and fl, are not given by (1.4) and (1.51, but they are, in general, the roots of a transcendental equation. Such systems of integral equations appear, for example, in problems of the theory of elasticity, where angular points exist. Thus, we arrive at such systems of generalized-Cauchy integral equations in the cases of branched cracks, of contact problems wlth friction between bodies presenting angular points of composite dissimilar materials meeting at angles [12] and so on.

We note also that Corollary 2, especially in the case where preassigned nodes do not exist, coincides with the results of Krenk [4] presented in a recent paper. But, his complicated demonstration is valid only for the particular case where a, b are constants. Also he has not proved a generalized-Gauss quadrature formula, but he has simply calculated the singular part of the integral equation by using a series expansion of the unknown function. For the demonstration of the convergence of the method. one must follow the same processes applied by Krenk [4].

5. NUMERICAL EXAMPLE Consider the singular integral equation:

which is solved under the additional condition:

where

C = tan (0.6671)'

In this problem o(t) has integrable singularities at x = + 1 and a, p are then determined by:

The solution is readily found to be of the form:

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

SINGULAR INTEGRAL EQUATIONS

TABLE I

Solution of (5.1) with a = -0.34, b= -0.66 and 11 = 15

.: j ( ~ ( x ! ~ ) L , , ( x ! ~ ) R, , ( x ;~ ) C: s

TABLE I1 Solution of (5.1) with a= -0.34, D = -0.66 and n=20

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012

52 P. S. THEOCARIS AND G. TSAMASPHYROS

The method described in paragraph 3 was used with n=15 or n=20 and with no preassigned nbdes. The roots t; of F(x) and the roots x: of P;,"(x) with the corresponding values cp(x:) are listed in Tables I and I1 for n= 15 and n=20 respectively. The roots were determined with an accuracy of 10- ".

References [I] N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen. Holland,

1967. r2] N. P. Vekua, Systems of Singular Equations, Noordhoff, Groningen, Holland, 1967. 131 F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral

equations. In: Methods qf Analysis und Solutions of Crack Problems (G. C. Sih, ed.), Noordhoff Leyden, 1973, pp. 368-425.

[4] S. Krenk, On quadrature formulas for singular integral equations of the first and the second kind, Quart. Appl. Math. 33 (19751, 225-232.

[5] P. S. Theocaris and N. I. Ioakimidis, Numerical integration methods for the solution of singular integral equations, Quart. Appl. Math. 35 (1977), 173-183.

[6] P. S. Theocaris. On the numerical solution of Cauchy-type singular integral, "Serdica". Bulgar'icae Muth. Publ. 2 (1976), 252-275.

[7] G. Tsamasphyros and P. S. Theocaris. Sur une methode generate de quadrature pour des integrales du type Cauchy, Symposium of Applied Mathematics, Salonica, August 1976 (to be published).

[8] N. I. Ioakimidis and P. S. Theocaris, On the numerical evaluation of Cauchy principal value integrals, Rec. Roum. Sci. Techn. S i r . Mec. Appl. 22 (1977), 803-818.

[9] A. H. Stroud and Don Secrest, Gaussian Quadrature Formula, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966.

[lo] G. Tsamasphyros and P. S. Theocaris, O n the convergence of a Gauss quadrature rule for the evaluation of Cauchy type singular integrals, BIT 17 (1977), 458-464.

1111 H . G . Tricomi, Integral Equations, Interscience, New York, 1957. [I21 G. Tsamasfyros, Contribution a l'etude de la repartition des contraintes et defor-

mations dans les multilames de longueur finie sous l'effet des variations dimensionnelles propres aux materiaux constitutifs en tenant compte des singularites aux extremites (These de Doctorat es Sciences Physiques, Universite Paris VI, 1973).

[I31 P. S. Theocaris and N. I . Ioakimidis. Numerical solution of Cauchy type singular integral equations, Transactions of fhe ilcaderny of .Athens 40 (1977). 1-39.

[I41 N. I. Ioakimidis and P. S. Theocaris, O n the numerical solution of a class of singular integral equations, Journal of Mathemaricrtl and Physical Sciences 11 (1977), 219-235.

[I51 P. S. Theocaris and N. I. Ioakimidis, On the numerical solution of Cauchy type singular integral equations and the determination of stress intenslty factors in case of complex singularities, Zeitschrift jiir unpewzndte Malhematik und Physik 28 (19771, 1085-1098.

Dow

nloa

ded

by [

Stan

ford

Uni

vers

ity L

ibra

ries

] at

09:

42 0

6 O

ctob

er 2

012