Course overview
This course introduces and develops the theory of integral transforms, with a focus on how they can be applied to solve problems in science and engineering. It introduces the Dirac delta function, a generalised function or distribution that can be used to represent a point source or instantaneous impulse. Students will develop a range of techniques for solving ordinary and partial differential equations using the Laplace and Fourier Transforms, including problems involving the Dirac delta-function. Students will also understand a range of other integral transforms, and how they are applied to solve specific types of problems.
Course learning outcomes
- Recall the definition of the Dirac-delta function, its properties, and its interpretation as a point source / sudden impulse, etc
- Recall the definitions of the Laplace and Fourier transforms, and be able to derive their basic properties (shifting, derivatives, convolution, etc.). Calculate transforms of functions using these properties. Recall the inversion formulas for the Laplace and Fourier transforms, and be able to compute the inverse transforms of a range of functions using the methods from the course
- Apply the Laplace and / or Fourier transforms to solve problems in science and engineering (including ODE, PDE and signal processing problems)
- Recall that a range of other integral transforms exists, and given the definition of a transform, be able to apply it to solve problems similar to those encountered in the course