Product Integration for the Generalized Abel … Integration for the Generalized Abel Equation By Richard Weiss Abstract. The solution of the generalized Abel integral equation gi')

MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 117, JANUARY, 1972

Product Integration for the Generalized Abel Equation

By Richard Weiss

Abstract. The solution of the generalized Abel integral equation

gi') = [kit, *)/« - s)"}f(s) ds, 0 < a < 1,

where kit, s) is continuous, by the product integration analogue of the trapezoidal method

is examined. It is shown that this method has order two convergence for a £ [at. 1) with

ai = 0.2117. This interval contains the important case a = \. Convergence of order two

for a £ (0, ai) is discussed and illustrated numerically. The possibility of constructing

higher order methods is illustrated with an example.

1. Introduction. The generalized Abel integral equation

(1.1) git) = f S\ f(s)ds, 0<«<l,0g(gr<«=,Jo {t — s)

where k(t, s) is continuous for 0 g s rg t ^ T and

(1.2) k{t, <)^0,

is of considerable importance in many fields. For example, consider the equation

(1.3) Y(y) = 2 f1 2 R(r\1/2 dr, 0 g ^ 1,K 0- — y )

which occurs frequently in mathematical physics. By a change of variables, (1.3)

can be transformed to

(1.4) git) = f . Ks\l/2ds, O^t^l,J0 (t — i)

which is an important special case of (1.1). Various numerical methods for per-

forming the inversion of (1.3) as well as (1.4) have been developed.-A bibliography

can be found in Minerbo and Levy [1].

As a second example, consider equations of the type

Or)-172 exp[-Ma(0}7']

(1.5)

= / {2x(f - SV1'2} exp[-i{a(0 - ais)}2/« - *)]/(*) ds,

where a(t) is a differentiable function with o(0) > 0. Such equations and systems of

Received May 10, 1971.

AMS 1970 subject classifications. Primary 65L05.Key words and phrases. Generalized Abel equation, numerical solution, product integration

methods.

Copyright © 1972, American Mathematical Society

177

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

178 richard weiss

similar structure appear in the analysis of diffusion processes [see Fortet [2]}. A

method for solving (1.5) has been proposed by Durbin [3].

However, none of the methods suggested for (1.4) as well as (1.5) can be applied

to the general case (1.1). Only methods based on the concept of product integration

{see Young [4]} appear to be applicable. Such methods have been suggested on

the basis of rigorous numerical experimentation by Linz [5] and Noble [6]. Inde-

pendent support for this can be found in [3], though the product integration type

method developed there is not generally applicable.

A theoretical justification for the use of product integration methods can be

found in Weiss and Anderssen [7]. They show that the midpoint product integration

method is convergent of order one if f(s) and k(t, s) are suitably smooth and con-

vergent of order at least (1 — a) if weaker smoothness conditions apply.

In this paper, we extend the result of [7] by examining the applicability of other

product integration methods to (1.1). Initially, we show that the use of trapezoidal

product integration leads to a method with second order convergence. The result

again depends on the smoothness of /(r) and k(t, s). In addition, the possibility of

determining higher order methods is discussed.

The existence and uniqueness of solutions of (1.1) is considered in Section 2.

In Section 3, we give a basic definition and a lemma required in the subsequent

analysis of Section 4, where we define the trapezoidal method and investigate its

convergence. In Section 5, we examine the effect of computational errors, and a

numerical example is presented in Section 6. A third order method is considered

in Section 7.

2. Analytic Solution of (1.1). The existence and uniqueness of solutions of

(1.1) is guaranteed under the following conditions:

Theorem 2.1 {Kowalewski [8, Section 1, pp. 80-82]}. The integral equation

(1.1) has a unique continuous solution /(r), t £ [0, T], if k(t, s) and g(f) satisfy

(i) k(t, s) and dk(t, s)/dt are continuous for 0^1^(51,

(ii) k(t, t) * Ofort £ [0, T],(iii) G(r) = d/dt f'Q (g(s)/(t - s)1'") ds is continuous for t £ [0, T].

