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An Adaptive Method for the Numerical Solution of Fredholm Integral Equations of the Second Kind. I. Regular Kernels
Beny Neta*
Naval Postgraduate School
Department of Mathematics
Code 53Nd
Monterey, CA 93943
and
Paul Nelson
Institute for Numerical Transport Theory
Department of Mathematics
Texas Tech University
Lubbock, Texas 79409
Transmitted by Melvin R. Scott
An adaptive method based on the trapezoidal rule for the numerical solution of Fredholm integral equations of the second kind is developed. The choice of mesh points is made automatically so as to equidistribute both the chauge in the discrete solution and its gradient. Some numerical experiments with this method are presented.
1. INTRODUCTION
Consider a Fredholm integral equation of the second kind, which is to say the problem of finding a function f(x) such that
f(x) =i’k’ x, y)f(y)dy + g(x)> XE [WI, (1)
for a given function g(.r) and a given kernel k(x, y). In the present paper we consider only regular kernels.
Fredholm integral equations of the second kind appear in many applica- tions, e.g., transport theory (see for example Case and Zweifel [2] and Wing
*Part of tllir work was performed while at the Institute for Nwwrical Transport Theory at Tew, Tech I’niverslty.
APPIJED MATHEMATICS AIVD COLMPI’TATIOI-V 21:171-184 (1987) 171
0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 MM-3003/87/$0:3.50
17” HENY NET.\ .\NI> I’.\ITL NELS( )A’
III tlti\ paper \\‘e develop ait adaptive niethod lmed OII the trapezoidal ndc
fot- 0l)taiiiiirg the tiiiirleric~al solution of (1 ). The idea is to start with a gi\,ett
t~~~tiil~~t~ of c’qudy qac.cd poiilts (or 3 givelr iiiesh). The wliitioir at thi\ stage
i\ ol~taiittd l~r\, \olvittg ;i littear \ystettt of algelmic~ eqrtatiotrs. ‘I’lre pr0fqat11 tiI(btt clt~c~itle\ if tlte ttttdt tiwtls to tw refitted aid \\hx~. Followiirg SIIIOO~\~~ IS], \I(’ tlo ttii\ iii such ;L bay that twth the chtrgt in the appmsilriatc sol~~tiol~
ar~tl it\ qtdi~t~t arc ccl\iidistl-it)lttetl. Ra5etl ~tpotr the esperieiic~e tlesc~ritxd 1))
II\\.> C’1. SIll00k and Kw \:;j for- tlotttl~l~l~y-\~~llll~ ptY~lkvlts. 011e pdl;Lp~ c’;ltl
c’\pt’c’t tltix oftett to ttc tttorc c.fficicttt tlimt the alterttatiws of cc~ttidist~ittrttitt~~
ai-c,-lcviqtli 01. 11~x1 tt~tttc~atiott wror. ~\tlapti\v itit~~letrte~tt~~tiotts of tltc Iattet
11x\ (5 INYVI tli\c~thwtt, for I~ot~t~d~~~y-\~~iltt~~ ptxdtleiii~. Ity \2’ltitc 1121 :IIICI 1)) I’(~tx~! r;l dilct Sc~\\.c~ll 171, tnpectivet),
‘l’tlc% Iitatt-i\ at (~LCY]L 9t:i,ge is trol recotttptttetl. ttttt oiil>. ttir. ii~ccswi~y txt\\.\ ;IIKI CY~I~III~II~ arc colliprittd. ‘l’tw ilen \ysteiri is ~tlved Ity the (httss-Scitlcl itcrati\ (’ tttc~ttto~l. ‘I‘tt(~ ittitial giiess at the previotts ttottrs is tahett to tw CY~IL~I
to tttc% ~oltttioti iii ttte ptxGotts stage. ,\t ttic ttt’u’ ttodrs \\‘c’ tahc tlte ittitial
gt~c’\\ (~(~tial to tltc avtmjitn of the ptwiott9 wlrttiott at tlita trt~igttttorittg poitth. III ttic ilca\t wcTioit bt‘ ttesdw ttw ttiettiott itt detail. 111 Sc~ctiolr :i \w
tx*pot? oti ttrc Iitittt~ri~al c~xperitrtettts perfontred rising oiir tttettrod. Iii the hst \c,c,tic)tl. \\Y’ (.ottcdttclc w?ttt rctttarks OII analyis of the tttetttott md ftrtrlrv \\ork.
