Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations

Journal of Computational and Applied Mathematics 115 (2000) 193–211 www.elsevier.nl/locate/cam

Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations a

E.A. Galperina , E.J. Kansab , A. Makroglouc; ∗ , S.A. Nelsond Department de Mathematiques et d’Informatique, Univ. du Quebec a Montreal, Case Postale 8888, Succ. Centre Ville, Montreal, Quebec, Canada H3C 3P8 b Embry-Riddle Aeronautical University, 7700 Edgewater Dr., Oakland, CA 94621, USA c School of Computer Science and Mathematics, University of Portsmouth, Mercantile House, Portsmouth PO1 2EG, UK d Department of Mathematics, Iowa State University, Ames, IA 50011, USA Received 23 September 1998; received in revised form 10 June 1999

Abstract A number of techniques that use variable transformations in numerical integration have been developed recently (cf. Sidi, Numerical Integration IV, H. Brass, G. Hammerlin (Eds.), Birkhauser, Basel, 1993, pp. 359 –373; Laurie, J. Comput. Appl. Math. 66 (1996) 337–344.). The use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule. In this paper the application of variable transformation techniques of Sidi and Laurie type to the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels is considered. Since the transformations are such that the end points of integration need not be used as mesh points, the methods introduced can be used for VIE with both continuous and weakly singular kernel in a uniform way. The methods have also the advantages of simplicity of application and of achieving high order of convergence. The application of the idea to Fredholm integral equations with continuous and weakly singular equations is also considered. Numerical results are included and they verify the expected increased order of convergence. They were obtained by using c 2000 Published by Elsevier Science B.V. All the trapezoidal formula for the evaluation of the transformed integrals. rights reserved. Keywords: Numerical solution; Variable transformations; Volterrra and Fredholm integral equations; Second kind; Continuous and weakly singular

∗

Corresponding author. Tel.:+44-1705-843046; fax: +44-1705-843106. E-mail address: [email protected] (A. Makroglou)

c 2000 Published by Elsevier Science B.V. All rights reserved. 0377-0427/00/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 2 9 7 - 6

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1. Introduction We consider the Volterra integral equation (VIE) of the second kind given by y(x) = f(x) +

Z

x

0

K(x; s; y(s))H (x; s) ds;

x ∈ I = [0; X ]

(1.1)

where f; K are given smooth functions and where H (x; s) is assumed to be equal to 1 in the case of equations with continuous kernel and given by H (x; s) =

1 ; (x − s)

0 ¡  ¡ 1; 06s6x6X;

(1.2)

in the case of equations with weakly singular kernel. We also consider Fredholm integral equations of the form y(x) = f(x) +

Z

b

0

K(x; s; y(s))H (x; s) ds;

06x6b;

(1.3)

where H (x; s) = 1 in the case of continuous kernel and H (x; s) =

1 ; |x − s|

0 ¡  ¡ 1; 06x; s6b

(1.4)

in the case of equations with weakly singular kernel. The numerical solution of VIEs and FIEs with weakly singular kernels has been considered by many authors. Existing methods for VIEs include the generalized Newton–Cotes formulae combined with product integration rules (cf. [11,17,26]); collocation methods in certain polynomial and nonpolynomial spline spaces with uniform and graded meshes [7,8,10,29,39]; fractional linear multistep methods [16,30]; rational basis functions with product integration methods [1]; sinc approximation methods [40]; mathematical programming methods [14]. Alternative approaches include the method of “subtracting” the nonsmooth behavior of the solution y(x) [13,20,21]; transformation of variables (cf. [12,35,36]) and a two-level approach where the solution is computed rst over a portion of the interval of integration and then continued with a standard step-by-step product integration method for the rest of the interval [37]. Existing methods for FIEs with weakly singular kernels include the use of generalized Newton– Cotes formulae combined with product integration rules, Chebyshev polynomials methods, collocation methods with uniform and graded meshes, Galerkin methods (cf. [3,22,23,38,42]) and the references therein. There are also quadrature methods based on the trapezoidal rule that are very eective for periodic FIEs of the rst and second kind with singular and weakly singular kernels [44]. For an extended bibliography we refer to the survey paper by Atkinson [4] and to the books by Baker [6] and Atkinson [5]. The bibliography of numerical methods applied to VIEs and FIEs with continuous kernel is very rich (cf. [5,9,15,28]). Graded meshes were introduced to deal with the problems of deriving high-order methods for equations with nonsmooth solutions. So, if collocation methods with polynomial splines of degree p − 1 are used with (graded) mesh given by xi = (i=N )r X; i = 0; : : : ; N; r = 1=(1 − ), for VIEs with weakly singular kernels, the order of convergence for the error e of the approximation is