3. Preliminaries. To solve (1.1) we introduce the grid {r, - /Ar, / = 0, 1, • • • ,

/ = T/At} with gridspacing At. An approximate solution, defined with respect to

these grid points, will be denoted by /,.

Definition 3.1. Let Ji(At) (i = 0, 1, ■ • •) denote the approximation to /(r,) ob-

tained by a finite difference method with gridspacing At. Then, the method is said

to be convergent if and only if

max 17, (AO - fitdl -* 0OSiS/

as Ar —> 0, I —> oo such that I At = T. Furthermore, the method is said to be con-

vergent of order p, if p is the largest value for which there exists a finite constant C

and a Ar0 such that

max I/,(Ar) - /(r,)l ^ CAr"0S.S7

for all 0 < At ^ At0.


the generalized abel integral equation 179

The convergence proof developed in Section 4 is based on the following lemma

which is a consequence of the standard results for regular infinite systems of algebraic

equations {see Kantorovich and Krylov [9, Section 2, p. 27]}.

Lemma 3.1. Let the real numbers xtii — 1» 2, • • •) satisfy

|*iI ^ K, \x2\ ?s K, K is const,

l**+»l =s E k.,i \xA + \b<\ a = 2,3, •••),

. withi

p. = i - Z k.,1 > o,i-l

IM ^ #p, 0" = 2, 3, •••).

/Aen |x4| g JT(f - 1, 2, •••).

Throughout the paper, we use the notation

In addition, the subscripted capitals Cu C2, • • • will always denote positive real

constants.

4. The Trapezoidal Product Integration Method. We replace s)f(s) in

Jo (r,. — i)

by the piecewise linear function

Mi,(tits) = + 1 - sykt.J, + (s - ^)£,,)+17, + I]/Af

[t, ^ s rS ti+1 0 = 0, l)},

and thus obtain the following lower triangular system for /< (/ = l, ■ • • , /):

(4.1) £ = gi/(A(1->(«)) - /0A:,.ofP.- (/ = 1, , /),

where

*(*) = —1-t-^— ,I — a 2 — a

<*(<*) A<2_° Jo fa0(a)

.(I+DAl

^' rfl + /0(a)

= (/ + l)2"* - 2l2~a + (I - 1)2~° (#—-.»,«•• . 7 — 1).

<6(a) A/ VJc,-nil ,« t" I


180 richard weiss

0(a) At ° f

/ _L_ (/-« _ (/ _ _ (L-Jl (/- _ (/ _ !)"«)}1,2 — a 1 — a J

= T71-;0(a) 1,2 — a

(I - 1. ••• . I).

For the starting value /0, we take

f0 = U - Hm (gW^^Xl ~ «)Ao.o,I-.0

which exists, whenever (1.1) has a continuous solution. It follows from (1.2) that

(4.1) is nonsingular.

The numerical process (4.1) defines the product integration analogue of the

trapezoidal rule applied to nonsingular first kind Volterra equations {see Linz

[10]}. We shall refer to it as the trapezoidal product integration method.

As in [7], we first investigate the convergence of (4.1) for the case k(t, s) = I.

Theorem 4.1. 7/7(0 's three times differentiable and f"(t) is boundedfor 0 g t g T,

then, for a £ [au 1) with ax = 0.2117, (4.1) is convergent of order 2.

Let /4 ,(s) denote the piecewise linear function interpolating f(s) at the grid points,

fitO) - [(',+> - tilt + (/- tdh.iVAt r, + I 0' = 0, •••,/- 1)},

and let

eA((i) = /a<(s) - /(*).

By subtracting

Ar1_o0(a) \Jo (/, - Jo (/, - a)« /

from (4.1), we obtain

(4.2) = --r^— f & " ^< (/ = 1, ••• , /)•At 0(a) Jo (,,. - S)

Finally, subtraction of (4.2) from (4.2), with /' replaced by i -f- 1, yields the basic

error equation

(4.3) e«+, - £ 6,fl,_, + fc, (j = 1, • • • , 7 - 1),j-i

with

«0 = Wo - Wi = 3 - 22-°, a, - IF, - r,+, (I - 1, •••,/- 1),

0(a) Af1 "

(4.4) ft = *-g p (—L_ - —-5"-^*« *.(/, + 1 — s) i-o J<, — s) (f<+, — s) /

ft2 = £o(fF, - Wi+1).



