Basics of Laplace Transforms

What you'll Learn

• Introduction to Laplace Transforms:
  • Understand the concept of Laplace Transforms and its historical development.
  • Learn the advantages of using Laplace Transforms over other methods in solving differential equations.
  • Explore the basics of complex variables and functions relevant to Laplace Transforms.
• Properties and Theorems:
  • Study the key properties of Laplace Transforms, such as linearity, time-shifting, scaling, and differentiation/integration in the s-domain.
  • Master important theorems like the Convolution Theorem and Initial Value Theorem, and understand their implications in practical scenarios.
• Inverse Laplace Transforms:
  • Learn techniques for finding inverse Laplace Transforms to convert functions from the s-domain back to the time-domain.
  • Practice using various methods such as partial fraction decomposition, completing the square, and contour integration.
• Applications in Engineering and Physics:
  • Apply Laplace Transforms to solve ordinary and partial differential equations arising in engineering and physics.
  • Understand how Laplace Transforms are used in modeling and analyzing systems in control theory, circuit analysis, mechanical vibrations, and more.
• Advanced Topics:
  • Delve into advanced topics including Laplace Transforms of periodic functions, delta functions, and distributions.
  • Explore Laplace Transforms in discrete systems and their relation to Z-transforms.
• Numerical Techniques and Computational Tools:
  • Gain practical experience in using computational tools (such as MATLAB, Mathematica, or Python) to perform Laplace Transform calculations and simulations.
  • Implement numerical techniques for efficiently computing inverse Laplace Transforms and evaluating complex functions.