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195

given by k e k∞ = O(N −p ) (cf. [7]). Compare also Schneider [42] for a corresponding result for FIEs with weakly singular kernel. A problem that may arise with the use of graded meshes is that it can sometimes create signi cant round-o errors in calculations over a long interval of integration, since a large number of these calculations is performed with a very small step size. To avoid problems associated with the use of graded meshes and still maintain optimal order of convergence of the methods used, the idea of the variable transformation methods was introduced. So, Norbury and Stuart [35,36] considered VIEs with weakly singular kernel of the form (1.1) – (1.2) with  = 0:5 and introduced two transformations of the independent variable given by s = x sin2 ; x =  2 . The use of these transformations then results in a transformed problem with regularized solution. Use of the trapezoidal quadrature rule with variable stepsize resulted in a method of order O(h1:5− );  ¿ 0. The convergence proof involved uses a generalized continuous Gronwall inequality introduced in the same papers. Diogo et al. [12], considered also VIEs with weakly singular kernel of the form (1.1) – (1.2) with  = 0:5 and introduced a single transformation of the form x = t 2 X to regularize the solution of the new transformed problem. Polynomial collocation methods of degree p − 1 were then used with constant stepsize which attained O(N −p ) order of convergence. This method was generalized to Volterra integro-dierential equations (VIDEs) with weakly singular kernels by Makroglou and Miller [32]. Recently, another type of smoothing transformation has been applied to weakly singular FEIs with input function f(x) that may have a nite number of jumps or singularities in one of its derivatives at a nite number of points in the interval of integration, by Monegato and Scuderi [33]. The limitations of small initial stepsizes associated with graded meshes were also mentioned in a paper by Hu [18] who introduced the so called -polynomial collocation methods. These methods were combined with a “higher-order postprocessor” and resulted in order of convergence O(hp+1− ). In this paper we consider the application of transformations of the independent variable of trigonometric form [43] and of polynomial form [25] combined with a trapezoidal quadrature rule applied to the transformed integral. The transformations are such that: (i) the end points of the interval of integration need not be used as mesh points. This allows uniform treatment of the integral equations for both the continuous and weakly singular kernel case, thus avoiding the use of possibly costly product integration rules for the evaluation of the integrals. (ii) when working with the new integrand, a number of the derivatives involved in the Euler– Maclaurin sum of the integration error formula are equal to zero. This results in an increase of the order of convergence of the method which is easy to achieve. Preliminary results for the case of VIEs and FIEs with continuous kernels were obtained in Makroglou [31]. The organization of this paper is as follows: In Section 2 some background material on the use of various transformations in numerical integration is presented. In Section 3 the description of the application of Laurie’s and Sidi’s [25,43] transformation to VIEs of the form (1.1) with continuous and weakly singular kernel is given. In Section 4 a convergence result is proved. In Section 5 the methods are extended to FIEs. Numerical results are presented in Section 6. These are obtained by using Laurie’s polynomial transformation and Sidi’s trigonometric transformation for equations with continuous and weakly singular kernels. For comparison reasons results are also obtained by using the polynomial type transformations of Korobov type. Finally Section 7 contains conclusions and ideas for further research.

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2. Background on the use of variable transformations in numerical integration (cf. [25; 43]) Consider I=

Z

b

f(x) d x

a

(2.1)

and its approximation by the trapezoidal rule IN = h

N X

wj f(jh);

(2.2)

j=0

w0 = wN = 0:5; wj = 1; j = 1; 2; : : : ; N − 1: An asymptotic expansion (Euler–Maclaurin summation formula) for the error is given by I − IN =

∞ X

k [f (2k−1) (b) − f (2k−1) (a)]h2k

(h → 0)

(2.3)

k−1

and k are constants (cf. [2]). By applying a change of variable x = (t); (a) = A; (b) = B, (2.1) is written I=

Z

B

A

0

f((t)) (t) dt ≡

Z

A

B

F(t) dt;

F(t) ≡ f((t))0 (t):

(2.4)

Since an integral over [a; b] can be easily transformed into one over [0; 1], the transformations considered in the literature are using a = 0; b = 1 and they map [0; 1] to [0; 1]. Since the aim is to make a number of the derivatives of the Euler–Maclaurin formula (2.3) equal to zero, the transformation (t) is chosen so that (t) has a sucient number of derivatives that vanish at t = 0 and t = 1. We also wish the transformation to be invertible, so 0 (t) ¿ 0 on (0; 1) is required. The trapezoidal rule then applied to the new integral (2.4) with xi = ih; i = 0; : : : ; N; h = 1=N , gives IN = h

N −1 X

f((jh))0 (jh):

(2.5)

j=1

Examples of transformations used in numerical integration include: 

Z

t 2m (t) = (2m + 1) [s(1 − s)]m ds m 0 m = 1; 2; : : : (Korobov [24], polynomial transformation),







1 c 1 1 (t) = 0:5 tanh − − + ; 2 t 1−t 2 ([41] with c = 1, the “tanh” transformation), Rt

(t) = R 01 0

w(s) ds w(s) ds



;

(2.6)

w(s) = exp −

c¿0

(2.7)



c ; t(1 − t)

c¿0

(2.8)

(Iri, Moriguti, Takasawa [19], the “IMT” transformation), 

 

1 1 1 − (t) = tanh a sinh b 2 t 1−t



1 + ; 2

a; b ¿ 0

(2.9)

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197

(Mori [34], the “DE” (double exponential) transformation), Rt

(t) = R 01 0

w(s) ds w(s) ds

;

w(s) = (sin s)m

(2.10)

m = 1; 2; : : : (Sidi [43], the “sinm ”-transformation) and Rt

(t) = R 01 0

w(s) ds w(s) ds

;

w(s) = w(1 − s);

w(s) v csm

(2.11)

c nonzero constant, m = 1; 2; : : : (Laurie [25], a special polynomial transformation). Remark. Sidi’s trigonometric transformation (2.10) is a member of a general class of transformations Sm (Sidi ([43], Section 2:1, p. 363)). A member (t) of Sm has the properties: (a) (t) ∈ C ∞ [0; 1] increases on [0; 1] and satis es (0) = 0; (1) = 1. (b) 0 (t) is symmetric with respect to t=1=2, i.e., 0 (t)=0 (1−t) and consequently (1−t)=1−(t). (c) 0 (t) has the asymptotic expansions: 0 (t) v

