Page 688 - Handbook Of Integral Equations

2 . By applying the Mellin transform, one can solve nonlinear integral equations of the form
◦
∞

β
y(x) – λ t y(xt)y(t) dt = f(x). (20)
0
The Mellin transform (see Table 2 in Section 7.3) reduces (20) to the following functional equation
for the transform ˆy(s)= M{y(x)}:
ˆ
ˆ y(s) – λ ˆy(s) ˆy(1 – s + β)= f(s). (21)
On replacing s by 1 – s + β in (21), we obtain the relationship
ˆ
ˆ y(1 – s + β) – λ ˆy(s) ˆy(1 – s + β)= f(1 – s + β). (22)
On eliminating the quadratic term from (21) and (22), we obtain
ˆ
ˆ
ˆ y(s) – f(s)= ˆy(1 – s + β) – f(1 – s + β). (23)
We express ˆy(1 – s + β) from this relation and substitute it into (21). We arrive at the quadratic
equation
ˆ
ˆ
ˆ
2

λ ˆy (s) – 1+ f(s) – f(1 – s + β) ˆy(s)+ f(s) = 0. (24)
On solving (24) for ˆy(s), by means of the Mellin inversion formula we can find a solution of the
original integral equation (20).
14.3-3. The Method of Differentiating for Integral Equations
1 . The equation
◦
b

y(x)+ |x – t|f t, y(t) dt = g(x). (25)
a
can be reduced to a nonlinear second-order equation by double differentiation (with subsequent
elimination of the integral term by using the original equation):
y +2f(x, y)= g (x). (26)

xx xx
For the boundary conditions for this equation, see Section 6.8 in the first part of the book (Eq. 35).
◦
2 . The equation
b

y(x)+ e λ|x–t| f t, y(t) dt = g(x). (27)
a
can also be reduced to a nonlinear second-order equation by double differentiation (with subsequent
elimination of the integral term by using the original equation):
2
2
y +2λf(x, y) – λ y = g (x) – λ g(x). (28)

xx xx
For the boundary conditions for this equation, see Section 6.8 of the first part of the book (Eq. 36).
◦
3 . The equations
b

y(x)+ sinh λ|x – t| f t, y(t) dt = g(x), (29)
a
b

y(x)+ sin λ|x – t| f t, y(t) dt = g(x), (30)
a
can also be reduced to second-order ordinary differential equations by means of the differentiation.
For these equations, see Section 6.8 of the first part of the book (Eqs. 37 and 38).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 671