18.3. Example Problems

Try these first

1. 1.

   What is the general form of a Volterra integral equation of the second kind?

   Hint

   Think about the elements involved in the equation structure.

2. 2.

   What does the kernel $K(t - \tau)$ represent?

   Hint

   Consider what role it plays in the behavior of the integral.

3. 3.

   What is an integral equation?

   Hint

   Think about the general definition of integral equations.

4. 4.

   Does the kernel in a Volterra equation affect the solution?

   • True
   • False

   Hint

   Consider the effect of the kernel on previous function values.

5. 5.

   Use Laplace Transforms to solve the Volterra integral equation: $f(t) = \cos(t) + \int_{0}^{t} \sin(t - \tau)f(\tau)d\tau$.

   Hint

   Focus on identifying the correct kernel and solving dimensions carefully.

6. 6.

   Derive the solution for the integral equation: $f(t) = 2 + \int_{0}^{t} (t - \tau)^2 f(\tau) d\tau$ using transformations.

   Hint

   Make sure to handle the squaring in the kernel when applying transformations.