Before we give a proof of the theorem, it is first necessary to obtain some preliminary

results.

Lemma 4.1.

\ß)\ is cAf—r- (/= l, •••,/ - i),

(4.5)

\ß2\ g cMrl~a 0 = l, •••,/- i), if |e0| ^ c3 a*2.

Proof. By a standard theorem on Lagrange interpolation

= Us - ',)(',+. - *)/"(/,) + r,(s)

[t, g s g rI+1 V = 0, •••,/- 1)},

with

(4.6) M»)| g 17*3Ar3, F3 = max /"'(/).ieio. t]

By substituting (4.5) into (4.4), we obtain

ß) - Ar3-°(/"(r,)7 - z HWfw) + *<•

where

1

2 At Jo

0, = -~- f s'At - s)(-) ds,2 At3-" Jo \l At + s)a ((/ + 1) Ar + s)a>

jjBr_i^_^irp-l_aJ« (r.+1 - *)- f=i J,t Xu - s)° (ti+1 - s)aJ

It can easily be verified that, by (4.6),

(4.7) \6i\ g C4 At*~a

and that

(4.8) 0 < 0o, 0 < 6, < CJ

(f - 1, 1),

».tI c.r'.

/ ^ i.

f ^ l.

(s) cfa.

(4.9)

By partial summation,

z /"(*,)««_*-, - f(f«-i) z »i + z (/"('.-,-,) - /"('<-»)) X z «ij-0 1-0 »-l !-»

(/ = 2, ■•• , / - 1)

and, using (4.8),

(4.10) z (T(t<-,-i) - /"('<-,)) X z »i rS C7 Ai/1"" (i = 2, • • • ,7-1).


182 RICHARD WEISS

Combining (4.9) and (4.10), we obtain

/"(/<>Y - Z /"(*,)*«-,-.

(4.11) ^ f"(tJy - Z 0«)\ 1-0 /

g F2C6ra + 7F3 A/ + C7 Afi

+ l/'U) - /'U-i)| X Z -5, + C7 A//1"

(/ = 1, ••• , / - 1), Fa = max /"(*)•• eio.ri

Since

Ar = T/I rg T/i (i = 1, ■ •• , /),

it follows from (4.7) and (4.11) that

\ß]\ £ c, A/3-ar" («• = i,... , / _ i).

The second inequality of Lemma 4.1 is an immediate consequence of the asymp-

totic expansion of (W, — W,+1). □

Proof of Theorem 4.1. Using Lemma 4.1 it is clear that

(4.12) ^ CsAfra (I - 1, •••,/- 1).

Returning to (4.3), we note that

(4.13)

(4.14)

(4.15)

Z*i « 1 - Wt\ 1.

a. > 0, / £ 1,

0 g a0 < 1. a G [«o, 1),• • ■

-1 < a0 < 0, a G (0, a0), a0 = 2 - == 0.4150.In 2

Substituting for e, in (4.3), we obtain

(4.16) e<+1 = Z + 6. + «o*<-i (i = 2, •••,/- 1),i-i

with

a0 = 0, a} = a, + a0a'-i, /SI,

and

al > 0, a G (0, 1),

a) ^ 0, Ü 2, a G t«i, 1), «i =4= 0.2117,

al < 0, a G (0, «,).

The latter is a consequence of the structure of {a,(a), / ^ Oj. From (4.13),

(4.17)

Z a\ = 1 - (W, + ao^-i), » ̂ 2,



and, since

eg

W, = a, + Wt+l = £ a„ i ^ 0,

it follows from (4.17) that

W< + flo^.-i = Z) (a. + «ofl,-i) = Z «i > 0, i ^ 2, a G 1).

Consequently,

(4.18) £a} g 11-0

since behaves asymptotically like

It follows from (4.2) that

(4.19) k| g C10Af2, Id -g C10A/2.

