The authors investigate a hypersingular integral equation which arises in the study of acoustic wave scattering by moving objects. A Galerkin method and two collocation methods are presented for solving the problem numerically. These numerical techniques are compared and contrasted in three test problems

Dunn, M. H.

St. John, R.

Tweed, J.

English

AbstractThe authors investigate a hypersingular integral equation which arises in the study of acoustic wave scattering by moving objects. A Galerkin method and two collocation methods are presented for solving the problem numerically. These numerical techniques are compared and contrasted in three test problems

John, R.St.

Dunn, M.H.

Elsevier - Publisher Connector 

Algorithms for the numerical solution of a finite-part integral equation 

Old Dominion University

Old Dominion UniversityODU Digital CommonsMathematics & Statistics Faculty Publications Mathematics & Statistics1999Algorithms for the Numerical Solution of a Finite-Part Integral EquationJ. TweedOld Dominion UniversityR. St. JohnOld Dominion UniversityM. H. DunnOld Dominion UniversityFollow this and additional works at: https://digitalcommons.odu.edu/mathstat_fac_pubsPart of the Applied Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion inMathematics & Statistics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contactdigitalcommons@odu.edu.Repository CitationTweed, J.; St. John, R.; and Dunn, M. H., "Algorithms for the Numerical Solution of a Finite-Part Integral Equation" (1999).Mathematics & Statistics Faculty Publications. 132.https://digitalcommons.odu.edu/mathstat_fac_pubs/132Original Publication CitationTweed, J., St John, R., & Dunn, M. H. (1999). Algorithms for the numerical solution of a finite-part integral equation. AppliedMathematics Letters, 12(3), 3-9. doi:10.1016/s0893-9659(98)00163-3PERGAMON Applied Mathematics Letters 12 (1999) 3-9 Applied Mathematics Letters Algor i thms  for the  Numerica l  Solut ion of a F in i te -Part  Integral Equat ion  J. TWEED, R. ST. JOHN AND M. H. DUNN Old Dominion University, Norfolk, VA 23529, U.S.A. (Received May 1998; revised and accepted July 1998) A b s t r a c t - - T h e  authors investigate a hypersingular integral equation which arises in the study of acoustic wave scattering by moving objects. A Galerkin method and two collocation methods are presented for solving the problem numerically. These numerical techniques are compared and contrasted in three test problems. © 1999 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - S i n g u l a r  integral equation, Finite-part integral, Hypersingular integral, Galerkin method, Collocation method. 1. I N T R O D U C T I O N  Investigations into the scattering of acoustic waves by moving objects have led the authors to consider the integral equation [1] fab ~(s)f(s) { 1 } (sSx)2÷Clnls-xt+D(s,x) ds=g(x), xe(a,b), (1) where C is a constant, D(s, x) is a nonsingular kernel, and A ( s )  = ~ / (b  - s ) ( s  - a ) .  (2) The integral equation is accompanied by two subsidiary conditions (linear functionals) Wk (f)  = gk, k = 0, 1, (3) either or both of which may be of the integral type fab ~ f (  Wk(f) = s) ds = gk in which the Wk(S) are prescribed functions, or of the end point type w k ( f )  = f ( c )  = ak in which c = a or b. The first integral in (1) is a finite-part integral defined by fab F(s) ds= d fab F(s) ds, ( s  - z )  - - - - - - - - ~  ~ s - z (4) (g) (6) 0893-9659/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(98)00163-3 Typeset by A~-TEX 4 J. TWEED et  al. where the integral on the right of (6) is a Cauchy principal value integral. Equations involving such integrals have been investigated by a number of