Chebyshev Iteration Methods for Integral Equations … OF COMPUTATION, VOLUME 24, NUMBER 110, APRIL. 1970 Chebyshev Iteration Methods for Integral Equations of the Second Kind By T.
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MATHEMATICS OF COMPUTATION, VOLUME 24, NUMBER 110, APRIL. 1970
Chebyshev Iteration Methods for Integral
Equations of the Second Kind
By T. W. Sag
Abstract. In this paper the numerical solution of Fredholm integral equations of the second
kind using an iterative method in which the solution is represented by a Chebyshev series is
discussed. A description of a technique of Chebyshev reduction of the norm of the kernel
for use in cases when the iterations converge slowly or not at all is also given. Finally, the
application of the methods to other types of second-kind equations is considered.
1. Introduction. Consider first the method of successive approximations (Tricomi
[1]) for the solution of Fredholm equations of the second kind, which have the form
(1.1) fix) = gix) + x(b Kix, y)fiy) dy,
where gix) sind the kernel Kix, y) are known functions, X is a known constant and
fix) is to be determined. The method is to obtain successive approximations /¡(x),
i = 1, 2,... to the solution fix) from the equation
(1-2) fiix) = gix) + A P Kix, v)/,- xiy) dy,
starting with the approximation foix) = gix).
It can be shown that if gix) is an L2-function, i.e., fj g\x) dx exists, and Kix, y) is an
L2-kernel, i.e., ||X|2 = \ba \ba K2(x, y) dy dx exists, then the successive approximations
converge almost uniformly* to the unique function fix) satisfying Eq. (1.1) for all
values of A inside the circle \X\ = 1/||.K||.This classical iteration procedure may be approximated by a matrix iteration by
replacing the iterates /¡(x) by truncated Chebyshev series approximations and eval-
uating the integral in Eq. (1.2) by a quadrature formula. Details of this procedure are
given in Section 2. It can be shown that the matrix iteration is equivalent to the
classical iteration for an integral equation with the functions gix) and X(x, y) per-
turbed. If the number of terms in the Chebyshev series approximations and the
number of quadrature points is sufficiently large, then the perturbations will be small
and so the condition for convergence of the matrix iteration will be almost the same
as that for the classical iteration. The perturbations are caused by errors arising from
the truncation of Chebyshev series, and from the quadrature formula, and so, if the
matrix iteration converges, it gives a solution which differs from the true solution by
a function depending on these errors.
Received February 26, 1968, revised June 23, 1969.
AMS Subject Classifications. Primary 4511, 4530, 6575; Secondary 6510, 6520, 6555.Key Words and Phrases. Fredholm iptegral equations, iteration, Chebyshev series approximation,
numerical quadrature, Chebyshev reduction of kernel, nonlinear integral equations.
* Convergence for all x e [a, b] for which \ba K2(x, y) dy is finite.
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CHEBYSHEV ITERATION METHODS 351
(c) f{x) = e~(x+6) + P (x + 3)eix+2)"-3fiy)dy.
In examples (a) and (b) the solutions of the equations are even and odd functions
respectively. The author has shown, [5], that it is possible to obtain the solution in
these cases by computing only even or odd Chebyshev coefficients respectively. Only
the odd columns ofthe iteration matrix BM are different for M = 1 and M = 2 and
so for Eq. (a) the computations are the same as for these reductions. Similarly, for
Eq. (b) the computations are the same for M = 0 and M = 1, and for M = 2 and
M = 3.For each example an accuracy limit of 10"6 was prescribed. The computations
are summarized in Tables 1, 2, and 3. The computed solutions of Eqs. (a) and (b)
were similar to those obtained by Elliott [3] who used the same accuracy limit. In
Table 3 computed solutions of Eq. (c) are tabulated along with the exact solution ex.
For this equation certain error estimates did not fall below the prescribed limit ; this
is indicated in the summary of computation, and the values ofthe estimates are given
below the summary. The maximum relative errors in the computed solutions are also
given for comparison.
Table 1. Love's eguation (a)
Order of Reduction 0 13
KM ||2 0.2076 0.0042 0.0005
Final number of
Chebyshev coefficients
Final number of
Quadrature Points
Number of iterations 12 2 2
Average ratio of successive Q ̂ Q(m QQ05
iteration error estimates
From the tables it appears that the effect ofthe higher order reductions in reducing
the number of iterations is not as marked as one might expect from examining the
norms ||XM|| ofthe reduced kernels. This is because at least one or two iterations
are required before the iterates settle down.
8. Application to Other Types of Second Kind Equations. The Chebyshev iterationmethod and the technique of Chebyshev reduction of the kernel have been applied
to systems of Fredholm equations ofthe second kind and to Volterra equations. Only
minor modifications to the methods given in Sections 2 and 6 are required. Details
are given in [5].
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352 T. W. SAG
Table 2. Lichtenstein-Gershgorin equation (b)
Order of Reduction 0 2
\\KM\\2 0.00861 0.00357
Final number of
Chebyshev Coefficients16 16
Final number of
Quadrature points30 30
Number of iterations
Average ratio of successive
iteration error estimates0.091 0.037
Nonlinear integral equations have also been solved by a Chebyshev iteration
method. The iteration equation in this case is not linear, but the practical procedure
of Section 5 may still be used. A nonlinear equation of the form
(8-!) /(*)= P K(x,y,fix),fiy))dy
may be solved under certain conditions [1], [5] by an iteration ofthe form
(8.2) fix) =■ i
K(x,y, fi-xix), fi-i(y))dy,
starting with the approximation /0(x) = 0. Assuming the iterate /¡_i(x) has been
determined as a truncated Chebyshev series, the iteration (8.2) can be carried out by
expanding the right-hand side into a truncated Chebyshev series by the Lanczos
curve fitting method discussed in Section 2, and the use of a quadrature formula to
evaluate the integrals. Values of fi-xix) required for computing values of the inte-
grand Kfoy./i.^xX/t-iGO) may be obtained by the method of Clenshaw (seeSection 1).
Effectively the same iteration may be carried out by representing the iterates /¡(x)
by their values at the points xt = cos kn/N, k = 0,..., N, since these are the values
used to calculate Chebyshev coefficients by the Lanczos method. With some manipu-
lation it can be shown that the truncated Chebyshev series (b{N\x) of a function 4>(x)
9. Acknowledgements. The author wishes to thank Dr. Joan Walsh of the Uni-
versity of Manchester for her helpful suggestions for the work and constructive
criticism of the text of this paper.
Discipline of Applied Mathematics
School of Physical Sciences
Flinders University of South Australia
Bedford Park, South Australia 5042
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CHEBYSHEV ITERATION METHODS 355
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