., _. I)l-:\‘E:I ,ol’IlE:r\‘l’ OF ‘I’IIP: hlE:‘I‘IIoL>
(ii\ C’II ;I ftittc.tiott g(I) atrtl ;i regritar lenid (k( s, y ). let ,/I .V ) Ite ;i fttttctioli clc*fittcttl OII IO. I] aiitl satisfying (1). Let 0 = .x1 < x., i . K .v, = I tw ;i \tlt)cli\i\ioil of [O. I] \vitlt /I) = xz , ~ .Xx,. .\ utrvcy of ttotratlaptiw tttttttericat ttrt~tttotl~ for ttw wltttiotl of (1) i\ givctt t)y !\tkittsott [I]. Kettle 151 ha c3)Ilc~c+t~tl ;i t tit~liograptiy OII the srd)jec4. Hew \2-e da~etop itii adaptive tttetttotl
t );iv’ct oti tttc, c~otttpo~itt~ ttapezoitlal ntlc. Tttc txxsott for this choice is t\~ufold. Fir4t. \\ (’ \I ;iitt ;t rtilc for \vhictt \j.t’ c’att cttoose the locatiort of the irotlc~.
S~c~tticl. \‘i (2 wattt ii twlc for \vliicti ttte ittsertiotl of ii tide Lrill affect ttic \\.ciqttts of 0111) tltc tieigttttoritrg poittts. ‘IIic reason is that me wotiltt trot \wirt to rrcoittptitr the etttries of the tttatrix at every stage.
‘l’tlc~ itrtegral tcrttt iii (1) is rcplacett tty ;i fitiite sttiii t)y III~;~II~ of ttitb c.c)riipo\itc ti~apt~7oidal r-tile
Numerical Solution of Fredholm Equations. I 173
\vhcre
k,(x) = k(x, x,), (:I)
f;=f(x,), (4)
T~IllaXl(k(X,Y)f(Y))(,yj. (3
Incorporating (2) and (1) and dropping the truncation error term, one o!,taills
Evaluating (6) at x I, one has a linear system of algebraic equations
(Z-KD)P=C, (7)
where the vectors F and G have components fi and g, = g( x I) respectively, the components of K are given by
(8)
and the diclgonal matrix D has entries
i
x2 - Xl
D&X X,-X
i = 1,
\-I i = ‘A’, (9)
X !+I -X, I 1 < i < N.
The system (7) is solved iteratively using the Gauss-Seidel iterative method, i.e.,
.A
c Qf.1 (11 Jh!, ) i=O,1,2 ,...,
“=j 4 I
(I())
where
Q=(Z-KD). (11)
RENY NETA AND P:\KTL NELSON 174
‘I’hcl initial approxinlatiotr is
1;‘“’ zz (;. ( I”)
After F is obtained one has to decide if the mesh should he refined. The procchire is to halve any interval 1 x,, x, , , ] for which one of the following is rrc~f 4atisfietl:
for a sndl nrml)er y < 1. This particular form of equidistril)lltion was used by Pearsol [fi] in solving scalar ~~oundary layer problems and 1,~ Dwyer, 5’1nooke. at~d lice [3] for heat and mass transfer proHems. The aim is to equidistril)utc the difference ill the components of the discrete solution and the difference irr the gradient of the components of the discrete solution between adjacent nodes (see [:I]). We also require that any interval larger than c (given) tinlcs the snullest illterval should be refined. This will ensure that the error \vill I)econre smaller. 111 order to obtain the integrals OII the left in (1.3) (13) \ve differentiate ( 1). This yields
j.‘(s)= ‘l;,(s,Y)fTY)rlY+~‘(x), I ( 15) * 0
f-“(*)= iill;,,(r.Y)f.(s)dy+R”(r). J (16)
.A\ X-( s. y ) and g( x ) are known, one can obtain an approximate value of .f“( a ) ant1 f“'( s ) 1)~ approximating (15)-(16). Thus
&'(.I. ) = ; "X ' (k:;f; + k:‘. , J / , 11~ + r;‘. (18) r=l
Numerical Solution of Fredholm Equations. I 175
k:I=k.y(xj> x,)>
k;; = kXX(xj, x,).