authors [2-6] but not that  given by (1) and (3). It is well known that  the shifted Chebyshev polynomials of the first and second kind, respec- tively, ( ) b'---a , Un(S)=Un 2 s - b - a  "---a ' a < s < b (7) are related via the Cauchy type integral [7] b--arab tn(S) { 0 ,  n = 0 ,  27r A(s)(s- x) ds = un-l(x), n > 1. (8) Therefore, by differentiation, we find that (b 85)2 ~b  tn(s) ds= { O, n=0,1, c2_2(x), n > 2, (9) where c2n-2(s) = C2n-2\(2s b-a--b-- a) (10) and the C~m(y) are Gegenbauer polynomials of degree m and parameter A. Equation (9) suggests that (1) and (3) may be solved by expanding f(s) in a series of shifted Chebyshev polynomials of the first kind. We may therefore seek to approximate f(s) by the finite sum N f(s) = E fntn(S). (11) n = 0  With this approximation the subsidiary conditions (3) take the form N E fnWnk = gk, k = O, 1, (12) 'l'z=O where = wk(t ). In the case of an integral condition (4), we have Wnk = Zb ~ t n ( s )  ds, while, for a left end condition, we have (la) (14) Wnk ---- ( -1)  n, (15) and for a right end condition W n k  ~ 1, k =0 ,1 ;  n=O, 1,. . . ,N. On substituting from (11) into (1) and making use of (9) and of the result [7] z b  In Is - x I ds = -~r#ntn(x) tn(s) (16) (lr) Finite-Part Integral Equation 5 in which (b_-~4a) - l ( n > l ) ,  #o=In , # n =  n '  we find t ha t  the  coefficients f~, n = 0, 1, 2 , . . . ,  N must  satisfy the  equat ion N E fnK. (x )  = g(x), x 6 (a, b), n=0 where and p ,q  = 2 , 3 , . . .  8 r  6no)(1 6nl)C2_2(X) 7rC/J, ntn(X) + Dn(x) Kn(x) - (b - a) 2(1 - - - c ° t,~(s) Dn(x) = Ja -~(s) D(s'x)ds" (18) (19) (20) (21) Knm = (~.m + C7,~m + Dnm, (25) g m -  r ( m 2 - 1 )  ~ - a  A3(x)a(x)C2m-2(x)dx' (26) ~ . ~  = (1 - ~.0)(1 - 6~1)~1--6_ 1 ~ ~ 3 ( x ) 4 _ : ( x ) c i _ ~ ( x )  dx 8 r  -- (b - a) 2 (i  - 6nO)(1 - 6nl)6nm, (27) "Tnm= m---T'~- I ~ - a  Aa(x)tn(X)C2m-2(x)dx' (28) n~m = , ( m 2  _ 1) t~(s)c2_2(x)D(s, x) ds dx, (29) n = 0 , 1 , . . . , N ;  m = 2 , 3 , . . . , N .  Now, by means of the  change of variable x = ((b - a)/2)y + (b + a)/2, we readily see tha t  "Tnm - - - -  (--8tZn/(m 2 - 1))Jnm, where /_1 J . m  = (1 - y2)3/2 Tn(y)C2_2(y) dy 1 = 1 / _ L  (1 - y2)W2 Tn(y)[yUm-l(y) - mTm(y)] dy 2 ~ (30) = ~ sin 0 cos nO [sin mO cos 0 - m sin 0 cos toO] dO lr -- 1--6 [(m + 1)(6rim-2 + ~,~060~-2) -- 2me,~,~ + (m - 1 ) ( e . ~ + 2  + &~0~om+2)] • where 2 .  T H E  G A L E R K I N  M E T H O D  In the  Galerkin  Method,  g(x) is approximated  by the sum N g(x) = ~ gmci_~(x).  (22) rn=2 The  or thogonal i ty  condit ions [7] J~a 3 2 2 7r A (x)c~_2(xlcq_2(x) dx = g (p2 _ 1) 6pq, (23) are then  used to  show tha t  (19) is equivalent to the  linear algebraic sys tem N E fnKnm = gin, m = 2, 3 , . . . ,  N, (24) n=0 6 J. TWEED et at. Thus "yo,,,=-~In e,,,=, ~ . , , = - ~ e , , , ~ , - ~ , , , , , = T L , , = _  1 n(n+l) n-'~---i)J '(311 n = 2 , 3 , . . . , N ;  m = 2 , 3 , . . . , N .  In order to evaluate the elements gm and Dnm, we recall the Gaussian quadrature formula [8] M ~ b F(s) , r E F(sk) ' (32) where M is the number of Gaussian points and b - a  [ (2k~M1)lr ] b + a  k = 1,2, M. (33) sk = - Z -  cos + - 7 '  ""' An application of this formula to (26) and (29) yields the results M 8 ( 2 )4 1 EA4(sk)g(sk)c2m_2(Sk), (34) g~=~-zET-1 ~ ~ = ~  8~ ( 2 )  4 1 M  A4(sk)tn(Sl)C2m_2(sk)D(sl, Dnm 8k) ,  (35) k,l=l n=O,  1 , . . . ,N ;  m =  2 ,3 , . . . ,N .  Lastly, we observe that the linear system (24) consists of N - 1 equations for the N + 1 unknowns fn, n = 0, 1 , . . . ,  N. These are supplemented by the two subsidiary conditions (12) thus yielding the ( N + I )  x (N+I)  system N E fnWnk = gk, k = O, 1 , . . . ,  N, (36) n ~ O  where S wnk, n = 0 , 1 , . . . , N ,  k = 0 , 1 ,  W,k (37) K.,.,k, n=0 ,1 , . . . ,N ,  k=2 ,3 , . . . ,N .  