Page 344 - Handbook Of Integral Equations

2 . Let us derive the boundary conditions for equation (3). We assume that the limits of
◦
integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we
obtain two consequences
b

y(a)+ (t – a)g(t)y(t) dt = f(a),
a
b
(4)
y(b)+ (b – t)g(t)y(t) dt = f(b).
a

Let us express g(x)y from (3) via y and f xx and substitute the result into (4). Integrating
xx
by parts yields the desired boundary conditions for y(x),

y(a)+ y(b)+(b – a)[f (b) – y (b)] = f(a)+ f(b),

x x
(5)

y(a)+ y(b)+(a – b)[f (a) – y (a)] = f(a)+ f(b).

x x
Note a useful consequence of (5),
y (a)+ y (b)= f (a)+ f (b), (6)

x
x
x
x
which can be used together with one of conditions (5).
Equation (3) under the boundary conditions (5) determines the solution of the original
integral equation. Conditions (5) make it possible to calculate the constants of integration
that occur in the solution of the differential equation (3).
b

37. y(x)+ e λ|x–t| g(t)y(t) dt = f(x), a ≤ x ≤ b.
a
1 . Let us remove the modulus in the integrand:
◦
x b
y(x)+ e λ(x–t) g(t)y(t) dt + e λ(t–x) g(t)y(t) dt = f(x). (1)
a x
Differentiating (1) with respect to x twice yields
x b

y (x)+2λg(x)y(x)+ λ 2 e λ(x–t) g(t)y(t) dt + λ 2 e λ(t–x) g(t)y(t) dt = f (x). (2)
xx xx
a x
Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary
differential equation for y = y(x),
2
2

y +2λg(x)y – λ y = f (x) – λ f(x). (3)
xx
xx
2 . Let us derive the boundary conditions for equation (3). We assume that the limits of
◦
integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we
obtain two consequences
b
λt
y(a)+ e –λa e g(t)y(t) dt = f(a),
a
b (4)
y(b)+ e λb e –λt g(t)y(t) dt = f(b).
a
Let us express g(x)y from (3) via y and f xx and substitute the result into (4). Integrating

xx
by parts yields the conditions
λb
λa
λb
λa

e ϕ (b) – e ϕ (a)= λe ϕ(a)+ λe ϕ(b),

x
x
ϕ(x)= y(x) – f(x).
e –λb ϕ (b) – e –λa ϕ (a)= λe –λa ϕ(a)+ λe –λb ϕ(b),

x x
Finally, after some manipulations, we arrive at the desired boundary conditions for y(x):

ϕ (a)+ λϕ(a)=0, ϕ (b) – λϕ(b)=0; ϕ(x)= y(x) – f(x). (5)
x
x
Equation (3) under the boundary conditions (5) determines the solution of the original
integral equation. Conditions (5) make it possible to calculate the constants of integration
that occur in solving the differential equation (3).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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