Laplace Transforms
7.0 Abstract
Sometimes an ODE can be really complex and hard to solve by any basic methods we have looked at in previous tutorials. For example, it may involve exponential functions, trigonometric functions, Heaviside functions, and anything else you can imagine. Hence, a famous French mathematician Pierre-Simon Laplace found a transform method, which converts the functions in "time-domain" to "complex number-domain", to overcome the problem. It is great for transforming the calculation of differentiation and integration to the simple algebraic calculation.
7.1 Concept
Here is a basic concept map from my course note when first learning Laplace transform:
$ \begin{bmatrix} ODE in y(t) \\ "Time-domain" \end{bmatrix} → \begin{bmatrix} Laplace Transform \\ L=[y(t)] \end{bmatrix} → \begin{bmatrix} Algebraic equation in Y(s) \\ "Complex number-domain" \end{bmatrix} $;
$ \begin{bmatrix} Algebraic equation in Y(s) \\ "Complex number-domain" \end{bmatrix} → \begin{bmatrix} Inverse Laplace Transform \\ L^{-1}[Y(s)] \end{bmatrix} → \begin{bmatrix} ODE in y(t) \\ "Time-domain" \end{bmatrix} $.
7.2 Linearity of Laplace Transform
7.3 Heaviside Unit Step Function for Discontinuous Functions
7.4 Exercises
7.5 References
Institute of Natural and Mathematical Science, Massey University. (2017). 160.204 Differential Equations I: Course materials. Auckland, New Zealand.
Robinson, J. C. (2003). An introduction to ordinary differential equations. New York, NY., USA: Cambridge University Press.
