Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS. 131
The use of a function analogous to the Green's function was also
pointed out, based on a particular solution analogous to log r*.
78. The Method of Particular Solutions. The method of particular solutions may be generalized not only to the integro-differential equations we are
now discussing, but also to those where the same variable appears in both
differentiation and integration. t Consider however, as an example, the
generalization of Laplace's equation which we have already mentioned:
(59') all 2 + aC22 a = 0,
which may for combinations of the second kind be written explicitly in the
form
a2U C32U r { 2U(xy I ts) a2U(xy I ts)
(59") - + +- + a () ~ All(rt) ) dt = O.,aX2 aY2 f" aX2 ay2
We proceed to obtain the solution of this equation, of form jU(xy rs), which
vanishes when x = 0, when x = c, and when y = d, and takes on the values
U(xO I rs) = f(xrs)
when y = 0.
If we write,812 = all, /22 = a22 where /1 = 1 + jBl(rs), (2 = 1 + jB2(rs),
the function
mnr(d-y) -mtr(d-y)
e c -e c. mrx
=m sin
mird -mird 1l
~_Z ~ "I -md____d ~c
e c -e c
in which
/3 =-/ = 1 + B(rs),
P2
will be a particular solution of (59') which vanishes for x = 0, x = c, y = d
and takes on the value sin m7rx/c for y = 0. The function
00
t = SmWmWm,
with
2 rc. mrx
Wm = c,o sin m- dx,
will be the solution which takes on given values for y = 0, assuming the
necessary conditions for convergence. Our problem is solved if we let
y=0 = jf(x, r, s).
* The special case in which I describe this function as reducing to the
ordinary Green's function is invalid (Rendiconti del Circolo Matematico di
Palermo, vol. 34, p. 25).
t Volterra, Rendiconti della R. Accademia dei Lincei, vol. 21, 2d semester,
p. 1; Evans, ibid., p. 25.