∞ X

i t m+2i

(t → 0+);

i (1 − t)

m+2i

i=0 0

 (t) v

∞ X

(2.12) (t → 1−);

i=0

where 0 ¿ 0. Since in this paper numerical results will be presented for: (i) Laurie’s special polynomial type transformation, (ii) Sidi’s trigonometric transformation, (iii) Korobov’s polynomial transformation, we state convergence results (Theorems 1–5) only for these transformations. Theorem 1 (Sidi ([43], p. 369)). The order of the trapezoidal rule transformed by the Korobov transformation (2:7) is m + 1 when m is odd and m + 2 when m is even. Theorem 2 (cf. Laurie ([25], p. 341)). The order of the trapezoidal rule transformed by the polynomial transformation (2:11) (Laurie’s) is m + 1 if m is odd. If m is even; then the order is 2m + 2 (best possible) if and only if w(s) satis es the 3m conditions: w( j) (0) = w( j) (1) = 0; ( j)

( j)

w (0) = w (1) = 0;

j = 1; 2; : : : ; m − 1; j = 1; 3; : : : ; 2m − 1:

(2.13)

Theorem 3 (Sidi [43], p. 365, 367). Consider the transformation (2:10). Let m = 2k − 1; k positive integer. Then for the error I −IN of the transformed trapezoidal rule (2:5) and f ∈ C 2p+1 [0; 1]; p¿k

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we have I − IN =

p X

 [F (2−1) (1) − F (2−1) (0)]h2 + O(h2p+1 )

=k

as h → 0; where F(t) = f((t))0 (t) is the transformed integrand and  are as in (2:3). Thus the error in IN is at worst O(hm+1 ) as h → 0. Let now m be even; m = 2k; k¿1. Assume f ∈ C 2p+1 [0; 1]; p¿q(m + 1); where q is the smallest integer ¿1 for which f (2q−1) (1) − f (2q−1) (0) 6= 0: Then I − IN =

p X

 [F (2−1) (1) − F (2−1) (0)]h2 + O(h2p+1 );

as h → 0:

=q(m+1)

Thus the error in IN is O(hq(2m+2) ) as h → 0. Theorem 4 (Sidi [43, p. 369], for functions with one end point singularity). Let f(x) = x g(x);  ¿ − 1; not an integer; g ∈ C 2p+1 [0; 1]; g(0) 6= 0. Then the error I − IN of the transformed trapezoidal rule (2:5) applied to (2:4) satis es I − IN = O(hw ) as h → 0 where; (

w=

min((m + 1)( + 1); m + 1); m odd; min((m + 1)( + 1); 2m + 2); m even:

Theorem 5 (Sidi [43, p. 370], for functions with singularities at both end points). Let f(x)=x (1− x) g(x);  ¿ − 1; ¿ − 1 not integers; g ∈ C 2p+1 [0; 1]; g(0) 6= 0. Then the error I − IN of the transformed trapezoidal rule (2:5) applied to (2:4) satis es (at worst) I − IN = O(hw )

as h → 0

where w = min((m + 1)( + 1); (m + 1)( + 1)): These theorems will be used in obtaining convergence results (Section 4). In next section a description of the methods applied to Volterra integral equations is given. 3. Description of methods We consider the application of variable transformation methods in combination with the trapezoidal quadrature rule to VIEs with continuous kernel (Section 3.1) and to VIEs with weakly singular kernel (Section 3.2). 3.1. Case of VIEs with continuous kernels Consider the VIE (1.1) with H (x; s) = 1, i.e. consider y(x) = f(x) +

Z

0

x

K(x; s; y(s)) ds;

06x6X:

(3.1)

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Two approaches have been considered: Approach 1. We write y(x) = f(x) + x

Z

1

0

K(x; xs; y(xs)) ds:

(3.2)

Using the transformation s = (t) and discretizing at x = xi = ih; i = 0; 1; : : : ; N; h = 1=N (X = 1 is used for simplicity), we obtain Z

y(xi ) = f(xi ) + xi

1

0

K(xi ; xi (t); y(xi (t)))0 (t) dt

(3.3)

or using the trapezoidal rule as given by (2.5), yi = f(xi ) + xi h

N −1 X

K(xi ; xi (xj ); y(xi (xj )))0 (xj );

(0 (t) = 0; at t = 0; 1)

(3.4)

j=1

or yi = f(xi ) + xi h

N −1 X



K xi ; xi (xj );

j=1

p X



l (xi (xj ))y−2+  0 (xj );

i = 1; 2; : : : ; N

(3.5)

=1

where  is such that xi (xj ) is in [h; ( + 1)h) (with the cases  = 0;  = N − 1 treated separately), ‘ (·) are the coecients of the Lagrange interpolation polynomial and p must be chosen as: p=

  m + 2;  

m even; (t) of Korobov’s type (2.6);

   m + 1;

m odd; (t) of Korobov’s, Sidi’s or Laurie’s type:

2m + 2;

m even; (t) of Sidi’s type (2.10), or Laurie’s type (2.11);

(3.6)

The disadvantage of this approach is that we need to solve a system of N equations instead of the one equation at a time of Approach 2. An advantage is that the application of the Lagrange interpolation method using more than 2 mesh points causes no problems. Approach 2. Consider Eq. (3.1). Discretizing at x = xi we obtain y(xi ) = f(xi ) + = f(xi ) +