3. T H E  C O L L O C A T I O N  M E T H O D  On approximating f (s)  by the finite sum (11), it was found that the integral equation (1) and its subsidiary conditions (3) yielded the equations and N fnw.k = gk, k = 0, 1 (38) n=o N ~,/.K.(=) = g(= ) ,  = ~ (a, b) (39) n..~O for the expansion coefficients fo, f l ,  f2 , . . . ,  fN. In the Collocation Method, N - 1 collocation points x2, x3, x4 , . . . ,  xg  E (a, b) are chosen and the fn are found by solving the N + I  simultaneous linear algebraic equations N E fnwnk = gk, k = O, 1, n-~O N E f n K n ( x k )  =g(xk), k = 2 , 3 , . . . , N .  n=O (40) Fin i t e -Par t  Integral  Equa t i on  7 In the numerical results which follow, two distinct collocation schemes are used. In the first of these, the collocation points x~ are taken to be the zeros of tg- l (X)  so that  (b + a), k = 2, 3, , N. (41) r 2k 3 ] 1 xo _- (b-o)cos 2"J + " In the second scheme, the collocation points x~ are taken to be the zeros of UN-l(x) so that  x~ = ( b - a ) c o s  [ ( k - 1 ) N  ] + ~(b+a),  k - -  2 , 3 , . . . , N .  (42) 4. N U M E R I C A L  R E S U L T S  In this section, we illustrate the use of the above algorithms by displaying the results obtained from their application to three test problems. PROBLEM 1. The first problem to be considered is that  of solving the integral equation / ~  f ( s )  { 1 } 1 lx/T'Z-~- s 2 (s - x) - ' - ' ~  ÷ In [ s - x[ + 2(s - x) 2 ds = g(x), x • ( -1 ,  1), (43) with subsidiary conditions and right-hand side 167r g ( x )  - (5 - 4x) 2 f ( - 1 )  = 0 and f(1)  = 0 (44) + w i n  ( 5  ~84---"-xx) +l-2Zr (201n2 _ 5 + 24x _ 16x2) (45 )  - 1 . 0 0  --0.80 - 0 . 6 0  - 0 . 4 0  --0.20 0.00 0.20 0.40 0.60 0.80 1.00 Elapsed  Values of f ( s )  Collocat ion Col locat ion Exac t  Galerkin wi th  x ° wi th  x I 0.0000000 -0 .2341463  -0 .4612613  -0 .6787879  -0 .8827586  -1 .0666667  -1 .2190476  -1 .3176471  -1 .3128205  -1 .0666667  0.0000000 T i m e  (Sec) 12.76 0.12 0.12 0.0000000 -0 .2341463  -0 .4612612  -0 .6787878  -0 .8827585  -1 .0666665  -1 .2190475  -1 .3176469  -1 .3128203  -1 .0666664  0.0000000 0.0000000 -0 .2341464  -0 .4612613  -0 .6787880  -0 .8827586  -1 .0666665  -1 .2190476  -1 .3176472  -1 .3128207  -1 .0666668  0.0000000 0.0000000 -0 .2341463  -0 .4612613  -0 .6787879  -0 .8827586  -1 .0666666  -1 .2190476  -1 .3176471  -1 .3128206  -1 .0666667  0.0000000 It is not too difficult to show that this problem has the exact solution 3 5 + 4 s  f(s)  - 5 - 4s 3 (46) Table 1 below exhibits the results obtained for Problem 1. It shows the exact and computed values of f(s)  for several values of s when 25 terms of the expansion are taken and 25 quadrature points used. This table also shows the time taken for each of the numerical schemes used. Table  1. Tes t  resu l t s  for P rob l em 1 wi th  25 series t e rms  and  25 q u a d r a t u r e  points .  8 J. TWEED et al. PROBLEM 2. Next  we cons ider  t he  p rob lem of solving the  in tegra l  equa t ion  /_, { 1 } : M ( s _ - x )  2 + l n [ s - x [ + 2 ( s - x )  2 ds=g(x),  with  subs id i a ry  condi t ions  ~ f ( s )  f ( - 1 ) = 0  and  - - d s = O  : lx/y-z-~_ s~ and  r i gh t -hand  side z e  ( - 1 , 1 ) ,  (47) (48) : 0 -  g ( x ) -  ( 5 - - ~ x ) 2  + T r l n  +Tr l n 2 +  . (49) In  th is  case t he  exac t  so lu t ion  is 3 3 + 2 s  f ( s )  - 5 -  4s 3 (50) Table  2 exh ib i t s  t he  resul ts  ob t a ined  for P rob l e m 2. I t  shows the  exac t  and  c o m p u t e d  values of f(s) for severa l  values of  s when  25 t e rms  of t he  expans ion  are  t aken  and  25 q u a d r a t u r e  po in ts  used. Th is  t ab l e  also shows the  t ime  t aken  for each of t he  numer ica l  schemes used.  