Page 315 - Handbook Of Integral Equations

4.7. Equations Whose Kernels Contain Combinations of
Elementary Functions

4.7-1. Kernels Containing Exponential and Hyperbolic Functions

b
1. y(x) – λ e µ(x–t) cosh[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.18 with g 1 (x)= e µx cosh(βx), h 1 (t)= e –µt cosh(βt),
g 2 (x)= e µx sinh(βx), and h 2 (t)= –e –µt sinh(βt).

b

2. y(x) – λ e µ(x–t) sinh[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.18 with g 1 (x)= e µx sinh(βx), h 1 (t)= e –µt cosh(βt),
g 2 (x)= e µx cosh(βx), and h 2 (t)= –e –µt sinh(βt).

b
3. y(x) – λ te µ(x–t) sinh[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.18 with g 1 (x)= e µx sinh(βx), h 1 (t)= te –µt cosh(βt),
g 2 (x)= e µx cosh(βx), and h 2 (t)= –te –µt sinh(βt).

4.7-2. Kernels Containing Exponential and Logarithmic Functions

b

4. y(x) – λ e µt ln(βx)y(t) dt = f(x).
a
µt
This is a special case of equation 4.9.1 with g(x) = ln(βx) and h(t)= e .

b
5. y(x) – λ e µx ln(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x)= e µx and h(t) = ln(βt).

b
6. y(x) – λ e µ(x–t) ln(βx)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x)= e µx ln(βx) and h(t)= e –µt .

b
7. y(x) – λ e µ(x–t) ln(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x)= e µx and h(t)= e –µt ln(βt).

b
8. y(x) – λ e µ(x–t) (ln x – ln t)y(t) dt = f(x).
a
µx
This is a special case of equation 4.9.18 with g 1 (x)= e µx ln x, h 1 (t)= e –µt , g 2 (x)= e , and
h 2 (t)= –e –µt ln t.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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