Z

xi

0

K(xi ; s; y(s)) ds

i−1 Z X j=0

= f(xi ) + h

xj+1

xj

i−1 Z X j=0

0

1

K(xi ; s; y(s)) ds K(xi ; xj + hs; y(xj + hs)) ds:

(3.7)

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Using the transformation s = (t) we nd i−1 Z X

y(xi ) = f(xi ) + h

j=0

1

0

K(xi ; xj + h(t); y(xj + h(t)))0 (t) dt:

(3.8)

Thus using the trapezoidal quadrature formula for the evaluations of the integrals, ˜ −1 i−1 N X X

h˜

yi = f(xi ) + h

j=0

K(xi ; xj + h(tk ); y(x ˜ j + h(tk )))0 (tk )

(3.9)

k=1

˜ k = 0; : : : ; N˜ , and y(x where h˜ = 1= N˜ ; tk = k h; ˜ j + h(tk )) are approximations to y(xj + h(tk )) in terms of y-values at mesh points given by p X

y(x ˜ j + h(tk )) =

‘ ((tk ))y−p+j+1 :

(3.10)

=1

Approach 2 allows for (3.9) to be solved in yi ; i = 1; 2; : : : ; N , one at a time. Diculties occur with nding approximations of suciently high order for the case i = 1 when only two y-values y0 ; y1 are available for use. In this case the block-by-block approach of Linz [27] will need to be used for large values of m. 3.2. Case of VIEs with weakly singular kernels Consider the VIE (1.1) with H (x; s) given by (1.2), i.e. consider y(x) = f(x) +

Z

x

0

k(x; s; y(s)) ds;

0 ¡  ¡ 1; 06x6X;

(3.11)

where k(x; s; y(s)) =

K(x; s; y(s)) ; (x − s)

0 ¡  ¡ 1; 06s6x6X:

(3.12)

Starting from Eq. (3.11) and proceeding as in Section 3.1, Approach 2, we arrive at the equation y(xi ) = f(xi ) + h

i−1 Z X j=0

0

1

k(xi ; xj + h(t); y(xj + h(t)))0 (t) dt

(3.13)

which gives the approximate equations, yi = f(xi ) + h

˜ −1 i−1 N X X

h˜

j=0

k(xi ; xj + h(tr ); y(x ˜ j + h(tr )))0 (tr )

(3.14)

r=1

where h˜ = 1= N˜ ;

˜ r = 0; 1; : : : ; N˜ ; tr = r h;

and y(x ˜ j + h(tr )) are approximations to y(xj + h(tr )) in terms of y-values at mesh points, de ned similarly to (3.10).

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We may note that the fact that 0 (t0 ) ≡ 0 (0); 0 (tN˜ ) ≡ 0 (1) = 0 allows the summation with respect to r in method (3.14) to go from r = 1 to r = N˜ − 1 and thus the evaluation of k(x; s; y(s)) at points where x = s to be avoided. This permits treatment of both the continuous and weakly singular case in a similar manner without complicated and costly calculations for the evaluation of the weakly singular integrals. In next section (Section 4) the convergence of the methods will be examined. 4. Convergence analysis The order of convergence of VIEs with continuous kernels will be examined in Section 4.1 and that of the VIEs with weakly singular kernels in Section 4.2. The analysis will be given for Approach 2. Results of Section 2 on numerical integration with variable transformations will be used. 4.1. Case of VIEs with continuous kernel Consider Eqs. (3.8) and (3.9), i.e. consider y(xi ) = f(xi ) + h

i−1 Z X 0

j=0

yi = f(xi ) + h

1

˜ −1 i−1 N X X

h˜

j=0

K(xi ; xj + h(t); y(xj + h(t)))0 (t) dt;

K(xi ; xj + h(tk ); y(x ˜ j + h(tk )))0 (tk ):

(4.1)

(4.2)

k=1

Subtracting, (4:1)–(4:2), gives y(xi ) − yi = h

i−1 Z X j=0

−h

1

0

˜ −1 i−1 N X X

h˜

j=0

+h

K(xi ; xj + h(t); y(xj + h(t)))0 (t) dt

k=1

˜ −1 i−1 N X X

h˜

j=0

K(xi ; xj + h(tk ); y(xj + h(tk )))0 (tk )

[K(xi ; xj + h(tk ); y(xj + h(tk )))

k=1

˜ j + h(tk )))]0 (tk ): − K(xi ; xj + h(tk ); y(x

(4.3)

Let ei = y(xi ) − yi :

(4.4)

Then |ei |6h

i−1 X j=0

˜ +h |Ti; j (h)|

i−1 X j=0

˜ 3 hL

˜ −1 N X k=1

|y(xj + h(tk )) − y(x ˜ j + h(tk ))| |0 (tk )|;

(4.5)

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˜ is the quadrature where L3 is a Lipschitz constant of K with respect to the 3rd variable and Ti; j (h) error. Using (3.10), inequality (4.5) gives  ˜ −1  p i−1 N X X X ˜ ˜ |ei | 6 hNT1 (h) + hhL3 M y(xj + h(tk )) − l ((tk ))y(x−p+j+1 )  j=0 k=1 =1  X  p + l ((tk ))[y(x−p+j+1 ) − y−p+j+1 ] ; =1 

