Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod*
In case K neither has a separable kernel nor is small, then the next resort is to approximate K with an operator which has a separable kernel.
Theorem. If
then there are kernels Kn and G such that
(1) K = Kn + G,
(2) Kn has a separable kernel,
and
In the succeeding pages, we show how to compute Kn and G. However, we first illustrate that the problem is - in theory - solved if we have such a resolution of K into Kn and G. We seek y such that
y = Ky + f = Kny + Gy + f
or y - Gy = Kny + f.
Use the resolvent for G:
(1-G)-1 = 1 + RG,
to get that
y = Kny + RG(Kny + f) + f
=[Kn + RGKn]y + (RGf + f).
Define z to be RGf + f, or, what is the same, solve the equation
z = Gz + f.
We can solve this equation because G is small. Now, we seek y such that
y = (Kn + RGKn)y + z.
Re-writing this as an integral equation, we seek y such that
where
What is astonishing is that this last integral equation is separable! To see this, suppose
Then
So, here is the conclusion. If K is Kn + G as in the above Theorem, in order to solve y = Ky + f, use the fact that
to form the resolvent for RG; then find z such that z = (1+R G)f. Finally, solve the separable equation y = (Kn + RGKn)y + z
QED
We now must address the question of how to achieve the decomposition of K into Kn + G. The ideas are familiar to persons knowledgeable about the techniques of Fourier series. In summary of those ideas, recall that if p and q are integers, then
We seek Apq such that
In fact, by integrating both sides of this last equation after multiplying by sin(m \pi x) sin(n \pi y), we have
From the theory of Fourier series,
in the sense that
as n -> * . Let n be an integer such that
Define Kn and G by
and G = K - Kn.
Then these three requirements are met:
(1) K = Kn + G,
(2) Kn is separable,
and
Thus, we have an analysis of an integral equation y = Ky + f where
The engineer will want to know about approximations. Here are two appropriate questions:
(a) Suppose one hopes to solve y = Ky + f and that Kn is separable and approximates K. How well does the solution u for u = Kn u + f approximate y?
(b) Suppose K = Kn + G and
Gp approximates RG. How well does the solution u for
u = [Kn+ SKn] u + [1+S ]f
approximate y?
Exercises XIII
XIII.1. With K, f, and an interval as given, solve the integral equation y = Ky + f.
(a) K(x,t) = 2x-t, f(x) = x2 on [1,2]. ans: y(x) = x2 - (75x - 61)/6.
(b) K(x,t) = x + 2xt, f(x) = x on [0,1]. ans: y(x) = -6x.
(c) K(x,t) = 2x2 -3t, f(x) = x on [0,1]. ans: y(x) = x +(6x2-13)/28.
(d) K(x,t) = t(t+x), f(x) = x on [0,1] ans: y(x) =(18+48x)/23.
(e) K(x,t) = xt2+1, f(x) = x on [0,1] ans: y(x) = -3.
(f) K(x,t) = 1/2 + x t, f(x) = 3x2-1 on [-1,1]. ans: y(x) =3x2 + c
(g) K(x,t) = x t, f(x) = exp(x) on [0, ln(7)]. ans: y(x) = ex+ax where a is3(7ln(7)-6)/(3-(ln(7)3)
(h) K(x,t) = x - t, f(x) = x on [0,1]. ans: y(x) = (18x-4)/13
(i) K(x,t ) = sin([[pi]]x) cos([[pi]]t), f(x) = sinh(x) on [0,1].
XIII.2. Show that if f is continuous and 1 + [[lambda]]/2 - [[lambda]]2/240 != 0,
then
y(x) = -[[lambda]] I(0,1, ) ( x2 t + x t2 ) y(t) dt + f(x)
has a solution.
XIII.3. (a) For what functions f does the equation
have a solution?
XIII.4. Solve the integral equation y = Ky + f where
and f(x) = x. (Hint: take the derivative of both sides.)
XIII.5. Let
(a) Show that if 0 <= x <= 1,
In fact,
(b) Toward solving y(x) = K[y](x) + x , compute [[phi]]0, [[phi]]1, and [[phi]]2.
(c) Give a bound on the error between the solution y and [[phi]]2.
ans: |y - [[phi]]2| <= 1/4
(d) Solve y(x) = Ky(x) + x in closed form for this K. (Reference
XIII.7. Suppose that
Give a formula for
XIII.8. Compute
for each K in the previous exercise set. ans: 1/12 and 1/6.
XIII.9. Let
For this K, find y such that y(x) = Ky(x) + x. Note that
What is the significance of this observation?
ans: x +1/8
XIII.10. Let
For this K, find y such that y(x) = Ky(x) + x. Note that
What is the significance of this observation?
ans: exp(x) - 1
5. Suppose that
so that the kernel of K is cos(x+t) and the kernel of H is
sin(x+t). What is the kernel of K[H]?
XIII.11. Find the kernel for the resolvent of the K
whose kernel is K(x,t) =
x t.
Ans: R(x,t) = K(x,t) + K2(x,t) + K3(x,t) + . = 3xt/2.
XIII.12. Consider the problem
(a) Explain how you know this problem is in the second alternative.
ans: y(x) = c is a non-trivial solution
to the non-homogeneous problem.
(b) Find linearly independent solutions for the equation y=K*(y).
(c) Let f1(x) = 3x - 1 and f2(x) = 3x2 - 1. For one of these there
is a solution to the equation y = K(y) + f, for the other there is not. Which
has a solution?
(answer)
(d) For the f for which there is a solution, find two.
(answer)
XIII.13. Consider the problem
(a) Show that the associated K is small in both senses of this
section.
(b) Compute \phi2 where f(x) = 1.
(answer)
(c) Give an estimate for how much \phi2 differs from the solution y of
y=K(y)+f.
(answer)
(d) Using the kernel k for K, compute the kernel k2 for
K2 and k2 for K3.
(answer)
(e) Compute the kernel for the resolvent of this problem.
(answer)
(f) What is the solution for y=Ky+f in case f(x) = 1.
(answer)
XIII.14. Consider the problem
(a) Compute the associated approximations \phi0, \phi1, \phi2, and
\phi3.
(answer)
(b) Give an estimate for how much \phi3 differs from the solution.
(c) Give the kernel for the resolvent of this problem.
(answer)
(d) Using the resolvent, give the solution to this problem.
(answer)
(e) Using the fact that the kernel of the problem separates, solve the
equation.
XIII.15. Suppose that
(a) Show that
(b) Solve the problem y = K[y] + 1.
(answer)
XIII.16. (a) Find a nontrivial solution for y = K[y] in L2[0,1]
where
K(x,t) = 1 + cos(\pi x) cos(\pi t).
(b) Find a nontrivial solution for z = K*[z].
(c) What condition must hold on f in order that
y = K[y] + f
shall have a solution? Does f(x) = 3 x2 meet this condition?
(answer)
Go to a test at this point of your studies.
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