Exact Solutions > Integral Equations > Fredholm Integral Equations of the Second Kind and Related Linear Integral Equations with Constant Limits of Integration

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4. Fredholm Integral Equations of the Second Kind

4-1. Integral equations whose kernels contain power-law functions

1. y(x) − λ (x − t) y(t) dt = f(x).
2. y(x) + A |x − t| y(t) dt = f(x).
3. Ay(x) + Bπ⁻¹ (t − x)⁻¹ y(t) dt = f(x).
4. y(x) − λ [(t − x)⁻¹ − (x + t − 2xt)⁻¹] y(t) dt = f(x).    Tricomi's equation.

4-2. Integral equations with kernels involving exponential or hyperbolic functions

5. y(x) + λ e^−|x−t| y(t) dt = f(x).
6. y(x) − λ e^−|x−t| y(t) dt = 0.    Lalesco--Picard equation.
7. y(x) + λ e^−|x−t| y(t) dt = f(x).
8. y(x) + A e^λ|x−t| y(t) dt = f(x).
9. y(x) + λ {cosh[b(x − t)]}⁻¹ y(t) dt = f(x).

4-3. Integral equations with kernels involving trigonometric functions

10. y(x) − λ cos(xt) y(t) dt = f(x).
11. y(x) − λ sin(xt) y(t) dt = f(x).
12. y(x) − λ sin(x − t)(x − t)⁻¹ y(t) dt = f(x).
13. Ay(x) − B(2π)⁻¹ cot(t/2 − x/2) y(t) dt = f(x).
14. y(x) − λ e^{μ(x − t)} cos(xt) y(t) dt = f(x).
15. y(x) − λ e^{μ(x − t)} sin(xt) y(t) dt = f(x).

4-4. Integral equations with kernels involving arbitrary functions

16. y(x) − K(x − t) y(t) dt = f(x).
17. y(x) − K(x − t) y(t) dt = f(x).    Wiener--Hopf equation of the second kind.