Solving high-level applications in engineering, physics, chemistry, and biology requires an understanding of modeling at a system level. To fully prepare a student, this course emphasizes system analysis. Crucial to modeling in the modern world is an understanding of computational modeling as well as mathematical formulation. Therefore a variety of numerical/computational methods will be reviewed in the first part of this course and extended for the purpose of understanding the computational methods required to do modeling in a modern setting. Subjects to be studied include error analysis, roots of non-linear equations, solving systems of linear equations, eigenvalues, eigenvectors, and eigenfunctions, optimization, curve fitting including splines, Fourier analysis, modeling, numerical differentiation and integration, and numerical solving of differential equations including, but not limited to, predictor-corrector methods and finite element analysis. It will be assumed that the student is at least partially familiar with these concepts from previous mathematics classes. Extra study may be required for a student lacking these skills. These concepts will be extended into computational methods that are useful in analyzing signals and systems. Topics will include representation of systems and signals, transfer functions, and filters. The relationship between linear systems and both discrete time and continuous time signals and sampling will be explored and used to better understand real-world applications. Practical issues of representation and sampling of signals will be explored with particular emphasis on best case solutions. This will be extended into the study and use of a number of filters, in particular digital filters. Topics will include OTFs, DFTs, Laplace transforms, Z-transforms, Radon transforms, and convolutions. Lastly, extensive surverys of a number of advanced subjects include molecular dynamics, percolation, and Monte Carlo simulation methods. Some new mathematical concepts will be introduced in the class. A number of software packages and languages important to engineering are surveyed with primary emphasis on mastering one high-level language such as MATLAB/Octave, C/gcc/g++, or Fortran/gfortran. This course, recognizing the fact that all engineers and scientists need the aforementioned topics, will emphasize a number of case studies in such areas as mechanical, civil, environmental, electrical, aerospace, chemical, and biological engineering, as well as in the sciences.
Physical Sci and Engineering