A Technique for the Numerical Solution of Certain Integral Equations of the First Kind*

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1 A Technique for the Numerical Solution of Certain Integral Equations of the First Kind* DAVID L. PHH~LIPSt Argonne National Laboratory, Argonne, Illinois Introduction The general linear equation may be written as h(x)f(x) --~ I p Ja b K(x, y)f(y) dy = g(x) (a<=x<=b) where the known functions h(x), K(x, y) and g(x) are assumed to be bounded and usually to be continuous. If h(x) ~- the equation is of first kind; if h(x) ~ for a -<_ x ~ b, the equation is of second kind; if h(x) vanishes somewhere but not identically, the equation is of third kind. If the range of integration is infinite or if the kernel K(x, y) is not bounded, the equation is singular. Here we will consider only nonsingular linear integral equations of the first kind: b ~ K(x,y)f(y) dy = g(x) (a ~ x ~ b) (l) There is extensive literature on equations of the second kind, but literature on linear equations of the first kind is sparse. However, several, methods for solving equations of the first kind numerically have been proposed [1-1]. No method has been very successful for arbitrary kernels when the function g(x) is known with only modest accuracy. The reason for this is inherent in the equation itself. Think of the equation as a linear operator, operating on f(y) to produce g(x). This operator does not have a bounded inverse (it may not even have an inverse, but we will assume here that it does) which can be seen as follows. Let f(y) be the solution to (1) and add to it the function fm = sin my. /, b For any integrable kernel it is known that gm= J. K(x, y) sin (my) dy -~ a as m -~ ~. Hence only an infinitesimal change g,~ in g causes s finite change f~ inf (i.e. the equation is unstable).,also, one would expect that g,~ -~ as m --~ faster for flat smooth kernels than for sharply peaked kernels (indeed if K(x, y) were the a-function, K(x, y) = a(x -- y), then g,~ = fm would not approach zero). Hence we conclude that the success in solving equation (1) by any method depends to a large extent on the accuracy of g(x) and the shape of K(x, y). * Received June, t Based on work performed under the auspices of the U. S. Atomic Energy Commission. 84

2 CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND 85 Description of Technique If the straightforward matrix approximation to equation (1) is used, it is found that as the mesh width decreases, the solutions at first become more accurate, but eventually begin to get worse. How soon the solutions begin to get worse depends on the accuracy of g(x). Larger errors in g cause the solutions to get worse sooner. Also, the error in each of the approximate solutions tends to be an oscillatory function of x. Since the function g(x) is not known accurately, we should state the problem as f v K(x, y)f(y) dy = g(x) + e(x) (a <- x <- b) (2) a where e(x) is an arbitrary function except for some condition on the size of ~(x), such as I ~(x) I =< M or p(x)e2(x) dx < 2~, p(x) >. Instead of a F a unique solution of (2) we get a family ~ of solutions. The real problem then is to pick out of the family of functions ~ the true solution f. This cannot be done without more information about the problem than is given in equation (2). We will assume here that the functional form of f is not known. If it were known we could use a least square fit to find a best fit to f. However, we will assume here that f is a reasonably smooth function. With this assumption the best approximation tof we can choose is probably the function f8 C ~ which is the smoothest in some sense. Of the various smoothness conditions, we choose the following (assuming the f have pieeewise continuous second derivatives): f b (f~t)2dx = min f b (ff)2dx. (3) In order to solve numerically (2) and (3) we make a matrix approximation to them. We subdivide the interval into n parts by the points a = x < xl < x2 < < x r, = b, and replace the integral equation (2) by the following linear system, i= wikiifi = gi + ei (j = O, 1,...,n) (4) where f~ = f(xi), gi = g(xi), ei = e(xj), lcii =- K(xi, x~), and the w~ are weight factors whose values depend on the quadrature formula used. However, for simplicity, we will assume that the x~ are uniformly spaced. For the condition on the magnitude of e(x) we take i~ ~ e ~ (5) 2 where the where d is a constant (more generally one can take ~= p~e~ = e 2 p~ = are weights). The anmogous problem to (3) is to look for the vector

