Foundations of Applied Mathematics

Course details, syllabus, and materials for MATH 3583.

Course Description

This course introduces students to the foundations of applied mathematics, emphasizing both the analytical techniques and their numerical implementations. Designed for undergraduate students with a solid understanding of calculus and differential equations, the course offers a deep dive into topics ranging from the fundamentals of numerical linear algebra to the modeling of physical phenomena through kinetics, diffusion, and more. Real-world applications will be showcased, providing students with a comprehensive understanding of how mathematical techniques are employed in various fields. Python (jupyter notebook) will be employed as the primary computational tool, making it a dual journey of mathematical rigor and computational exploration.

By the end of this course, you will be able to:

  • Understand and apply Taylor expansion techniques to real-world problems.
  • Solve complex problems using numerical linear algebra, appreciating the matrix multiplication, eigenvalues, and transformations.
  • Understand the fundamental principles of dimensionality reduction, applying them to real-world problems.
  • Analyze and model dynamic systems with kinetics equations, understanding the underlying principles of chemical reactions, ecological dynamics, and conservation laws.
  • Dive deep into the diffusion equation, grasping both discrete models like random walks and continuum theories.
  • Understand the fundamental principles behind traffic flow, applying mathematical models to predict and analyze traffic dynamics.
  • Tackle Laplacian equations in 2D, appreciating their significance and solving them using finite difference methods.

Content

  1. Python basic
  2. Vectors: vector addition, scaler multiplication, linear combination, linear independence, vector space
  3. Inner product: dot product, inner product, Taylor expansion, polynomial approximation
  4. Matrices: matrix multiplication, Echelon form, Gauss Jordan elimination, inverse
  5. Dimensional Analysis dimension reduction
  6. Kinetics: kinetic equations, conservation law, steady states
  7. Diffusion: random walks, continuous limit
  8. Traffic flow: density, flux, characteristic
  9. Laplacian: 2D Cartesian, boundary condition, numerical solutions
  10. Projects