Schedule:

START OF CLASSES

• September 22: Lecture 1 [Slides] [Notes] [Laplace tables]
  Introduction. Ordinary differential equations (ODEs), and initial and boundary conditions. Solution of homogeneous and inhomogeneous ODEs. Eigenvalue and eigenvector analysis, and multimode analysis. Linear time-invariant systems, impulse response, and transfer function.
  
  
• September 23: Lecture 2, tutorial and problem solving session [Slides] [Notes] [Matlab code]
  Introduction to Matlab for linear systems, ODEs and PDEs. Analytical and numerical techniques. Solution to example problems, using paper and pencil, and verified by numerical simulation.

WEEK 1

• September 27: Lecture 3 [Notes] [Matlab code]
  Introduction to PDEs. One-dimensional heat equation, and its equivalents in electrical and chemical transport with applications to biomedical engineering. Flux through membranes. One-dimensional wave equation in an electrical transmission line, with open and short circuit termination. Finite difference PDE approximations.
  
  
• September 29: Lecture 4 [Notes] [Fourier tables]
  Solutions to PDEs over bounded and unbounded domains. Separation of variables. Boundary value problem and solution of the x dependent equation. Product solution of the PDEs with specified boundary conditions, and Fourier series expansions of initial conditions. Solutions over infinite domains using Fourier transforms.
  
  
• September 30: Problem solving session--student presentations

WEEK 2

• October 4: Lecture 5 [Notes]
  Heat equation. Temperature, thermal energy, and flux. Diffusion of thermal energy, and boundary conditions on temperature and flux. Thermal equilibrium.
  
  
• October 6: Lecture 6 [Notes]
  Analytical solution to the inhomogeneous heat equation with space varying source and boundary conditions. Decomposition of the solution into a particular steady-state solution, and Fourier series eigenmodes of the homogeneous solution. Fourier series expansions of initial conditions revisited.
  
  
• October 7: Problem solving session--student presentations

WEEK 3

• October 11: Lecture 7 [Notes]
  Analytical solution to inhomogeneous PDEs using Green's functions. Relationship to impulse response of linear time and space invariant systems. Green's solution to the inhomogeneous heat equation with time-varying and space-varying heat source.
  
  
• October 13: Lecture 8 [Notes] [Green's examples]
  Extended Green's solution to the inhomogeneous heat equation with time-varying value and flux boundary conditions. Solutions on infinite domains using Laplace and Fourier transforms.
  
  
• October 14: Problem solving session--student presentations

WEEK 4

• October 18: Lecture 9 [Notes]
  Heat and diffusion equation. Time-dependent solutions.
  
  
• October 20: Lecture 10 [Notes] [Sample midterm] [Solutions]
  Review and sample midterm.
  
  
• October 21: Problem solving session--student presentations

WEEK 5

• October 25: Midterm [Midterm] [Solutions] [Laplace tables]
  
  
• October 27: Lecture 11 [Notes]
  Review of vector calculus. Gradients, divergence, curl, and Laplacian. Transformation between Cartesian, cylindrical, and radial coordinates. Fields and potentials. Divergence theorem, and Stokes theorem.
  
  
• October 28: Problem solving session--student presentations

WEEK 6

• November 1: Lecture 12 [Notes]
  Gradient descent optimization. First-order and higher-order methods for null-finding and function minimization. Introduction to linear and nonlinear control systems in bioengineering.
  
  
• November 3: Lecture 13 [Notes] [Matlab code]
  Numerical solution to PDEs using finite element methods. Orthogonal, non-orthogonal, and triangular elements. Practical applications in bioengineering.
  
  
• November 4: Problem solving session--student presentations

WEEK 7

• November 8: Lecture 14 [Notes]
  Brownian motion, and diffusion. Theory for one-dimensional displacement. Scaling of diffusion in space and time. Viscous flow, and Reynolds number.
  
  
• November 10: Problem solving session--student presentations
  
  
• November 11: Veterans Day

WEEK 8

• November 15: Lecture 15 [Notes]
  Electrostatics. Coulomb's law. Electric field and potential. Work and moving charge. Equivalence of surface/field product and enclosed charge. Gauss's law, and Poisson's and Laplace's equation. Electric field near and in conductors. Dielectric phenomena. Capacitance.
  
  
• November 17: Lecture 16 [Notes] [Supplement]
  Introduction to electromagnetism using Maxwell's equations. Wave propagation in homogeneous and inhomogeneous media. Far and near field. RF telemetry and power delivery. Tissue absorption.
  
  
• November 18: Problem solving session--student presentations

WEEK 9

• November 22: Lecture 17 [Notes]
  The one dimensional wave equation. The vibrating string as a boundary value problem. Vibrating string clamped at both ends. Standing waves and summation of traveling waves.
  
  
• November 24: Thanksgiving
  

WEEK 10

• November 29: Lecture 18 [Notes]
  Sound. Transmission of waves in gases. Pressure variation in a sound wave.
  
  
• December 1: Lecture 19 [Notes] [Sample final exam] [Solutions]
  Course review and sample final exam.
  
  
• December 2: Problem solving session--student presentations

FINAL EXAM

• December 7: 11:30am-2:29pm, PFBH 161 [Solutions]
  

Reading and reference materials:

• Richard Haberman, Applied Partial Differential Equations (4th Edition), Pearson-Prentice Hall, 2004.
• H. M. Schey, Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (4th Edition), Norton Press, 2005.
• Richard Fitzpatrick, Classical Electromagnetism: An intermediate level course, Univ. Texas, 2006.
• Albert Einstein, Investigations on the theory of the Brownian movement, Dover Publications Inc., 1956 (translation from the 1905 original).
• Wikipedia, the free encyclopedia:
  • Ordinary differential equations
  • Partial differential equations
  • Vector calculus
  • Optimization
  • Classical electrodynamics
  • Diffusion
• Gerard Michon, Final Answers on Maxwell's Equations, Numericana, 2009.