We are now in a position to apply Lemma 3.1 to (4.16). On the basis of (4.19) together

with (4.17), (4.18) and (4.12), the conditions of the lemma are satisfied and the requiredresult

W ^ C„A? (/ = 0, ••• , I), a G [«„ 1)

follows. □

Note 4.1. Using Lemma 3.1 and (4.13)—(4.15), we can prove second order con-

vergence for (4.1) with a G [«o, 1) without making a substitution of the type (see

(4.16)} introduced in the proof of Theorem 4.1. On the other hand, if we substitute

for e,-_2 instead of e, in (4.16) using (4.3), we have

i

(4.20) ei + 1 = £ e,ö2_, + bt + «o0,-i + «5*,-, (i = 4, •••,/- 1)i-i

with

2 A 2 1 2n 2 111 ■ *w 1aa = 0, «! = au a2 = 0, a( = at + a2"(-3. ' s 3.

It can be shown numerically that (4.20) satisfies the conditions of Lemma 3.1 for

a G [a2, 1) with a3 == 0.1292. It follows that, if the proof of order 2 convergence

for all a, 0 < a < 1, by means of Lemma 3.1 is possible, it will involve a complex

inductive argument. However, because our proof already covers the case a «■ J,

which is the most important one in application, an extension of the result of Theorem

4.1 will not be pursued further in this paper. Numerical experimentation indicates

that second order convergence holds for all a G (0, 1). This is illustrated in Section 6.

Since the extension of Theorem 4.1 to a general k(t, s) can be established using a

technique analogous to that of [7] for midpoint product integration, we only state

the result.

Theorem 4.2. If(i) /(/) satisfies the conditions of Theorem 4.1,

(ii) k(t, s) is twice continuously dijferentiable with respect to t and s for

0 ?S s -g t ^ T,and

- Col" (i = 2, 1), a G [a,, 1),


184 RICHARD WEISS

(iii) d3k(t, s)/ds3 and d3k(t, s)/dt ds2 are bounded for 0 rg s rg t rg T, then, for

a G Wi, 1), (4.1) is convergent of order 2.

5. Computational Errors. During the actual application of algorithm (4.1),

the values {/„, • ■ • , //} instead of {7o, • • • , //}, which satisfy /„ = f0 + t;0,

£ Jfc.i Wt-t = gi/(Atl~a<p(a)) - }jc(,*Wi + Vi a = 1, • • •, D,1-1

are determined, where the perturbations {Vi, i = 0, • • • , I) represent the effect of

rounding and truncation errors. It follows from Lemma 3.1 that

(5.1) I/,- - 7.| ^ + h d = 0, • • • , 7), a G \fii,i 1),

where = const > 0 and |%| £ 9 (/ ■» 0, • •J), However, since the application

of Lemma 3.1 cannot yield a sharp estimate in this case, (5.1) represents a conserva-

tive bound. Numerical experimentation indicates that

max \ f( — f{\ g K2ri,os.sr

where K% is a positive constant depending on T. This is the usual behaviour associated

with computational errors in the numerical solution of Volterra integral equations

of the second kind; equations of the first kind tend to be less well conditioned.

6. A Numerical Example. The trapezoidal product integration method was

applied to the following equation:

n ' v,. IxFiiU 2 - a; it) - ^,(1; 2 - <x; -it)] = f /C?) - ds,(6.1) v '

0 £ * £ «r,

where iFj is Kummer's hypergeometric function {see Erd61yi [11, Section 13, p.

189]}. (/ is the imaginary unit.)

The solution is f(t) = sin t.

The errors obtained for a = 0.5, for which convergence is guaranteed by Theorem

4.1, and for a = 0.05 are listed in Table 6.1. In both cases, the error satisfies the

predicted At2 dependence.

Remark. All numerical results presented in this paper were computed in double

precision arithmetic on the IBM 360/50 computer at the Australian National Uni-

versity.

7. A Third Order Method. The existence of higher order methods for (1.1) is

not immediately obvious. So far, convergent methods of order greater than 2 have

not been found for the first kind Volterra equation

(7.1) g(0 = f k(t,s)Ks)ds,Jo

where k(t, s) satisfies (1.2) {see [10]}. Equation (7.1) can be considered as a special

case of (1.1) with a = 0. Linz [5] has shown numerically that fourth order methods

can be constructed for the equation


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188 RICHARD WEISS

g(t) = f 2 m,1/a ds, 0 rg t g r <Jo (' — i )

However, it is not possible to infer the existence of higher order methods for (1.1) on

the basis of this existence, since the integral operator

AM = f , 1(S\adS> a<l>Jo (t — S)

is a compact operator from C[0, T] to C[0, 7], while the operator

is not.