˜ = maxi; j=0;1;:::; N |Ti; j (h)|, ˜ and |0 (t)|6M; t ∈ [0; 1]. Thus, where T1 (h) ˜ + hhL ˜ 3M |ei |6hNT1 (h)

 ˜ −1  i−1 N X X

T2 (h) +

j=0 k=1

where T2 (h) =

max

j=0;1;:::; N −1 k=1;2;:::; N˜ −1

Let L = max06t61;



p X

 

|l ((tk ))| |e−p+j+1 | ;

(4.6)



=1

p X y(xj + h(tk )) − l ((t ))y(x )  k −p+j+1 : =1

k=1; 2;:::;p

(4.7)

|lk (t)|. Then,

˜ + h(N˜ − 1)hL ˜ 3M |ei | 6 XT1 (h)

 i−1  X

T2 (h) + L

j=0

˜ + hL3 M 6 XT1 (h)

 



NT2 (h) + L



p i−2 X X

p X

 

|e−p+j+1 |



=1

p−1

|e−p+j+1 | + L

j=0 =1

X =1

 

|e−p+i | : 

Therefore, ˜ + hL3 MNT2 (h) + hL3 ML |1 − hL3 ML| |ei | 6 XT1 (h) ˜ + L3 MXT2 (h) + hL3 MLp 6 XT1 (h)

 p i−2 X X 

j=0 =1

i−1 X

|ej |:

p−1

|e−p+j+1 | +

X =1

 

|e−p+i |



(4.8)

j=0

Using the discrete Gronwall inequality we obtain the result q max |ei | = O(h˜ ) + O(hp )

i=1;:::; N

(4.9)

where q is the order of the quadrature rule and p is the order of the interpolation error given by (4.7).

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Remark. If we use Korobov’s transformation with m even then m+2 max |ei | = O(h˜ ) + O(hp ):

(4.10)

If we use Sidi’s or Laurie’s transformation with m even then 2m+2 ) + O(hp ): max |ei | = O(h˜

(4.11)

i=1;:::; N

i=1;:::; N

4.2. Case of VIEs with weakly singular kernels Consider Eqs. (3.13) and (3.14), i.e., consider the equations (with x = xi ) i−1 Z X

y(xi ) = f(xi ) + h

j=0

where k(x; s; y(s)) =

0

1

k(xi ; xj + h(t); y(xj + h(t)))0 (t) dt;

K(x; s; y(s)) ; (x − s)

(4.12)

0 ¡  ¡ 1; 06s6x6X;

(4.13)

k(xi ; xj + h(tr ); y(x ˜ j + h(tr )))0 (tr );

(4.14)

and yi = f(xi ) + h

˜ −1 i−1 N X X

h˜

j=0

where y(x ˜ j + h(tr )) =

r=1

p X

l ((tr ))y−p+j+1

(4.15)

=1

and l (·) are the coecients of the Lagrange interpolating polynomial. q Using the usual add and subtract method we obtain that max16i6N |ei | = O(h˜ h1− ) + O(hp ) where q is the order of the quadrature rule and p is the order of the interpolation with the (implicit) assumption that y(x) ∈ C k [0; X ] for k suciently large. Thus for (t) being the Sidi transformation min[2m+2; (m+1)(+1)] h1− ) + O(hp ) as given by (2.10) using Theorem 4 we nd that max16i6N |ei | = O(h˜ for m even. 5. Extension to FIEs The application of the methods to FIEs with continuous and weakly singular kernel is described in Sections 5:2 and 5:3, respectively. Without loss of generality, we will consider Eq. (1.3) with b = 1. 5.1. Case of FIEs with continuous kernel Consider the FIE (1.3) with H (x; s) = 1, i.e., consider, y(x) = f(x) +

Z

0

1

K(x; s; y(s)) ds;

06x61:

(5.1)

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We proceed as in Section 3.1 except that we apply the same transformation to the x variable as well. Thus we arrive at the equation y((w)) = f((w)) +

Z

1

0

K((w); (t); y((t)))0 (t) dt:

(5.2)

˜ Setting Y (w) ≡ y((w)); f(w) = f((w)), Eq. (5.2) is written as Z

˜ Y (w) = f(w) +

1

0

K((w); (t); Y (t))0 (t) dt;

(5.3)

which is FIE of the second kind in Y (w). Discretization of Eq. (5.3) using the trapezoidal quadrature rule for the evaluation of the integrals then follows. Results for the choices (t) given by the Korobov, Sidi and Laurie transformations are given in Section 6. 5.2. Case of FIEs with weakly singular kernel We consider equations of the form (1.3) with H (x; s) given by (1.4), i.e. we consider y(x) = f(x) + We then write y(x) = f(x) +

Z

1

K(x; s; y(s)) ds; |x − s|

x

K(x; s; y(s)) ds + (x − s)

0

Z 0

0 ¡  ¡ 1; 06x6X = 1: Z

1

x

K(x; s; y(s)) ds: (s − x)

(5.4)

(5.5)

Using change of variable s = xt and s = x + (1 − x)t respectively for the two integrals, we arrive at the equations y(x) = f(x) + x

1−

+(1 − x)1−

Z

1

K(x; xt; y(xt)) dt (1 − t)

1

K(x; x + (1 − x)t; y(x + (1 − x)t)) dt: t

0

Z

0

(5.6)

The last equation contains two integrals with end point singularities. Then we use a transformation of the form t = (w) and proceed as before. 6. Numerical results Numerical results are obtained for 6 examples, two VIEs with continuous kernel (Examples 1 and 2) two VIEs with weakly singular kernel (Examples 3 and 4), one FIE with continuous kernel (Example 5) and one FIE with weakly singular kernel (Example 6). All examples were solved for 06x61. Rx

