BENG 221 Mathematical Methods in Bioengineering
Fall 2016
START OF CLASSES
 September 22: Lecture 1
[Slides]
[Notes]
[LTI systems and ODEs]
[Laplace
tables and more]
Introduction. Ordinary differential equations
(ODEs), and initial and boundary conditions. Solution of homogeneous and
inhomogeneous ODEs. Eigenvalue and eigenvector analysis, and multimode
analysis. Linear timeinvariant 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. Onedimensional heat equation, and its equivalents in
electrical and chemical transport with applications to biomedical engineering.
Flux through membranes. Onedimensional wave equation in an electrical
transmission line, with open and short circuit termination. Finite difference
PDE approximations.
 September 29: Lecture 4
[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.
 September 30: Problem solving sessionstudent presentations
[Reports]
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 steadystate solution, and Fourier series eigenmodes of the
homogeneous solution. Fourier series expansions of initial conditions
revisited.
 October 7: Problem solving sessionstudent presentations
[Reports]
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 timevarying
and spacevarying heat source.
 October 13: Lecture 8
[Notes]
[Green's examples]
Extended Green's solution to the inhomogeneous heat equation with timevarying
value and flux boundary conditions. Solutions on infinite domains using
Laplace and Fourier transforms.
 October 14: Problem solving sessionstudent presentations
[Reports]
WEEK 4
 October 18: Lecture 9
[Notes]
Heat and diffusion equation in space and time. Separation of variables for
cartesian separable boundary conditions. Bounded, infinite, and semiinfinite
domains.
 October 20: Lecture 10
[Notes]
[Practice midterm]
[Solutions]
[Laplace tables]
[Fourier tables]
Review and practice midterm.
 October 21: Problem solving sessionstudent presentations
[Reports]
WEEK 5
 October 25: Guest Lecture  Dr. Intaglietta
[Lecture notes]
Brownian motion, and diffusion. Theory for onedimensional
displacement. Scaling of diffusion in space and time. Viscous flow, and
Reynolds number.
 October 27: Midterm
[Midterm]
 October 28: Problem solving sessionstudent presentations
[Reports]
WEEK 6
 November 1: 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.
 November 3: Lecture 12
[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. FourierBessel series
expansian of initial conditions.
 November 4: Problem solving sessionstudent presentations
[Reports]
WEEK 7
 November 8: Lecture 13
[Notes]
Gradient descent optimization. Firstorder and higherorder methods for
nullfinding and function minimization. Introduction to linear and nonlinear
control systems in bioengineering.
 November 10: Lecture 14
[Notes]
[Matlab code]
Numerical solution to PDEs using finite element methods. Orthogonal,
nonorthogonal, and triangular elements. Practical applications in
bioengineering.
 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 sessionstudent presentations
[Reports]
WEEK 9
 Novermber 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 2425: 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]
 December 2: Problem solving sessionstudent presentations
[Reports]

Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri, Numerical
Mathematics, Texts in Applied Mathematics 37, Springer, 2000 (2nd Ed., 2007).

Richard Haberman, Applied Partial Differential
Equations (4th Edition), PearsonPrentice 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:

Gerard Michon,
Final Answers
on Maxwell's Equations, Numericana, 2009.