Page 491 - Handbook Of Integral Equations

9.4-4. Solution of Operator Equations of Polynomial Form

The method described in Subsection 9.4-3 can be generalized to the case of operator equations of
polynomial form. Suppose that the solution of the linear nonhomogeneous equation (21) is given
by formula (22) and that the corresponding homogeneous equation has only the trivial solution.
Let us construct the solution of the more complicated equation with polynomial left-hand side
with respect to the operator L:
n
k k k–1
y(x) – A k L [y]= f(x), L ≡ L L , (30)
k=1

where A k are some numbers and f(x) is an arbitrary function.
We denote by λ 1 , ... , λ n the roots of the characteristic equation

n
n
λ – A k λ n–k = 0. (31)
k=1
The left-hand side of Eq. (30) can be expressed in the form of a product of operators:

n n
k
y(x) – A k L [y] ≡ 1 – λ k L [y]. (32)
k=1 k=1
The solution of the auxiliary equation (26), in which we use the substitution w → y n–1 and λ 2 → λ n ,
is given by the formula y n–1 (x)= Y (f, λ n ). Reasoning similar to that in Subsection 9.4-3 shows
that the solution of Eq. (30) is reduced to the solution of the simpler equation

n–1

1 – λ k L [y]= y n–1 (x), (33)
k=1
whose degree is less by one than that of the original equation with respect to the operator L. We can
show in a similar way that Eq. (33) can be reduced to the solution of the simpler equation

n–2

1 – λ k L [y]= y n–2 (x), y n–2 (x)= Y (y n–1 , λ n–1 ).
k=1
Successively reducing the order of the equation, we eventually arrive at an equation of the form (28)
whose right-hand side contains the function y 1 (x)= Y (y 2 , λ 2 ). The solution of this equation is given
by the formula y(x)= Y (y 1 , λ 1 ).
The solution of the original equation (30) is defined recursively by the following formulas:

y k–1 (x)= Y (y k , λ k ); k = n, ... , 1, where y n (x) ≡ f(x), y 0 (x) ≡ y(x).

Note that here the decreasing sequence k = n, ... , 1 is used.

9.4-5. A Generalization
Suppose that the left-hand side of a linear (integral) equation

y(x) – Q [y]= f(x) (34)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 473