3 86 DAVID L. PHILLIPS f~ = (H, fl ~, "'", f~) such that 2 " ~ ~ ~ (f,+l - 2 f,~-l) (6) i= f~3* i= where ~* is the set of vectors satisfying (4) and (5) (here we assume f is zero Outside the interval (a, b) and set f-1 = f~+l = ). LeE us introduce the following matrix notation: A =- (w~lcj,), A -~ =- (a,j). Equation (4) can be written in the form f = A-lg + A-I~. (7) This shows that the f~ are linear functions of the ~i and that of, _ ~ (i,j = o, 1,..., n). (s) Oei From (6), (8), and the constraint (5), we see that the conditions f, must satisfy ean be written as follows: i= El = e i= (f~+l -- 2H + ]~-1) (ai+l.~ -- 2aii + ai-l,s) + 7-1ej = (j = O, 1,..., n) (9) where ~-1 is the Lagrangian multiplier. It will be much simpler to solve these equations for f, if we take ~, as known and e 2 as an unknown instead of the other way around. This is equivalent to replacing the conditions (5) and (6) by the single condition that f, be the vector (satisfying (4)) that minimizes the expression ~' ~-~ (S,+~ -- 2f, +f,-~)2 + ~ ~,2 (1) i~ i~ where ~/is a given constant. From (1) we see tha~ ~/should be non-negative and that for ~ = the whole problem reduces to just the straightforward matrix approximation to (1), Af = g. Equations (9) can be written in the matrix form where B = (~,) and ~,BL+ ~ = (11) fhk = ak-2,z -- 4ak--l.l + 6akz -- 4ak+l,, + ak+~,l (k,l =, 1, "'., n). (12) In (12) we define a_2,~ = --aoa, a-la =, a~+t,z =, and a~+%z = -a~t. Using (11) and the fact that f, satisfies (7) we solve for f, and eand gee f. = (A + ~,B)-~g, r = -~,Bf,. (13)

4 CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND 87 From equations (13), the matrix method described here merely replaces A by A + ~B where B is a certain matrix whose elements depend only on A and ~/ is an arbitrary non-negative parameter which controls the amount of smoothing. Increasing ~ produces greater smoothing. It follows from the second equation in (13) and (5) that e is approximately proportional to ~/so that only a very few v~flues of ~ need to be used in order to find one giving e the desired magnitude. However, one additional matrix inversion is needed for each new value of - used. The value of e is determined from the accuracy of the g~-.,vzamples Example 1. Let the problem be the following: where ff K(x - 3 X)f(z) dx = g(x) 7rz K(z) = 1+ cosy, l z] =< 3, =, lzl>3; ( 1 3) 9 ~rk g(z) = (6+X) 1--~cos - ~sin-~-, Izl =< 6, =, Izl>6. The solution to this problem is f(x) = K(x). Hence we can easily check the numerical solution against the true solution. Let us first take n = 12 (13 points) and use Simpson's rule for the quadrature formula. The truncation error is about.4. The values of g are rounded off so that the maximum error in g(kj) is.5. Table I and Figure 1 give the comparison between the true solution and the numerical solution for several choices of % Since the solution is symmetric TABLE I k true vaiues of/,y ~, =.11 ~'=.11 "t' ~.3 ~, =.1 T ~.5 ~'= ~ 6~.1.(.{].G [ [ [ ) ) ) ave. 1~5 I max. ]e~- [

5 ~ DAVID L. t'h[liaps about zero, outy uonmegative vaues of ;', are listed in the ta, blc, ;\1~(), t t.~ average ej i and maximum %. i are listed for each % Here the trtmc::~tion error, which acf, s much like an error i~.q, is bound(.,d by.4. He~ee we expect to get, the best results for some value of 7 for which avg. ei i <.4. Here v... 3 is the best choice even t, hough a, ve.. I ei] =.26, which is eov, siderabty sma, ller Wan.4. Now consider the same problem with -n (25 points). Here the truncation error is bmmded by.25. Table II "TRUE CURVE, ~.y =... Y:.11 j,~j " +7 :.@3,'=.5 I j '\ -a -4 J 2 I _ I I t 1 J I ]" I I FIG, 1 TABLE II [Pde values off ~ = v=.25 ~ =.2 7 =.1 v =.5 ~ = I 2, (~}.5 1.8(~ ~g fN 2.5O % i.;~ ( ~) (K) O , , ,312 2, (X), , , ,8531 1,48(51 1.9, ,18 --,531 --, E , t!.t7.3r P.155 i.28 max. '~ ~i i....(x)27.59,122 i k :... :...

6 CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND 89 =, = '1 I ] t- + + ~ ' t I I i I! I t I I I I I FIG. 2 TABLE III X true values off ~, = ~'=.1 ~ =.1 3' ~.1 '~ =.1 ~' = ave. [ ej" [ max. [ ej I and Figure 2 give the comparison between the true solution and the numerical solution for several values of ~. Here the smoothing effect of ~, on the oscillation due to the ill-condition of the matrix is much greater than the smoothing effect on the solution. Thus here it is possible to practically eliminate the oscillation due to the ill-conditioned matrix without appreciably affecting the solution. Now consider the same problem where the g's have been further rounded off.

7 9 DAVID L. PHILLIPS In this case the maximum error in g is.4 and the average error in g is.14. Table IIt and Figure 3 give the comparison between the true solution and the numerical solution for severm values of 7. 7: + 7 "=.1 TRUE I I I FIG. 3 TABLE IV X true values of f 7 = 7 =.5 7 =.5 7 =.5 7 = , i , ave. I e~ I... } I max. I ~J!... ] ,84

8 t5 --!4 -- I3 -- CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND ~ Z =o +~'= II -- I I I i FIG. 4 Notice that for 1' = there is considerably more oscillation than in the previous case. However, the oscillation is smoothed out just as before by using a suitable value of % Now consider the same problem where the g's have been even further rounded off. In this case the average error in I g I is.2 and the maximum error is.41. Table IV and Figure 4 give the comparison between the true solution and the numerical solution for several values of % Notice that for 7 = the solution is completely dominated by oscillation. The errors in g have been magnified in f so much that the solution is completely useless. However, for ~, =.5 the solution is reasonably good, especially near the peak which is the most important region. Also notice that the average ] ej] for =.5 is of the same order of magnitude as the average error of the g's.

