Page 162 - Handbook Of Integral Equations

x

28. y(x)+ A sinh(kt)+ B + AB(x – t) sinh(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A sinh(kt).
Solution:
x
y(x)= f(x)+ R(x, t)f(t) dt,
a
G(t) B 2 x B(t–s) A
R(x, t)= –[sinh(kt)+ B] + e G(s) ds, G(x)=exp cosh(kx) .
G(x) G(x) t k

∞ √

29. y(x)+ A sinh λ t – x y(t) dt = f(x).
x
√
This is a special case of equation 2.9.62 with K(x)= A sinh λ –x .
2.3-3. Kernels Containing Hyperbolic Tangent

x

30. y(x) – A tanh(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A tanh(λx) and h(t)=1.
Solution:
x cosh(λx) A/λ
y(x)= f(x)+ A tanh(λx) f(t) dt.
a cosh(λt)
x

31. y(x) – A tanh(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = tanh(λt).
Solution:
x A/λ
cosh(λx)
y(x)= f(x)+ A tanh(λt) f(t) dt.
a cosh(λt)
x

32. y(x)+ A tanh(λx) – tanh(λt) y(t) dt = f(x).
a
This is a special case of equation 2.9.5 with g(x)= A tanh(λx).
Solution: x
1

y(x)= f(x)+ Y (x)Y (t) – Y (x)Y (t) f(t) dt,
2
1
1
2
W
a
where Y 1 (x), Y 2 (x) is a fundamental system of solutions of the second-order linear ordinary
2
differential equation cosh (λx)Y + AλY =0, W is the Wronskian, and the primes stand for
xx
the differentiation with respect to the argument specified in the parentheses.
As shown in A. D. Polyanin and V. F. Zaitsev (1996), the functions Y 1 (x) and Y 2 (x) can
be represented in the form
e dξ

λx x
Y 1 (x)= F α, β,1; , Y 2 (x)= Y 1 (x) , W =1,
2
1+ e λx a Y (ξ)
1
where F(α, β, γ; z) is the hypergeometric function, in which α and β are determined from
the algebraic system α + β =1, αβ = –A/λ.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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