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Exact Solutions > Integral Equations > Volterra Integral Equations of the First Kind and Related Linear Integral Equations with Variable Limit of Integration

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1. Volterra Integral Equations of the First Kind

1-1. Integral equations with kernels involving power-law functions

  1. (x − t) y(t) dt = f(x).
  2. (Ax + Bt + C) y(t) dt = f(x).
  3. (x − t)n y(t) dt = f(x),     n = 1, 2, ...
  4. (x − t)1/2 y(t) dt = f(x).
  5. (x − t)−1/2 y(t) dt = f(x).     Abel equation.
  6. (x − t)λ y(t) dt = f(x),     0 < λ < 1.
  7. (x − t)−λ y(t) dt = f(x),     0 < λ < 1.     Generalized Abel equation.

1-2. Integral equations with kernels involving exponential functions

  1. eλ(x−t) y(t) dt = f(x).
  2. eλx+βt y(t) dt = f(x).
  3. [eλ(x−t) − 1] y(t) dt = f(x).
  4. [eλ(x−t) + b] y(t) dt = f(x).
  5. [eλ(x−t) − eμ(x−t)] y(t) dt = f(x).
  6. (eλx − eλt)−1/2 y(t) dt = f(x).

1-3. Integral equations with kernels involving hyperbolic functions

  1. cosh[λ(x − t)] y(t) dt = f(x).
  2. {cosh[λ(x − t)] − 1} y(t) dt = f(x).
  3. {cosh[λ(x − t)] + b} y(t) dt = f(x).
  4. cosh2[λ(x − t)] y(t) dt = f(x).
  5. sinh[λ(x − t)] y(t) dt = f(x).
  6. {sinh[λ(x − t)] + b} y(t) dt = f(x).
  7. sinh[λ(x − t)1/2] y(t) dt = f(x).

1-4. Integral equations with kernels involving logarithmic functions

  1. ln(x − t) y(t) dt = f(x).
  2. [ln(x − t) + A] y(t) dt = f(x).
  3. (x − t)[ln(x − t) + A] y(t) dt = f(x).

1-5. Integral equations with kernels involving trigonometric functions

  1. cos[λ(x − t)] y(t) dt = f(x).
  2. {cos[λ(x − t)] − 1} y(t) dt = f(x).
  3. {cos[λ(x − t)] + b} y(t) dt = f(x).
  4. sin[λ(x − t)] y(t) dt = f(x).
  5. sin[λ(x − t)1/2] y(t) dt = f(x).

1-6. Integral equations with kernels involving special functions

  1. J0(λ(x − t)) y(t) dt = f(x).
  2. J0(λ(x − t)1/2) y(t) dt = f(x).
  3. I0(λ(x − t)) y(t) dt = f(x).
  4. I0(λ(x − t)1/2) y(t) dt = f(x).

10-7. Integral equations with kernels involving arbitrary functions

  1. [g(x) − g(t)] y(t) dt = f(x).
  2. [g(x) − g(t) + b] y(t) dt = f(x).
  3. [g(x) + h(t)] y(t) dt = f(x).
  4. K(x − t) y(t) dt = f(x).
  5. [g(x) − g(t)]1/2 y(t) dt = f(x).
  6. [g(x) − g(t)]−1/2 y(t) dt = f(x).

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