(19)
(20)
The computation of the right sides of (17)-( 18) is organized in such a way that one column of k .( k,,) is computed at a time and used everywhere it is Ileeded. When the column is no longer needed we compute the next one. This saves on storage and calls to the functions evaluating k ,( k x ). The user should supply the first and second derivatives of g and first and second partial derivatives of k( x, y ) with respect to x. The values maxIFj/ in (1:3) are o\)tained from the solution, and maxi df/dxJ is obtained from (17). The computer program subdivides the interval (xi, xjil) if either (13) or (14) is not satisfied or if the interval is larger than c times the smallest one in the previous stage. Once this is done one can have at most 212’- 1 intervals, where S is the number of intervals in the previous stage. The matrices K and I> are modified. The vectors F and G are modified; values of g at the new nodes are computed and inserted in their proper place. The values of F at new nodes are obtained by averaging the values of F at neighboring points (klmwn from the solution at the previous stage). This vector F is used as a starting vector for the Gauss-Seidel process to solve the new system. Note that this choice reduces the number of iterations used by Gauss-Seidel in subse- quent stages. This process is continued until either there is no need to refine the mesh or the maximum allowable number of nodes is used. In the first case we say that convergence occurs.
:3. NUMERICAL EXPERIMENTS
In our first experiment we solved the following integral equation:
(21)
whose exact solution is f(x) = x. The maximum absolute error between the exact and approximate solution is given in Table 1 for various values of the parameters y, 6, and c.
Note that when y and 6 are larger or when c is larger, the error is larger too. When y and S are reduced to 0.0001, even though c is increased, the error is small. Note that in the cases where y = 6 < 0.1 the process is stopped because the number of nodes allowed is used.
BENT NETA AND PAlrL NELSON
TABLE 1
Initial Allowed lJ5etl y=s C’ Errol
5
.5
.I
.I
.Ol
.OOl
.OOOl
.Ol
.OOl
4
10 4
10 10 10 10 10 10
Number of nocles
TABLE 2
Initial Allowed LJsetl
1:3 1s 30 40 79 70 40 40 40 90
.Fi
.F,
.I
.l
.l
.l
.Ol
.OOl
.OOOl
.000001
4
10 4
10 4
10 10 10 10 10
111 Talk 2 we give the results for solving the problem
f‘(x)= / ‘(X-o.5)(y-o.5)f(y)tly+f’ “‘(’ ‘)+. (22) 0
whose exact solution is the bell shaped function
f(x) = (, lot1 IW’, (23)
Note the surprisingly excellent results for y = 6 = 0.5 compared to the remits for this choice of y, 6 in the first example. The program concentrated on
Numerical Solution of Fredholm Equations. I 177
the itltervals with sharp gradients. Note also that when y = 6 = 0.000001 the reslllts are not very good. The associated mesh did not concentrate on the intervals with sharp gradient, because of the smalhless of these parameters. Note the improvement in the result when we increase the allowed number of nodes ( y = 6 = 0.1 and c = 4 or 10): 39 additional nodes resulted in 8 more digits of accuracy.
In the next experiment we solve a problem whose exact solution is patched continuously from a constant function and an exponentially decaying fnnc- tioll. The equation is
where
g(x) = f(x) - x(0.185 - 0.1X’),
and the exact solution is
(~~5)
The results are given in Table 3. Note how the increase in c and reduction in y, 6 improved the results.
III Table 4 we show how the nodes were distributed at the end of the process for the case y = S = 0.1 and c = 10.
TABLE 3
Number of nodes
Initial Allowed Used y=S c Error
40 40
40 40 40 40
40 90 90 90
9
9
40 29 40 40 40 41 90 90
.5 4
.5 10
.l 4
.1 10
.Ol 10
.OOl 10
.OoOl 10
.l 4
.Ol 10 ,001 10
.2619 ~ 2
.2619 - 2
.2129 - 3 3391 ~ 3 .7898 - 4 .9284 ~ 4 .9284 ~ 4 .1966 - 3 .29x3 ~ 3 .1,525 ~ 4
178 BENY NETA AND PAlTL NELSON
TABLE 4
Interval Mesh spacing
Note that h is large for the interval (0,(X5), where f s ) is a constant and
the interval (0.75, l), where the slope is moderate. See Figure 1. In the next experiment we have taken a hiquadratic kernel. The equation
solved is
f(x)= ‘(x-o.5)~(y-o.5)2f(y)dy+x- /^
(.x - 0.5f 24 ’
(27) 0
n d
3
Q
d
i.0
Numerical Solution of Fredholm Equations. I 179
whose exact solution is f(x) = X. The results are given in Table 5. Note that, as ill Table 1 (the problems have different kernels but the same exact solution), the increase in y, S from 0.1 to 0.5 caused a loss of two digits of accuracy.