Table 2. Test results for Problem 2 with 25 series terms and 25 quadrature points. Values of f(s) Collocation Collocation Exact Galerkin with x~ with x 1 0.0000000 -0.1008130 -0.1945946 -0.2787880 -0.3494253 -0.3999999 -0.4190476 -0.3843139 -0.2461540 0.1333332 1.3333333 0.0000000 -0.1008130 -0.1945946 -0.2787879 -0.3494253 -0.4000000 -0.4190476 -0.3843138 -0.2461539 0.1333333 1.3333333 0.0000000 -0.1008130 -0.1945946 -0.2787879 -0.3494253 -0.4000000 -0.4190476 -0.3843137 -0.2461538 0.1333334 1.3333332 - 1 . 0 0  --0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 Elapsed Time (Sec) 0.0000000 -0.1008130 -0.1945946 -0.2787879 -0.3494253 -0.4000000 -0.4190476 -0.3843137 -0.2461538 0.1333333 1.3333333 12.74 0.12 0.12 PROBLEM 3. : } (s._-x) 2 + l n l s - x l + 2 ( s - x )  2 ds=g(x) ,  with  subs id i a ry  condi t ions  /_: f(s) and r i gh t -hand  side Last ly ,  we consider  t he  p rob lem of solving the  in tegra l  equa t ion  x e ( - 1 , 1 ) ,  f --ds=O and  sf(s___._~) ds=O 167r g(x)  = (5 - 4x)2 + T r l n  +Tr l n 2 + ~ + x  . (51) (52) (53) Finite-Part  Integral Equation Table 3. Test results for Problem 3 with 25 series terms and 25 quadrature  points. Values of f (s)  Collocation Collocation Exact Galerkin with x ° with x~ --1.00 -0 .80  -0 .60  --0.40 --0.20 0.00 0.20 0.40 0.60 0.80 1.00 Elapsed Time (Sec) 0.3333333 0.1658537 0.0054054 -0.1454545 -0.2827586 -0.4000000 -0.4857143 -0.5176471 -0.4461538 -0.1333333 1.0000000 0.3333333 0.1658536 0.0054054 -0.1454546 -0.2827586 -0.4000000 -0.4857143 -0.5176470 -0.4461538 -0.1333333 0.9999998 0.3333333 0.1658536 0.0054054 -0.1454546 -0.2827586 -0.3999999 -0.4857143 -0.5176472 -0.4461540 -0.1333335 1.0000000 0.3333333 0.1658537 0.0054054 -0.1454546 -0.2827586 --0.4000000 -0.4857143 -0.5176471 -0.4461539 -0.1333334 1.0000000 12.77 0.13 0.13 In this case the exact solution is 3 f ( s )  = 5 - 4 s  1 - s .  (54) Table 3 exhibits the results obtained for Problem 3. It shows the exact and computed values of f ( s )  for several values of s when 25 terms of the expansion are taken and 25 quadrature points used. This table also shows the time taken for each of the numerical schemes used. The computations associated with the problems discussed above were performed on a 233 Mhz Pentium II computer with 64 Mb of RAM. Each of the proposed algorithms converged quickly providing high accuracy with relatively few terms in the series approximation. For the problems considered, the collocation algorithms were clearly more efficient than the Galerkin algorithm. R E F E R E N C E S  1. M.H. Dunn, J. Tweed and F. Farassat, The prediction of ducted fan engine noise via a boundary integral equation method, AIAA Paper 96-1770, (April 1996). 2. A. Frenkel, A Chebyshev expansion of singular integrodifferential equations with a a 2 In Is - tl/asat kernel, J. Comp. Phys. 51, 335-342 (1983). 3. M.A. Golberg, The convergence of several algorthms for solving integral equations with a finite part integral, J. Integral Equations 5, 329-340 (1983). 4. M.A. Golberg, The  convergence of several algorthms for solving integral equations with a finite part  inte- grals II, Applied Math. f~ Comp. 21, 283-293 (1987). 5. A.C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. X L V  (1), 105-122 (1987). 6. A.C. Kaya and F. Erdogan, On the solution of integral equations with a generalized Cauchy kernel, Quart. Appl. Math. X L V  (3), 455-469 (1987). 7. W. Magnus, F. Oberhet t inger  and R.P. Soni, Formulas and Theorems for the Special Functions of Mathe- matical Physics, Springer-Verlag, New York, (1966). 8. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, (1965). 