We verify numerically the existence of higher order methods for (1.1). In fact,

we construct a third order one. For simplicity, we take k(t, s) = 1. The generalization

is straightforward. We assume I = TAt is even and, generalizing the technique used

in the development of the trapezoidal method, we approximate j(s) by a function

MÄ t(s) which is a quadratic polynomial in each of the intervals [t2j, t2i+2] (J = 0, • • • ,

1/2 - 1):

A/a,(s) = f2i +s — hi

(J21 + 1 /21) ~T"Q — t2i)(s — f2f+i)

(?2i + 2 2f2j + l + /2;,

rg s rg f2i+2 0" = o, ••• , 1/2 - i)j.

By substituting M4,(s) for /(s) in (1.1), we obtain

Jo ('21 + 1

i + 2 — /Jo

(7.2)MAt(s)

ds

ds

(j = 0, • • • , 7/2 - 1)

('21 + 2 s)

which, for each j, is a nonsingular two by two system for J2i+l and /2l+2. Thus, (7.2)

is a block by block method in the sense of [10]. Evaluation of the integrals in (7.2) and

division by At1'" yields

£_,/2 + F.Ji + G-J0 = gjAt1-",

£0/2 + Fjx + Go/o = g2/Atl~",

E-if2i+2 + F.iJu+i + Z Ji Fu+I_, + f0G2i-i = g2i + 1/At1'

0 = 1. 1/2 - 1),

E0f2i+2 + Foht+i + Z h Wu+x-t + foG2i = g2i+2/A/1 '>'-i

70 = lim GKOf-'Xl - a),1-.0

F2j = F2,_i (/ ^ 1); F2i + i — G21-1 + £21 + 1 (/ ^ 0);


THE GENERALIZED ABEL INTEGRAL EQUATION 189

W2l = F2l (/ ^ 1); W2M = G2l + E2l+2 (/ ^ 0);

1 2 \3 - a T2 - a/'G

= LX<J, a)(l + fy + i/) - L2a, a)(§ + j) + \UO, a) 0 S 0)

F, = -IM a)(2j + /) + L2(J, a)(2 + 2/J - L3(j, a) (j £ 0)

G,- = 1,0, a)G7 + If) ~ L2(J, a)(| + j) + \L3(J, a) <J ̂ 0)

1*0, a) = (0' + 2)'-" - - a);

L2U, a) = (0' + 2)2-" - /""°)/(2 - a);

£30, = (0 + 2)3~a - /-)/(3 - a).

This method was applied to (6.1). The errors for the cases a = 0.9, a = 0.5 and

a = 0.1 are tabulated in Table 7.1. Third order convergence holds for a = 0.9 and

a = 0.5. For a = 0.1, a strict At3 dependence is no longer apparent. However, the

process is still convergent—no explosion in the errors is observed. The numerical

approximations oscillate about the true solution, and the oscillations increase as

a tends to zero. The simplest interpretation that can apply to such behaviour is

one of increasing potential instability as a decreases.

A rigorous analysis of the convergence properties of this method cannot be given

here. The problem is the lack of accurate estimates for the solution of the lower

triangular system of equations which determines the behaviour of the error.

Acknowledgment. The author wishes to thank Dr. R. S. Anderssen for his as-

sistance and helpful comments during the preparation of this paper.

The Australian National University

Computer Centre

Canberra, A.C.T. 2601, Australia

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6. B. Noble, Lecture Notes, 1970.7. R. Weiss & R. S. Anderssen, "A product integration method for a class of singular

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190 RICHARD WEISS

9. L. V. Kantorovic & V. I. Krylov, Approximate Methods of Higher Analysis, 3rded., GITTL, Moscow, 1950; English transl., Interscience, New York; Noordhoff, Groningen,1958. MR 13, 77; MR 21 #5268.

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