Volterra Integral Equations, Second Kind - EqWorld - The World of ...

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Volterra Integral Equations, Second Kind - EqWorld - The World of ...

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Exact Solutions > Integral Equations > Linear Volterra Integral Equations

of the Second Kind and Related Integral Equations with Variable Limit of Integration

2. Volterra Integral Equations of the Second Kind

2-1. Integral equations whose kernels contain power-law functions

.1. y(x) – λ

.2. y(x) + λ

.3. y(x) + λ

.4. y(x) + λ

.5. y(x) + A

∫ x

a

∫ x

a

∫ x

a

∫ x

a

∫ x

a

∫ x

.6. y(x) + λ

a

∫ x

.7. y(x) – λ

0

y(t) dt = f(x). .

(x – t)y(t) dt = f(x). .

(x – t) 2 y(t) dt = f(x). .

(x – t) 3 y(t) dt = f(x). .

(x – t) n y(t) dt = f(x). .

y(t) dt


x – t

= f(x). Abel equation of the second kind. .

y(t) dt

(x – t)

α

= f(x). Generalized Abel equation of the second kind. .

2-2. Integral equations whose kernels contain exponential functions

.8. y(x) + A

.9. y(x) + A

.10. y(x) + A

∫ x

a

∫ x

a

∫ x

a

e λ(x–t) y(t) dt = f(x). .

[

e λ(x–t) – 1 ] y(t) dt = f(x). .

(x – t)e λ(x–t) y(t) dt = f(x). .

2-3. Integral equations whose kernels contain hyperbolic or spacial

functions

.11. y(x) + A

.12. y(x) + A

.13. y(x) – λ

∫ x

a

∫ x

a

∫ x

0

cosh[λ(x – t)]y(t) dt = f(x). .

sinh[λ(x – t)]y(t) dt = f(x). .

J 0 (x – t)y(t) dt = f(x). .

1


2 VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

2-4. Integral equations whose kernels contain arbitrary functions

.14. y(x) –

.15. y(x) +

.16. y(x) +

.17. y(x) +

∫ x

a

∫ x

a

∫ x

a

∫ x

a

g(x)h(t)y(t) dt = f(x). .

(x – t)g(x)y(t) dt = f(x). .

(x – t)g(t)y(t) dt = f(x). .

K(x – t)y(t) dt = f(x). Renewal equation. .

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http://eqworld.ipmnet.ru/en/solutions/ie/ie-toc2.pdf

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