Advanced mathematics and mechanics applications using MatLab

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Wilson & Turcotte & Halpern - Advanced mathematics and mechanics applications using MatLab.pdf
This book uses MATLAB
R to analyze various applications in mathematics and me-
chanics. The authors hope to encourage engineers and scientists to consider this
modern programming environment as an excellent alternative to languages such as
FORTRAN or C++. MATLAB1 embodies an interactive environment with a high
level programming language supporting both numerical and graphical commands for
two- and three-dimensional data analysis and presentation. The wealth of intrinsic
mathematical commands to handle matrix algebra, Fourier series, differential equa-
tions, and complex-valued functions makes simple calculator operations of many
tasks previously requiring subroutine libraries with cumbersome argument lists.
We analyze problems, drawn from our teaching and research interests, empha-
sizing linear and nonlinear differential equation methods. Linear partial differential
equations and linear matrix differential equations are analyzed using eigenfunctions
and series solutions. Several types of physical problems are considered. Among
these are heat conduction, harmonic response of strings, membranes, beams, and
trusses, geometrical properties of areas and volumes, ßexure and buckling of inde-
terminate beams, elastostatic stress analysis, and multi-dimensional optimization.
Numerical integration of matrix differential equations is used in several examples
illustrating the utility of such methods as well as essential aspects of numerical ap-
proximation. Attention is restricted to the Runge-Kutta method which is adequate to
handle most situations. Space limitation led us to omit some interesting MATLAB
features concerning predictor-corrector methods, stiff systems, and event locations.
This book is not an introductory numerical analysis text. It is most useful as a ref-
erence or a supplementary text in computationally oriented courses emphasizing ap-
plications. The authors have previously solved many of the examples in FORTRAN.
Our MATLAB solutions consume over three hundred pages (over twelve thousand
lines). Although few books published recently present this much code, comparable
FORTRAN versions would probably be signifcantly longer. In fact, the conciseness
of MATLAB was a primary motivation for writing the book.
The programs contain many comments and are intended for study as separate en-
tities without an additional reference. Consequently, some deliberate redundancy
exists between program comments and text discussions. We also list programs in a
style we feel will be helpful to most readers. The source listings show line numbers
adjacent to the MATLAB code. MATLAB code does not use line numbers or permit
goto statements. We have numbered the lines to aid discussions of particular pro-
gram segments. To conserve space, we often place multiple MATLAB statements on
the same line when this does not interrupt the logical ßow.
All of the programs presented are designed to operate under the 6.x version of
MATLAB and Microsoft Windows. Both the text and graphics windows should be
simultaneously visible. A windowed environment is essential for using capabilities
like animation and interactive manipulation of three dimensional Þgures. The source
code for all of the programs in the book is available from the CRC Press website at
http://www.crcpress.com. The program collection is organized using an
independent subdirectory for each of the thirteen chapters.
This third edition incorporates much new material on time dependent solutions of
linear partial differential equations. Animation is used whenever seeing the solution
evolve in time is helpful. Animation illustrates quite well phenomena like wave
propagation in strings and membranes. The interactive zoom and rotation features in
MATLAB are also valuable tools for interpreting graphical output.
Most programs in the book are academic examples, but some problem solutions
are useful as stand-alone analysis tools. Examples include geometrical property cal-
culation, differentiation or integration of splines, Gauss integration of arbitrary order,
and frequency analysis of trusses and membranes.
A chapter on eigenvalue problems presents applications in stress analysis, elastic
stability, and linear system dynamics. A chapter on analytic functions shows the
efÞciency of MATLAB for applying complex valued functions and the Fast Fourier
Transform (FFT) to harmonic and biharmonic functions. Finally, the book concludes
with a chapter applying multidimensional search to several nonlinear programming
problems.
We emphasize that this book is primarily for those concerned with physical appli-
cations. A thorough grasp of Euclidean geometry, Newtonian mechanics, and some
mathematics beyond calculus is essential to understand most of the topics. Finally,
the authors enjoy interacting with students, teachers, and researchers applying ad-
vanced mathematics to real world problems.The availability of economical computer
hardware and the friendly software interface in MATLAB makes computing increas-
ingly attractive to the entire technical community. If we manage to cultivate interest
in MATLAB among engineers who only spend part of their time using computers,
our primary goal will have been achieved.
Howard B. Wilson hwilson@bama.ua.edu
Louis H. Turcotte turcotte@rose-hulman.edu
David Halpern david.halpern@ua.edu





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