In our last experiment the solution is patched from two exponentially increasing functions and three constant functions. One of the gradients is much sharper than the other, and we show that the program concentrated on refining the interval where the gradient is sharpest. The problem has a different type of kernel:
where
(28)
(29)
with the constants given by
1
a = 3(e’/‘2 - I) ’
e1/12 -2 b=
3(e l/12 - 1) ’
(:30) 13
c = 3( e:3/X _ el/4) ’
2e:‘/H - 15e’/4 d=
3(e 30 - e’/”
1’
180 BENY NET:\ AND PAITL NELSON
TABLE .i
Number of nodes
TABLE 6
Nmnher of nodes
Initial Allowetl l~StKl y=S
.5
..5
.l
.l
.Ol
.OOl
.OOOl
.l
.I
.Ol
.OOl
.000001
i
<El461 g(r)=f(x-)-x’
I85<,” 12 _ 216 __ _ 3456 864( e ‘,‘I’ - 1)
C’ Err01
-4 .17X5 IO 17:3Fi 4 .16”H ~-1
10 .:3fm :i 10 .20.5-l :3 10 m1.5 :?I 10 .1;57.5 :I 4 .1627 4
10 .:3mo :i 10 .:33(N) :3 10 .lfj”-l -I 4 .6800 1
The results are summarized in Table 6. Note that the case y = S = 0.5 gave poor results. For y = S = 0.1 one more
digit of accuracy is gained b) decreasing c, at the expense of almost doubling the number of nodes. In order to show that the program concentrated its effort on refining the subinterval [ 1, z], we have plotted in Figure 2 the exact
Numerical Solution of Fredholm Equations. I 181
9 .:. o-
9 o7
0.0 0.1 01 0.) 0.4 0.0 0.0 0.7 0.0 0.0
EXACT souJTI0Nxor (28)
0
0.0 0.1 ox ox 0.1 0.6 0.0 0.7 0.0 0.9 1.0
x _ Frc. 2. The exact solution of (28) given by (29) and the location of the nodes at each of the
Seven stages.
sollltion and the location of the nodes at each stage for y = 6 = 0.1, c = 4, and muml)er of nodes used = 43.
4. CONCLUDING REMARKS
In this section, we give some information relating the parameters y and 6 in ( 1.‘3) -( 14) to known quantities.
182 RENT NET.\ AND PAIX. ?JE:L,SON
THEOREM. IA M,= max,lf(r)l, n~~=min~~~f(x)~, untf
h”‘(x)= ’ 2(x.!/) rly, Jl () dr’ i = 0. 1.2
/I = min(xl, , -I,). i
Llailrg ( 1 ) ogle can show that
(32)
( 33 )
(:;A)
( 33 )
Numerical Solution of Fredholm Equations. 1 183
Combining (35)-(36), one obtains (32). In a similar fashion, one can show that
M&7 rng” -
1 - M,,,, M
‘(‘) h.
To bo~md M,.,, one can use (15) to show
for all x,
and thus M,-. satisfies the same inequality,
(37)
Combining (38) and (37), one obtains (33). This completes the proof.
IKote that the values of m,,, mL+ M,, M,,, M,lil, M,,,, M,z, can be sup- plied by the user or evaluated numerically at each step by the program.
HEMARKS.
( 1) These results were extended to the case of weak singular kernels k( X, y ) in paper II of this series [15].
(2) The computer program uses the Gauss-Seidel iterative method to solve the system of algebraic equations. One can replace this by any iterative method such as the Jacobi or SOR. It is also possible to use extrapolated iterative methods (see [14] for references). Computer routines for each of the iterative methods and its extrapolated version can be found in [14].
This reseurch was supported by the Center for Energy Research of Texas
Tech lrniwrsity and by the U.S. National Science Foundation under &‘.S.F.
Grmt XI. CPE8007396.
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184 HENY NETA AND PAIJL NELSON
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