(1) y(x) = x + cos(x) − 1 + 0 Rsin(y(s)) ds; y(x) = x (constructed) x x (2) y(x) = ex − 0:5(e2x − R1) + 0 y2 (s) √ ds; y(x) = e (constructed) √ x 3 2 1=2 3 (3) y(x) = 8 x + x + 0 (−y (s)= x − s) ds; y(x) = x (Te Riele [39])

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Table 1 Example 1 (VIE with continuous kernel), Approach 1 (A system of N × N equations is solved in yi ; i = 1; : : : ; N ) Korobov’s

m=2

Sidi’s

m=2

Laurie’s

m=2

x

h = 0:1

h = 0:05

h = 0:1

h = 0:05

h = 0:1

h = 0:05

0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0

0:53D − 6 0:21D − 5 0:49D − 5 0:90D − 5 0:14D − 4 0:21D − 4 0:28D − 4 0:37D − 4 0:46D − 4 0:56D − 4

0:32D − 7 0:13D − 6 0:31D − 6 0:56D − 6 0:88D − 6 0:13D − 5 0:18D − 5 0:23D − 5 0:29D − 5 0:35D − 5

0:28D − 10 0:45D − 9 0:23D − 8 0:74D − 8 0:18D − 7 0:38D − 7 0:71D − 7 0:12D − 6 0:19D − 6 0:29D − 6

0:10D − 12 0:56D − 11 0:34D − 10 0:11D − 9 0:28D − 9 0:58D − 9 0:11D − 8 0:18D − 8 0:29D − 8 0:43D − 8

0:52D − 8 0:22D − 7 0:52D − 7 0:98D − 7 0:16D − 6 0:25D − 6 0:36D − 6 0:51D − 6 0:68D − 6 0:89D − 6

0:81D − 10 0:34D − 9 0:81D − 9 0:15D − 8 0:25D − 8 0:39D − 8 0:56D − 8 0:78D − 8 0:10D − 7 0:14D − 7

Table 2 Example 2 (VIE with continuous kernel), Approach 1 (A system of N × N equations is solved in yi ; i = 1; : : : ; N ) Korobov’s

m=2

Sidi’s

m=2

Laurie’s

m=2

x

h = 0:1

h = 0:05

h = 0:1

h = 0:05

h = 0:1

h = 0:05

0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0

0:12D − 4 0:31D − 4 0:60D − 4 0:11D − 3 0:18D − 3 0:29D − 3 0:49D − 3 0:82D − 3 0:14D − 2 0:25D − 2

0:76D − 6 0:20D − 5 0:38D − 5 0:67D − 5 0:11D − 4 0:19D − 4 0:31D − 4 0:52D − 4 0:90D − 4 0:16D − 3

0:66D − 8 0:34D − 7 0:11D − 6 0:30D − 6 0:71D − 6 0:15D − 5 0:31D − 5 0:62D − 5 0:12D − 4 0:23D − 4

0:96D − 10 0:45D − 9 0:16D − 8 0:44D − 8 0:11D − 7 0:23D − 7 0:46D − 7 0:92D − 7 0:18D − 6 0:35D − 6

0:12D − 6 0:27D − 6 0:47D − 6 0:71D − 6 0:95D − 6 0:12D − 5 0:13D − 5 0:11D − 5 0:27D − 6 0:20D − 5

0:18D − 8 0:44D − 8 0:76D − 8 0:11D − 7 0:16D − 7 0:20D − 7 0:24D − 7 0:24D − 7 0:18D − 7 0:46D − 8

√ √ Rx √ (4) y(x) = 1= 1 + x + =8 − 14 sin−1 ((1 − x)=(1 + x)) − 14 0 (y(s)= x − s) ds; y(x) = 1= 1 + x (Linz [26]) R1 (5) y(x) = x3 + 13 (cos(1) − 1) + 0 s2 sin(y(s)) ds; y(x) = x3 (constructed) p √ R1 (6) y(x) = f(x) + 0 (y(s)= |x − s|) ds; f(x) = x − 2x 1 − x − 23 (1 − x)3=2 − 43 x3=2 ; y(x) = x (constructed). The entries in Tables 1– 6 (for VIEs) correspond to absolute errors |y(xi )−yi |; i=1; 2; : : : ; N , where y(xi ) represents the true solution evaluated at x = xi and yi the corresponding approximate solution, and the entries in Tables 7a, 7b and 8 (for FIEs) correspond to absolute maximum errors |y((wi ))−yi |; wi = ih; i = 1; 2; : : : ; N . Finally, Table 9 gives an example of values of ve transformations for ten equidistant mesh points in [0; 1]. Results in Tables 1 and 2 verify orders of convergence O(h4 ); O(h6 ) and O(h6 ) respectively for the Korobov, Sidi and Laurie transformations as expected from the convergence result of

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Table 3 Example 3 (VIE with weakly singular kernel), Sidi’s transformation, Approach 2 (solves one equation at a time for yi ) m=3

m=4

x

N = 20; N˜ = 20

N = 40; N˜ = 40

N = 20; N˜ = 20

N = 40; N˜ = 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0:46D − 3 0:17D − 3 0:18D − 3 0:17D − 3 0:17D − 3 0:17D − 3 0:18D − 3 0:18D − 3 0:18D − 3 0:18D − 3