Algorithms for the Numerical Solution of a Finite-Part Integral Equation

PERGAMON Applied Mathematics Letters 12 (1999) 3-9 
Applied 
Mathematics 
Letters 
Algor i thms for the Numerical  Solution 
of a Finite-Part Integral Equat ion 
J. TWEED, R. ST. JOHN AND M. H. DUNN 
Old Dominion University, Norfolk, VA 23529, U.S.A. 
(Received May 1998; revised and accepted July 1998) 
Abst rac t - -The  authors investigate a hypersingular integral equation which arises in the study 
of acoustic wave scattering by moving objects. A Galerkin method and two collocation methods 
are presented for solving the problem numerically. These numerical techniques are compared and 
contrasted in three test problems. © 1999 Elsevier Science Ltd. All rights reserved. 
Keywords - -S ingu lar  integral equation, Finite-part integral, Hypersingular integral, Galerkin 
method, Collocation method. 
1. INTRODUCTION 
Investigations into the scattering of acoustic waves by moving objects have led the authors to 
consider the integral equation [1] 
fab ~(s)f(s) { 1 } (sSx)2÷Clnls-xt+D(s,x) ds=g(x), xe(a,b), (1) 
where C is a constant, D(s, x) is a nonsingular kernel, and 
A(s )  = ~/(b - s ) ( s  - a).  (2) 
The integral equation is accompanied by two subsidiary conditions (linear functionals) 
Wk (f) = gk, k = 0, 1, (3) 
either or both of which may be of the integral type 
fab ~f (  Wk(f) = s) ds = gk 
in which the Wk(S) are prescribed functions, or of the end point type 
wk( f )  = f (c )  = ak 
in which c = a or b. 
The first integral in (1) is a finite-part integral defined by 
fab F(s) ds= d fab F(s) ds, 
(s  - z )  - - - - - - - -~  ~ s - z 
(4) 
(g) 
(6) 
0893-9659/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved. 
PII: S0893-9659(98)00163-3 
Typeset by A~-TEX 
4 J. TWEED et al. 
where the integral on the right of (6) is a Cauchy principal value integral. Equations involving 
such integrals have been investigated by a number of authors [2-6] but not that given by (1) 
and (3). 
It is well known that the shifted Chebyshev polynomials of the first and second kind, respec- 
tively, 
( ) b'---a , Un(S)=Un 2s -b -a  "---a ' a < s < b (7) 
are related via the Cauchy type integral [7] 
b--arab tn(S) {0,  n=0,  
27r A(s)(s- x) ds = un-l(x), n > 1. (8) 
Therefore, by differentiation, we find that 
(b 85)2 ~b tn(s) ds= { O, n=0,1, 
c2_2(x), n > 2, (9) 
where 
c2n-2(s) = C2n-2\(2s b-a--b-- a) (10) 
and the C~m(y) are Gegenbauer polynomials of degree m and parameter A. Equation (9) suggests 
that (1) and (3) may be solved by expanding f(s) in a series of shifted Chebyshev polynomials 
of the first kind. We may therefore seek to approximate f(s) by the finite sum 
N 
f(s) = E fntn(S). (11) 
n=0 
With this approximation the subsidiary conditions (3) take the form 
N 
E fnWnk = gk, k = O, 1, (12) 
'l'z=O 
where 
= wk(t ). 
In the case of an integral condition (4), we have 
Wnk = Zb ~tn(s )  ds, 
while, for a left end condition, we have 
(la) 
(14) 
Wnk ---- (-1) n, (15) 
and for a right end condition 
Wnk ~ 1, 
k =0,1;  n=O, 1,...,N. 