9 92 DAVID L. PHILLIPS Example 2, The problem is the following: 3 dz f t<(x - x)f(z) = g(x) 3 where K(z), g(z) and f(z) are given in tabular form in "'t' ta)m V. K(z) and g(z) are also plotted in ]~,gure -a. o 5. The values of g were obtained from tile tabulated values of f(z) and K(z) by numerical integration. The values of g have errors which average about.t in magnitude. The maximum error in g is about,.2 Let us first take n - 14 (15 points) and use Simpson's rule for the quadrature formula and solve for f(z) assuming K(z), g(z) given. The truncation error is less than.66. Table VI and Figure 6 give the comparison between the true solution and the numerical solution for several choices of 1/. g (z) K (z) I TABLE V f(z) g(z) K(z) l(z) K(z) = for l z i > ~K (z) 3 - I IO 2 5 FIG. 5

10 CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND 93 TABLE VI z true values off ~= --I i 7 = ~/ = = "/= = 1? = ave. [ ej : max. ] ej s: /I~.~ = O, n = I --.I TRUE +x:l, n:3 _j-y= O, n=14 14, O FzG. 6

11 - -.] 94 DAVID L. PHILLIPS The poor results here are due to a large truncation error. The parameter 3' has to be chosen so large that the true shape of the function is noticeably af- fected. Hence t.he oscillation and the true shape cannot be separated. Now consider the same problem where we take n = 3 (31 points). Here the upper bound for the truncation error is.2, which is the same order of mag- nitude as the errors in g. Table VII and Figure 6 give the comparison between the true solution and the numerical solution for several choices of 7. TABLE VII z true values off O ] = 7 =.1 7 =.1 i =2 7~i ave. l~s I... i max. t osl... i --O oo i.4.2 i

12 CERTAIN INTEGRAL EQUATIONS OF TIIE FIRST KIND 95 TABLE VIII I trueo~llles 3,= 7=.1 7=I 3,~2 3,=5 T=IO O OO28.5 l I , : O ave. I ej { max. l ej! Notice that the best values of 3' are the ones for which the max I e~] is approximately the same as the maximum error in g. Also notice that the peak is rounded off a small amount. This is unavoidable with thepresent method. Finally, consider the same problem where the g's are given less accurately. Let the errors be about 5 per cent of the peak value (This simulates a certain problem from experimental physics ) The maximum error in g is about 15. Table VIII and Figure 7 give the comparison between the true solution (for accurate g) and the numerical solution for several values of 7.

13 96 DAVID L. PtlI:LLIP~ = +y: TRUE I , l I -.t8 }-- -zo ~ I -28 I I I 1. I I I t t I 1 _ -2 < Fro. 7 Notice that for T = the solution is dominated by oscillation. The best values of v are 1 and 2. For these values the error in the peak value is about 7per cent, a reasonable value since the maximum error in g is about 5 per cent. For the best values of T the max l eji is about.1, somewhat lower than the maximum error in g. This is probably because of the unusually large errors in ~?. Sumrr~ary The method described here should work reasonably well on problems where the unknown function f is assumed to be a relatively smooth function. The truncation error should be as small a.s, or smaller than the errors in g. The most difficult task is to choose v. When the errors in g are relatively small, ~/should probably be chosen such that. e is approximately the same magtfitude as t:hese errors. When the errors in g are relatively large, as they are in the last part of example 2, ~/should probably be chosen such that e is somewhat smaller l~han

14 CERTAIN INTEGRAL EQUATIONS OF THE FIRST KIND 97 these errors. In any case several values of 7 should be tried and the best value should be the one that appears to take out the oscillation without appreciably smoothing the function f. REFERENCES 1. C~ou% P. D. J. Math. Phys. 19 (194), CROUT, P. D.; AND HILDEBRAND, F.B. J. Math. Phys. 2 (1941), DIXON, W. R.; AND AITKEN, J. t-i. Canad. J. Phys. 36 (1958), FOX, T.; AND GOODWIN, E.J. Philos. Trans. Roy. Soc. London. A 245 (1953), GOLD, R.; AND SCOFIELD, N. E. Bull. Amer. Phys. Soc. 2 (196), t~reisel, G. Proc. Roy. Soc. London. A 197 (1949), N~(STnO~, E.J. Acta Math. 5~ (193), R~z, A. Ark. Mat. Astr. Fys. 29A No. 29 (1943). 9. VAN DE t-iulst, H. C. Bull. Astr. Inst. Netherlands 1 (1946), YouNo, A. Proc. Roy. Soc. London. A224 (1954), 561.

ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS

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