Schedule:

START OF CLASSES

• September 27: Lecture 1 [Slides] [Notes]
  Introduction. Ordinary differential equations (ODEs), and initial and boundary conditions. Solution of homogeneous and inhomogeneous ODEs.
  
  
• September 28: Lecture 2, tutorial and problem solving session [Notes] [Laplace tables and more]
  Introduction to Matlab for linear systems. Laplace and Fourier transforms. Linear time-invariant systems, impulse response, and transfer function.

WEEK 1

• October 2: Lecture 3 [Notes]
  Eigenvalue and eigenvector analysis, and multimode analysis. Principal component analysis and singular value decomposition.
  
  
• October 4: Lecture 4 [Slides] [Notes] [Matlab code]
  Analytical and numerical techniques for solving ODEs. Introduction to Matlab for ODEs.
  
  
• October 5: Problem solving session--student presentations [Reports] Solution to example problems, using paper and pencil, and verified by numerical simulation.

WEEK 2

• October 9: Lecture 5 [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.
  
  
• October 11: Lecture 6 [Notes] [Fourier tables] [Matlab code]
  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.
  
  
• October 12: Problem solving session--student presentations [Reports]

WEEK 3

• October 16: Lecture 7 [Notes]
  Heat equation. Temperature, thermal energy, and flux. Diffusion of thermal energy, and boundary conditions on temperature and flux. Thermal equilibrium.
  
  
• October 18: Lecture 8 [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 19: Problem solving session--student presentations [Reports]

WEEK 4

• October 23: Lecture 9 [Notes] [Green's blues]
  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 25: Lecture 10 [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 26: Problem solving session--student presentations [Reports]

WEEK 5

• October 30: Lecture 11 [Notes]
  Heat and diffusion equation in space and time. Separation of variables for cartesian separable boundary conditions. Bounded, infinite, and semi-infinite domains.
  
  
• November 1: Lecture 12 [Notes] [Practice midterm] [Solutions] [Laplace tables] [Fourier tables]
  Review and practice midterm.
  
  
• November 2: Problem solving session--student presentations [Reports]

WEEK 6

• November 6: Midterm [Midterm]
  
  
• November 8: Lecture 13 [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.
  
  
• November 9: Problem solving session--student presentations [Reports]

WEEK 7

• November 13: Lecture 14 [Notes] [Matlab code]
  Diffusion in polar and cylindrical coordinates. Analytical solution using Bessel functions. Value and flux boundary conditions in terms of roots and extrema of Bessel functions. Fourier-Bessel series expansian of initial conditions.
  
  
• November 15: Lecture 15 [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 16: Problem solving session--student presentations [Reports]

WEEK 8

• November 20: Lecture 16 [Notes] [Matlab code] [FEniCS platform]
  Numerical solution to PDEs using finite element methods. Orthogonal, non-orthogonal, and triangular elements. Practical applications in bioengineering.
  
  
• November 22-23: Thanksgiving
  

WEEK 9

• November 27: Lecture 17 [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 29: Lecture 18 [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 30: Problem solving session--student presentations [Reports]

WEEK 10

• December 4: Lecture 19 [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. Sound and transmission of waves in gases.
  
  
• December 6: Lecture 20 [Notes]
  Course review
  
  
• December 7: Problem solving session--student presentations [Reports]

FINALS WEEK

• December 15: Final project reports due

Reading and reference materials:

• Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri, Numerical Mathematics, Texts in Applied Mathematics 37, Springer, 2000 (2nd Ed., 2007).
Matlab examples
• Richard Haberman, Applied Partial Differential Equations (4th Edition), Pearson-Prentice Hall, 2004.
• Y.J. Shin and L. Bleris, "Linear Control Theory for Gene Network Modeling," PLoS ONE, vol. 5 (9), e12785 (doi:10.1371/journal.pone.0012785). 2010.
• 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.