0:37D − 4 0:34D − 4 0:31D − 4 0:30D − 4 0:30D − 4 0:30D − 4 0:31D − 4 0:31D − 4 0:32D − 4 0:32D − 4

0:48D − 3 0:12D − 3 0:10D − 3 0:79D − 4 0:62D − 4 0:52D − 4 0:45D − 4 0:40D − 4 0:37D − 4 0:35D − 4

0:33D − 4 0:25D − 4 0:18D − 4 0:13D − 4 0:96D − 5 0:77D − 5 0:64D − 5 0:56D − 5 0:51D − 5 0:47D − 5

Table 4 Example 4 (VIE with weakly singular kernel), Sidi’s transformation, Approach 2 (solves one equation at a time in yi ) m=3

m=4

x

N = 20; N˜ = 20

N = 40; N˜ = 40

N = 20; N˜ = 20

N = 40; N˜ = 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0:18D − 3 0:15D − 3 0:14D − 3 0:13D − 3 0:12D − 3 0:11D − 3 0:10D − 3 0:96D − 4 0:91D − 4 0:86D − 4

0:30D − 4 0:27D − 4 0:24D − 4 0:22D − 4 0:21D − 4 0:19D − 4 0:18D − 4 0:17D − 4 0:16D − 4 0:15D − 4

0:26D − 4 0:22D − 4 0:20D − 4 0:18D − 4 0:17D − 4 0:15D − 4 0:14D − 4 0:14D − 4 0:13D − 4 0:12D − 4

0:30D − 5 0:26D − 5 0:24D − 5 0:22D − 5 0:20D − 5 0:19D − 5 0:18D − 5 0:17D − 5 0:16D − 5 0:15D − 5

Section 4.1. Comparing results for methods of the same order of convergence, we may note that Sidi’s transformation with m = 2 produced more accurate results than Laurie’s for Example 1. For Example 2, the same behavior was observed, but errors for the former method increased more rapidly than for the latter towards the second half of the inverval of integration. Tables 3,4 contain results for Examples 3 and 4 using Sidi’s transformation with m = 3, m = 4 and approach 2. The true solution was used to nd y1 . Results were also obtained for both Examples 3 and 4 by using Laurie’s transformation with m=2. These were compared with corresponding ones obtained using Sidi’s transformation with m = 2 and were found of comparable accuracy. From Tables 3,4 we may note that orders of convergence O(h2:5 ) for m = 3 and O(h3 ) for m = 4 are obtained, in agreement with the convergence result of Section 4.2. Tables 5,6 contain results for Examples 3 and 4 using Sidi’s transformation with m = 3, m = 4 and approach 1.

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Table 5 Example 3 (VIE with weakly singular kernel), Sidi’s transformation, Approach 1 (a system of N × N equations is solved in yi ) m=3

m=4

x

h = 0:05

h = 0:025

h = 0:05

h = 0:025

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0:10D − 3 0:11D − 3 0:20D − 3 0:32D − 3 0:43D − 3 0:53D − 3 0:61D − 3 0:70D − 3 0:79D − 3 0:84D − 3

0:24D − 5 0:26D − 4 0:53D − 4 0:80D − 4 0:11D − 3 0:13D − 3 0:15D − 3 0:17D − 3 0:19D − 3 0:21D − 3

0:13D − 3 0:13D − 4 0:23D − 4 0:72D − 4 0:46D − 4 0:82D − 4 0:97D − 4 0:92D − 4 0:10D − 3 0:12D − 3

0:45D − 5 0:29D − 5 0:43D − 5 0:78D − 5 0:10D − 4 0:15D − 4 0:15D − 4 0:18D − 4 0:21D − 4 0:21D − 4

Table 6 Example 4 (VIE with weakly singular kernel), Sidi’s transformation, Approach 1 (a system of N × N equations is solved in yi ) m=3

m=4

x

h = 0:05

h = 0:025

h = 0:05

h = 0:025

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0:24D − 3 0:31D − 3 0:35D − 3 0:38D − 3 0:39D − 3 0:41D − 3 0:42D − 3 0:43D − 3 0:43D − 3 0:43D − 3

0:60D − 4 0:77D − 4 0:87D − 4 0:94D − 4 0:99D − 4 0:10D − 3 0:10D − 3 0:11D − 3 0:11D − 3 0:11D − 3

0:34D − 4 0:44D − 4 0:50D − 4 0:54D − 4 0:57D − 4 0:59D − 4 0:60D − 4 0:61D − 4 0:62D − 4 0:63D − 4

0:61D − 5 0:78D − 5 0:89D − 5 0:96D − 5 0:10D − 4 0:10D − 4 0:11D − 4 0:11D − 4 0:11D − 4 0:11D − 4

Table 7a Example 5 (FIE with continuous kernel) Korobov’s,

m=2

h = 0:1 0:53D − 4

h = 0:05 0:36D − 5

h = 0:025 0:23D − 6

Laurie’s,

m=2

h = 0:1 0:23D − 5

h = 0:05 0:34D − 7

h = 0:025 0:53D − 9

For Examples 3 and 4, Laurie’s and Sidi’s transformations with m = 2 were also used. The results were of comparable accuracy. Laurie’s transformation with Approach 1 for example, at x = 1:0 with h = 0:05; h = 0:025 for Example 3, gave errors equal to 0:44D − 2 and 0:15D − 2, respectively. For Example 4 the corresponding errors were: 0:23D − 2 and 0:80D − 3. The errors for Sidi’s method

