Foundations of Applied Mathematics

Course materials for MATH 35803, Fall 2026.

Course Offering

Fall 2026 · MWF 10:45–11:35 AM · CHPN 421 · University of Arkansas

Syllabus (PDF) · Weekly schedule (PDF)

Course Description

Introduction to applied mathematical modeling using deterministic and stochastic frameworks. The course develops dimensional analysis, perturbation methods, kinetics, traffic flow, continuum mechanics, Newtonian fluids, and diffusion as a sequence of increasingly rich deterministic models, then bridges to a stochastic module — random walks and Brownian motion — before ending with an introduction to financial mathematics: the binomial asset-pricing model, American options, and the Black–Scholes equation. Emphasis throughout is on formulating, analyzing, and interpreting mathematical models arising from real-world systems; the course is analytical, with no programming component.

Topics

  1. Dimensional Analysis — units, dimensional homogeneity, Buckingham Π, similarity, and nondimensionalization
  2. Perturbation Methods — regular and singular expansions, dominant balance, boundary layers
  3. Kinetics — mass action, equilibria, linear stability, nullclines, phase-plane dynamics, bifurcation
  4. Traffic Flow — conservation laws, characteristics, shocks, Rankine–Hugoniot, rarefaction fans
  5. Continuum Mechanics — material and spatial descriptions, balance laws, strain, stress, elastic and viscoelastic bars
  6. Newtonian Fluids — Navier–Stokes equations, Reynolds number, Poiseuille and Couette flow
  7. Diffusion — random walks, the heat equation, boundary conditions, Fourier methods, Langevin dynamics
  8. Probability, Random Walk, and Brownian Motion — the bridge from deterministic to stochastic modeling
  9. Binomial Model and No-Arbitrage — replication, risk-neutral probability, multi-period pricing
  10. American Options — early exercise, backward induction, optimal stopping
  11. Black–Scholes — the continuous limit, geometric Brownian motion, the pricing PDE

Assessment

Exam 1, Exam 2, and Exam 3 each 20% · Homework 20% · Quizzes 20%. Three 50-minute exams cover the material in topic blocks (Topics 1–4, 5–6, and 7–11); weekly homework and short weekly quizzes track the material as it is taught.

Course Materials

Homework, quizzes, and their solution keys are distributed through Blackboard. If you are not enrolled but are interested in the course, please email me at jiahuic@uark.edu with any questions.

Archive

Materials from earlier offerings of this course are available here.