Numerical Methods

for Calculus & DE

	MA3457/CS4033 Project # 2: Approximation

B'14 Term 2014-2015 Sections B01/B01 Mon, Tue, Thu, Fri - 2:00 pm

This project consists of two parts. The first one contains two practical problems: one comes from the interdisciplinary area of food sciences and microwave engineering, another from the history of music. For the second part, you should identify, formulate, and solve an original applied problem from the area of your concentration/professional interest.

When working on this project, rely on your knowledge of approximation theory and practice and use of the Method of Least Squares Approximation implemented in corresponding MATLAB procedures. Include essential elements of the MATLAB scripts and outputs in your report along with concise but comprehensive comments on your solution.

Note: It is expected that you would be limited to using the functions in the course Library of MATLAB Procedures (Lin_LS, Quad_LS, Cubic_LS, and polyfit for n = 4, 5, 6), however, if you wish (and think are able) to use more advanced methods (e.g., other MATLAB functions based on more sophisticated techniques) you can do that.

Problem 1

Background: In accordance with one of the basic design criteria employed in microwave power engineering, parameters of systems for microwave heating of dielectric materials should be adjusted to their dielectric properties. The designs of industrial devices for thermal processing of blocks of frozen meat were of limited success until the time when dielectric properties (and other media parameters) of meat had been investigated at different temperatures. It turned out that the dielectric constant of different sorts of meat dramatically changes over the narrow temperature range around 0 C.

While this experimental data is available in a tabulated format, some modern CAD tools (e.g., COMSOL Multiphysics) require representation of this data in a functional form.

Data: The set of experimental values of the dielectric constant of raw beef as function of temperature was provided by P.O. Risman, Microtrans AB, Sweden, in 2000; they can be presented as follows:

Questions:

• Apply the method of Least Squares to the given data. Visualize the experimental points along with the approximating functions; comment on the quality of approximation.
• Suggest a simple engineering solution significantly improving the initial approximation. Compose a MATLAB script implementing the entire solution; give an explicit analytical form of the obtained approximation.

Problem 2

Background and Data: Wolfgang Amadeus Mozart (1756-1791) was one of the most prolific composers of all time. In 1862, the German musicologist Ludwig von Köchel made a chronological list of Mozart's musical work. This list is the source of the Köchel numbers, or "K numbers", that now accompany the titles of Mozart's pieces (e.g., Sinfonia Concertante in E-flat major, K. 364). The table below gives the Köchel numbers and composition dates of 17 of Mozart's works.

Questions:

• Construct the linear, quadratic, and cubic Least Squares Approximations for the data and determine the best one.
• Mozart's String Quartet K. 173 was composed in 1773. What applied models predict the correct date?
• If Mozart had lived, when would he have finished his next 100 works, i.e., in what year, in accordance with the linear, quadratic, and cubic models, would he have finished K. 726? Which number out of the three seems to be most reliable? Why?

Problem from Your Field

Background, Data, Question

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Last modified: Fri, Nov 14, 2014