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Table 7b Example 5 (FIE with continuous kernel) Sidi’s,

m=3

h = 0:1 0:72D − 4

h = 0:05 0:45D − 5

h = 0:025 0:28D − 6

Sidi’s,

m=4

h = 0:1 0:21D − 5

h = 0:05 0:50D − 10

h = 0:025 0:46D − 13

Table 8 Example 6 (FIE with weakly singular kernel), Sidi’s transformation m=3

m=4

x

h = 0:05 N = N˜ = 20

h = 0:025 N = N˜ = 40

h = 0:05 N = N˜ = 20

h = 0:025 N = N˜ = 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0:38D − 2 0:44D − 2 0:29D − 2 0:16D − 4 0:33D − 2 0:60D − 2 0:78D − 2 0:69D − 2 0:45D − 2 0:18D − 2

0:95D − 3 0:11D − 2 0:73D − 3 0:18D − 4 0:79D − 3 0:15D − 2 0:18D − 2 0:17D − 2 0:12D − 2 0:49D − 3

0:55D − 3 0:64D − 3 0:41D − 3 0:42D − 5 0:47D − 3 0:86D − 3 0:11D − 2 0:98D − 3 0:64D − 3 0:26D − 3

0:97D − 4 0:11D − 3 0:75D − 4 0:24D − 5 0:80D − 4 0:15D − 3 0:19D − 3 0:18D − 3 0:12D − 3 0:50D − 4

Table 9 Transformation values: (xi ); i = 1; 2; : : : ; N = 10 Korobov, m = 2

Laurie, m = 2

Sidi, m = 2

Sidi, m = 3

Sidi, m = 4

0.00856 0.05792 0.16308 0.31744 0.50000 0.68256 0.83692 0.94208 0.99144 1.00000

0.00681 0.05055 0.15197 0.30915 0.50000 0.69085 0.84803 0.94945 0.99319 1.00000

0.00645 0.04863 0.14863 0.30645 0.50000 0.69355 0.85137 0.95137 0.99355 1.00000

0.00177 0.02561 0.10993 0.27561 0.50000 0.72439 0.89007 0.97439 0.99823 1.00000

0.00050 0.01377 0.08259 0.25004 0.50000 0.74996 0.91741 0.98623 0.99950 1.00000

with m = 2 at x = 1 with stepsizes h = 0:05 and h = 0:025 were equal to 0:43D − 2 and 0:15D − 2 for Example 3 and 0:22D − 2; 0:77D − 3 for Example 4 respectively. Tables 7a,7b contain results for Example 5 (FIE with continuous kernel). Table 8 contains results for Example 6 (FIE with weakly singular kernel) using Sidi’s transformation with m = 3; 4. The observed orders of convergence here are O(h4 ); O(h6 ) (Table 7a), O(h4 ); O(h10 ) (Table 7b) and O(h2 ); O(h2:5 ) (Table 8). Results with m = 2 using Laurie’s and Sidi’s transformations for

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Examples 5 and 6 gave almost identical results. So, for Example 6 at x = 1 with N = N˜ = 20 and N = N˜ = 40 the errors were 0:31D − 1 and 0:65D − 2 for Sidi’s transformation and very similar for the Laurie one. Convergence results similar to those in Sections 4.1 and 4.2 are expected to be valid for FIEs too. This needs further investigation. Table 9 shows the distribution of the transformed points xi ; i = 1; 2; : : : ; N = 10 under 4 dierent transformations. The transformed points appear to cluster towards the end points. 7. Conclusions In this paper the application of transformations of Korobov, Laurie and Sidi type to Volterra integral and Fredholm integral equations was considered in combination with the trapezoidal quadrature rule, extending corresponding results of numerical integration. The main advantages of the approach are that the methods can achieve high order of convergence easily by just changing the value of the transformation parameter m and that they can be applied to both equations with continuous and weakly singular kernels. It is believed that for equations with weakly singular kernels to fully bene t from the approach, preprocessing of the equations with solution regularization transformations of the type presented for example in Norbury and Stuart [35,36] or Diogo et al. [12] is needed rst. The idea can be applied to other types of equations and also to approximation of functions and it deserves further investigation with other quadrature rules and other types of transformations or combinations of them. Acknowledgements The third author (Makroglou) wishes to ackowledge that her part of the work of this paper began at the Department of Computer Science and Mathematics of the University of Portsmouth, UK, continued during a visit to the Department of Mathematics of the Iowa State University in July 1998, completed during a visit to the Department of Mathematics of the Aegean University, Samos, Greece and revised at the University of Portsmouth in May 1999. She wishes to thank all three Departments involved. Thanks are also due to Mrs. Ruth Deboer, Dept. of Mathematics, Iowa State University for her careful typing that she kindly oered to do and to Mr. Mike Fletcher, former Computer Specialist of the same Department for his valuable assistance. All four authors would like to thank NATO, for the collaborative grant (CRG 940076) without which their collaboration would not have been possible. They also wish to thank an anonymous referee for his helpful comments and for bringing to their attention an important additional reference. References [1] S. Abelman, D. Eyre, A rational basis for second kind Abel integral equations, J. Comput. Appl. Math. 34 (1991) 281–290. [2] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematical Series, Vol. 55, Dover, Washington, 1970. [3] K.E. Atkinson, The numerical solution of Fredholm integral equations of the second kind with weakly singular kernels, Numer. Math. 19 (1972) 248–259.

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