On substituting from (11) into (1) and making use of (9) and of the result [7] 
zb  In Is - x I ds = -~r#ntn(x) tn(s) 
(16) 
(lr) 
Finite-Part Integral Equation 5 
in which 
(b_-~4a) - l (n>l ) ,  #o=In , #n= n'  
we find that  the coefficients f~, n = 0, 1, 2 , . . . ,  N must satisfy the equation 
N 
E fnK.(x)  = g(x), x 6 (a, b), 
n=0 
where 
and 
p,q = 2 ,3 , . . .  
8r  6no)(1 6nl)C2_2(X) 7rC/J, ntn(X) + Dn(x) Kn(x) - (b - a) 2(1 - - - 
c ° t,~(s) 
Dn(x) = Ja -~(s) D(s'x)ds" 
(18) 
(19) 
(20) 
(21) 
Knm = (~.m + C7,~m + Dnm, (25) 
gm-  r (m2-1)  ~-a  A3(x)a(x)C2m-2(x)dx' (26) 
~.~ = (1 - ~.0)(1 - 6~1)~1--6_ 1 ~ ~3(x)4_ : (x )c i _~(x)  dx 
8r  
-- (b - a) 2 (i - 6nO)(1 - 6nl)6nm, (27) 
"Tnm= m---T'~- I ~-a  Aa(x)tn(X)C2m-2(x)dx' (28) 
n~m = , (m2 _ 1) t~(s)c2_2(x)D(s, x) ds dx, (29) 
n = 0 ,1 , . . . ,N ;  m = 2 ,3 , . . . ,N .  
Now, by means of the change of variable x = ((b - a)/2)y + (b + a)/2, we readily see that 
"Tnm - - - -  (--8tZn/(m 2 - 1))Jnm, where /_1 
J .m = (1 - y2)3/2 Tn(y)C2_2(y) dy 
1 
= 1/_L  (1 - y2)W2 Tn(y)[yUm-l(y) - mTm(y)] dy 
2 ~ (30) 
= ~ sin 0 cos nO [sin mO cos 0 - m sin 0 cos toO] dO 
lr 
-- 1--6 [(m + 1)(6rim-2 + ~,~060~-2) -- 2me,~,~ + (m - 1) (e .~+2 + &~0~om+2)] • 
where 
2. THE GALERKIN  METHOD 
In the Galerkin Method, g(x) is approximated by the sum 
N 
g(x) = ~ gmci_~(x). (22) 
rn=2 
The orthogonal ity conditions [7] 
J~a 3 2 2 7r A (x)c~_2(xlcq_2(x) dx = g (p2 _ 1) 6pq, (23) 
are then used to show that  (19) is equivalent o the linear algebraic system 
N 
E fnKnm = gin, m = 2, 3 , . . . ,  N, (24) 
n=0 
6 J. TWEED et at. 
Thus 
"yo,,,=-~In e,,,=, ~. , ,=-~e, , ,~, -~, , , , ,=TL , ,=_  1 n(n+l) n-'~---i)J '(311 
n = 2,3, . . . ,N;  m = 2,3, . . . ,N.  
In order to evaluate the elements gm and Dnm, we recall the Gaussian quadrature formula [8] 
M ~ b F(s) , r E F(sk) ' (32) 
where M is the number of Gaussian points and 
b-a  [ (2k~M1)lr ] b+a k = 1,2, M. (33) sk = -Z -  cos + -7 '  ""' 
An application of this formula to (26) and (29) yields the results 
M 8 ( 2 )4 1 EA4(sk)g(sk)c2m_2(Sk), (34) 
g~=~-zET-1 ~ ~=~ 
8~ (2)  41M 
A4(sk)tn(Sl)C2m_2(sk)D(sl, Dnm 8k), (35) 
k,l=l 
n=O, 1, . . . ,N;  m= 2,3, . . . ,N.  
Lastly, we observe that the linear system (24) consists of N - 1 equations for the N + 1 
unknowns fn, n = 0, 1,. . . ,  N. 
These are supplemented bythe two subsidiary conditions (12) thus yielding the (N+I)  x (N+I)  
system 
N 
E fnWnk = gk, k = O, 1,.. . ,  N, (36) 
n~O 
where 
S wnk, n=0,1 , . . . ,N ,  k=0,1 ,  
W,k (37) 
K.,.,k, n=0,1 , . . . ,N ,  k=2,3 , . . . ,N .  
