BENG 221 Mathematical Methods in Bioengineering
START OF CLASSES
- September 27: Lecture 1
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
tables and more]
Introduction to Matlab for linear systems. Laplace and Fourier
transforms. Linear time-invariant systems, impulse response, and
- October 2: Lecture 3
Eigenvalue and eigenvector analysis, and multimode analysis.
Principal component analysis and singular value decomposition.
- October 4: Lecture 4
Analytical and numerical techniques for solving ODEs. Introduction to
Matlab for ODEs.
- October 5: Problem solving session--student presentations
Solution to example problems, using paper and pencil, and verified by
- October 9: Lecture 5
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
- October 11: Lecture 6
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
- October 12: Problem solving session--student presentations
- October 16: Lecture 7
Heat equation. Temperature, thermal energy, and flux. Diffusion of thermal
energy, and boundary conditions on temperature and flux. Thermal equilibrium.
- October 18: Lecture 8
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 19: Problem solving session--student presentations
- October 23: Lecture 9
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
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
- October 30: Lecture 11
Heat and diffusion equation in space and time. Separation of variables for
cartesian separable boundary conditions. Bounded, infinite, and semi-infinite
- November 1: Lecture 12
Review and practice midterm.
- November 2: Problem solving session--student presentations
- November 6: Midterm
- November 8: Lecture 13
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
- November 13: Lecture 14
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
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
- November 20: Lecture 16
Numerical solution to PDEs using finite element methods. Orthogonal,
non-orthogonal, and triangular elements. Practical applications in
- November 22-23: Thanksgiving
- November 27: Lecture 17
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
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
- December 4: Lecture 19
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
- December 7: Problem solving session--student presentations
- December 15: Final project reports due
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.
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:
on Maxwell's Equations, Numericana, 2009.