BENG 221 Mathematical Methods in Bioengineering
START OF CLASSES
- September 22: Lecture 1
[LTI systems and ODEs]
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 time-invariant systems,
impulse response, and transfer function.
- September 23: Lecture 2, tutorial and problem solving session
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.
- September 27: Lecture 3
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
- September 29: Lecture 4
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
- September 30: Problem solving session--student presentations
- October 4: Lecture 5
Heat equation. Temperature, thermal energy, and flux. Diffusion of thermal
energy, and boundary conditions on temperature and flux. Thermal equilibrium.
- October 6: Lecture 6
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
- October 7: Problem solving session--student presentations
- October 11: Lecture 7
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
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
- October 18: Lecture 9
Heat and diffusion equation in space and time. Separation of variables for
cartesian separable boundary conditions. Bounded, infinite, and semi-infinite
- October 20: Lecture 10
Review and practice midterm.
- October 21: Problem solving session--student presentations
- October 25: Guest Lecture - Dr. Intaglietta
Brownian motion, and diffusion. Theory for one-dimensional
displacement. Scaling of diffusion in space and time. Viscous flow, and
- October 27: Midterm
- October 28: Problem solving session--student presentations
- November 1: Lecture 11
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
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 4: Problem solving session--student presentations
- November 8: Lecture 13
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 10: Lecture 14
Numerical solution to PDEs using finite element methods. Orthogonal,
non-orthogonal, and triangular elements. Practical applications in
- November 11: Veterans Day
- November 15: Lecture 15
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
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
- Novermber 22: Lecture 17
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 24-25: Thanksgiving
- November 29: Lecture 18
Sound. Transmission of waves in gases. Pressure variation in a sound wave.
- December 1: Lecture 19
- December 2: Problem solving session--student presentations
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), 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:
on Maxwell's Equations, Numericana, 2009.