3. THE COLLOCATION METHOD 
On approximating f(s) by the finite sum (11), it was found that the integral equation (1) and 
its subsidiary conditions (3) yielded the equations 
and 
N 
fnw.k = gk, k = 0, 1 (38) 
n=o 
N 
~,/.K.(=) = g(=),  = ~ (a, b) (39) 
n..~O 
for the expansion coefficients fo, fl, f2, . . . ,  fN. In the Collocation Method, N - 1 collocation 
points x2, x3, x4,. . . ,  xg E (a, b) are chosen and the fn are found by solving the N+I  simultaneous 
linear algebraic equations 
N 
E fnwnk = gk, k = O, 1, 
n-~O 
N 
EfnKn(xk)  =g(xk), k=2,3 , . . . ,N .  
n=O 
(40) 
Finite-Part  Integral Equat ion 7 
In the numerical results which follow, two distinct collocation schemes are used. In the first of 
these, the collocation points x~ are taken to be the zeros of tg- l(X) so that 
(b + a), k = 2, 3, , N. (41) r 2k 3 
] 1 
xo _- (b-o)cos 2" J  + " 
In the second scheme, the collocation points x~ are taken to be the zeros of UN-l(x) so that 
x~ = (b -a )cos  [ (k -1 )N  ] + ~(b+a), k--  2,3 , . . . ,N .  (42) 
4. NUMERICAL  RESULTS 
In this section, we illustrate the use of the above algorithms by displaying the results obtained 
from their application to three test problems. 
PROBLEM 1. The first problem to be considered is that of solving the integral equation 
/~  f(s)  { 1 } 
1 lx/T'Z-~- s 2 (s - x) - ' - '~  ÷ In [ s - x[ + 2(s - x) 2 ds = g(x), x • ( -1,  1), (43) 
with subsidiary conditions 
and right-hand side 
167r 
g(x)  - 
(5 - 4x) 2 
f ( -1 )  = 0 and f(1) = 0 (44) 
+win  (5  ~84---"-xx) +l-2Zr (201n2 _ 5 + 24x _ 16x2) (45) 
-1 .00  
--0.80 
-0 .60  
-0 .40  
--0.20 
0.00 
0.20 
0.40 
0.60 
0.80 
1.00 
Elapsed 
Values of f ( s )  
Collocation Collocation 
Exact Galerkin 
with x ° with x I 
0.0000000 
-0.2341463 
-0.4612613 
-0.6787879 
-0.8827586 
-1.0666667 
-1.2190476 
-1.3176471 
-1.3128205 
-1.0666667 
0.0000000 
T ime (Sec) 12.76 0.12 0.12 
0.0000000 
-0.2341463 
-0.4612612 
-0.6787878 
-0.8827585 
-1.0666665 
-1.2190475 
-1.3176469 
-1.3128203 
-1.0666664 
0.0000000 
0.0000000 
-0.2341464 
-0.4612613 
-0.6787880 
-0.8827586 
-1.0666665 
-1.2190476 
-1.3176472 
-1.3128207 
-1.0666668 
0.0000000 
0.0000000 
-0.2341463 
-0.4612613 
-0.6787879 
-0.8827586 
-1.0666666 
-1.2190476 
-1.3176471 
-1.3128206 
-1.0666667 
0.0000000 
It is not too difficult to show that this problem has the exact solution 
3 5+4s  
f(s) - 5 - 4s 3 (46) 
Table 1 below exhibits the results obtained for Problem 1. It shows the exact and computed 
values of f(s) for several values of s when 25 terms of the expansion are taken and 25 quadrature 
points used. This table also shows the time taken for each of the numerical schemes used. 
Table 1. Test results for Problem 1 with 25 series terms and 25 quadrature  points. 
8 J. TWEED et al. 
PROBLEM 2. Next we consider the problem of solving the integral equat ion 
/_, { 1 } 
: M ( s_ -x )  2 +ln[s -x [+2(s -x )  2 ds=g(x), 
with subsid iary condit ions 
~ f (s )  
f ( -1 )=0 and - -ds=O 
: lx/y-z-~_ s  
and r ight-hand side 
ze  ( -1 ,1 ) ,  (47) 
(48) 
:0 -  
g(x) -  (5 - -~x)2  +Tr ln  +Tr ln2+ . (49) 
In this case the exact solut ion is 
3 3+2s  
f ( s )  - 5 -  4s 3 (50) 
Table 2 exhibits the results obtained for Problem 2. It  shows the exact and computed values 
of f(s) for several values of s when 25 terms of the expansion are taken and 25 quadrature  points 
used. This table also shows the t ime taken for each of the numerical  schemes used. 
