18.1.1. Convolution Theorem

Try these first

1. 1.

   What is an integral equation?

   Hint

   Think about the form of equations you see that include an integral.

2. 2.

   What is the kernel in a Volterra Integral Equation?

   Hint

   Consider how functions interact in the integral.

3. 3.

   What does the Convolution Theorem state?

   • It states that convolution can be ignored in computation.
   • It relates the convolution of functions to their Laplace Transforms.
   • It is only applicable in discrete mathematics.

   Hint

   Think about how we can simplify problems in calculus.

4. 4.

   True or False: The Laplace Transform converts integrals directly into sums.

   • True
   • False

   Hint

   Consider the nature of transforms versus sums.

5. 5.

   Solve the integral equation f(t) = 3 + ∫_{0}^{t} (t − τ)f(τ)dτ where f(0) = 1.

   Hint

   Consider initial conditions when applying Laplace.

6. 6.

   Derive f(t) if f(t) = 2 + ∫_{0}^{t} cos(t − τ)f(τ)dτ.

   Hint

   Start by applying the Laplace Transform and resetting the cos function using transformations.