Table 2. Test results for Problem 2 with 25 series terms and 25 quadrature points. 
Values of f(s) 
Collocation Collocation 
Exact Galerkin 
with x~ with x 1 
0.0000000 
-0.1008130 
-0.1945946 
-0.2787880 
-0.3494253 
-0.3999999 
-0.4190476 
-0.3843139 
-0.2461540 
0.1333332 
1.3333333 
0.0000000 
-0.1008130 
-0.1945946 
-0.2787879 
-0.3494253 
-0.4000000 
-0.4190476 
-0.3843138 
-0.2461539 
0.1333333 
1.3333333 
0.0000000 
-0.1008130 
-0.1945946 
-0.2787879 
-0.3494253 
-0.4000000 
-0.4190476 
-0.3843137 
-0.2461538 
0.1333334 
1.3333332 
- 1 .00  
--0.80 
-0.60 
-0.40 
-0.20 
0.00 
0.20 
0.40 
0.60 
0.80 
1.00 
Elapsed Time (Sec) 
0.0000000 
-0.1008130 
-0.1945946 
-0.2787879 
-0.3494253 
-0.4000000 
-0.4190476 
-0.3843137 
-0.2461538 
0.1333333 
1.3333333 
12.74 0.12 0.12 
PROBLEM 3. 
: } (s._-x) 2 + ln l s -x l+2(s -x )  2 ds=g(x), 
with subsid iary condit ions 
/_: f(s) 
and r ight-hand side 
Lastly, we consider the problem of solving the integral equat ion 
x e ( -1 ,1 ) ,  
f - -ds=O and sf(s___._~) ds=O 
167r 
g(x) = (5 - 4x)2 
+Tr ln  +Tr ln2+~+x . 
(51) 
(52) 
(53) 
Finite-Part Integral Equation 
Table 3. Test results for Problem 3 with 25 series terms and 25 quadrature points. 
Values of f(s) 
Collocation Collocation 
Exact Galerkin 
with x ° with x~ 
--1.00 
-0.80 
-0.60 
--0.40 
--0.20 
0.00 
0.20 
0.40 
0.60 
0.80 
1.00 
Elapsed Time (Sec) 
0.3333333 
0.1658537 
0.0054054 
-0.1454545 
-0.2827586 
-0.4000000 
-0.4857143 
-0.5176471 
-0.4461538 
-0.1333333 
1.0000000 
0.3333333 
0.1658536 
0.0054054 
-0.1454546 
-0.2827586 
-0.4000000 
-0.4857143 
-0.5176470 
-0.4461538 
-0.1333333 
0.9999998 
0.3333333 
0.1658536 
0.0054054 
-0.1454546 
-0.2827586 
-0.3999999 
-0.4857143 
-0.5176472 
-0.4461540 
-0.1333335 
1.0000000 
0.3333333 
0.1658537 
0.0054054 
-0.1454546 
-0.2827586 
--0.4000000 
-0.4857143 
-0.5176471 
-0.4461539 
-0.1333334 
1.0000000 
12.77 0.13 0.13 
In this case the exact solution is 
3 
f ( s )  = 5 - 4s  1 - s.  (54) 
Table 3 exhibits the results obtained for Problem 3. It shows the exact and computed values 
of f ( s )  for several values of s when 25 terms of the expansion are taken and 25 quadrature points 
used. This table also shows the time taken for each of the numerical schemes used. 
The computations a sociated with the problems discussed above were performed on a 233 Mhz 
Pentium II computer with 64 Mb of RAM. Each of the proposed algorithms converged quickly 
providing high accuracy with relatively few terms in the series approximation. For the problems 
considered, the collocation algorithms were clearly more efficient han the Galerkin algorithm. 
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2. A. Frenkel, A Chebyshev expansion of singular integrodifferential equations with a a 2 In Is - tl/asat kernel, 
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3. M.A. Golberg, The convergence of several algorthms for solving integral equations with a finite part integral, 
J. Integral Equations 5, 329-340 (1983). 
4. M.A. Golberg, The convergence of several algorthms for solving integral equations with a finite part inte- 
grals II, Applied Math. f~ Comp. 21, 283-293 (1987). 
5. A.C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. 
Appl. Math. XLV (1), 105-122 (1987). 
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Algorithms for the Numerical Solution of a Finite-Part Integral Equation

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