Solving Applied Mathematical Problems with MATLAB

1 Solving Applied Mathematical Problems WITH MATLAB 19/19/08 4:21:15 PMThis page intentionally left blank Solving Applied Mathematical Problems WITH MATLAB Dingy XueYangQuan 39/19/08 4:21:15 PMChapman & Hall/CRCT aylor & Francis Group6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa businessNo claim to original Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-8250-0 (Hardcover)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can-not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained.

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3 Engineering mathematics--Data processing. 2. MATLAB . 3. Numerical analysis--Data processing. 4. Mathematical optimization--Data processing. I. Chen, YangQuan, 1966- II. Title. 5133--dc22 2008025953 Visit the Taylor & Francis Web site the CRC Press Web site 49/19/08 4:21:15 PMContentsPrefacexi1 Computer Mathematics Languages An Computer Solutions to Mathematics Problems .. Why should we study computer mathematics language? Analytical solutions versus numerical solutions .. Mathematics software packages: an overview .. Summary of Computer Mathematics Languages .. A brief historic review of MATLAB .. Three widely used computer mathematics languages . Introduction to free scientific open-source softwares .. Outline of the Book..8 Exercises ..92 Fundamentals of MATLAB Fundamentals of MATLAB Programming .. Variables and constants in MATLAB .. Data structure.

4 Basic structure of MATLAB .. Colon expressions and sub-matrices extraction .. Fundamental Mathematical Calculations .. Algebraic operations of matrices .. Logic operations of matrices .. Relationship operations of matrices .. Simplifications and presentations of analytical results . Basic number theory computations .. Flow Control Structures of MATLAB Language.. Loop control structures .. Conditional control structures .. Switch structure .. Trial structure .. Writing and Debugging MATLAB Functions.. Basic structure of MATLAB functions .. Programming of functions with variable Inline functions and anonymous functions .. Two-Dimensional Graphics .. Basic statements of two-dimensional plotting .. Other two-dimensional plotting statements .. Implicit function plotting and applications ..36vviSolving Applied Mathematical Problems with Graphics decorations.

5 Three-Dimensional Graphics.. Plotting of three-dimensional curves .. Plotting of three-dimensional surfaces .. Viewpoint setting in 3D graphs ..43 Exercises ..443 Calculus Analytical Solutions to Calculus Problems .. Analytical solutions to limit Problems .. Analytical solutions to derivative Problems .. Analytical solutions to integral Problems .. Series Expansions and Series Evaluations.. Taylor series expansion .. Fourier series expansion .. Series .. Sequence product .. Numerical Differentiation .. Numerical differentiation algorithms .. Central-point difference algorithm .. Gradient computations of functions with two Numerical Integration Problems .. Numerical integration from given data using trapezoidalmethod .. Numerical integration of single variable functions .. Numerical solutions to double integrals .. Numerical solutions to triple integrals.

6 Path Integrals and Line Integrals .. Path integrals .. Line integrals .. Surface Integrals .. Scalar surface integrals .. Vector surface integrals ..84 Exercises ..854 Linear Algebra Inputting Special Matrices.. Numerical matrix input .. Defining symbolic matrices .. Fundamental Matrix Operations .. Basic concepts and properties of matrices .. Matrix inversion and generalized inverse of a matrix . Matrix eigenvalue Problems .. Fundamental Matrix Transformations .. Similarity transformations and orthogonal matrices .. Triangular and Cholesky decompositions .. Jordan transformations .. Singular value decompositions .. Solving Matrix Equations .. Solutions to linear algebraic equations .. Solutions to Lyapunov equations .. Solutions to Sylvester equations .. Solutions to Riccati equations .. Nonlinear Functions and Matrix Function Evaluations.

7 Element-by-element computations .. Matrix function evaluations ..127 Exercises ..1335 Integral Transforms and Complex Variable Laplace Transforms and Their Inverses .. Definitions and properties .. Computer solution to Laplace transform Problems .. Fourier Transforms and Their Inverses .. Definitions and properties .. Solving Fourier transform Problems .. Fourier sine and cosine transforms .. Discrete Fourier sine, cosine transforms .. Other Integral Transforms .. Mellin transform .. Hankel transform solutions .. Z Transforms and Their Inverses .. Definitions and properties of Z transforms and Computations of Z transform .. Solving Complex Variable Function Problems .. Complex variable functions and mapping Concept and computation of residues .. Partial fraction expansion for rational functions .. Inverse Laplace transform using PFEs.

8 Computing closed-path integrals ..160 Exercises ..1626 Nonlinear Equations and Numerical Optimization Nonlinear Algebraic Equations .. Graphical method for Solving nonlinear equations .. Quasi-analytical solutions to polynomial-type equations Numerical solutions to general nonlinear equations .. Nonlinear matrix equations .. Unconstrained Optimization Problems .. Analytical solutions and graphical solution methods .. Numerical solution of unconstrained optimization usingMATLAB .. Global minimum and local minima .. Solving optimization Problems with gradients .. Optimization Problems with bounded constraints .. Constrained Optimization Problems .. Constraints and feasibility regions ..184viiiSolving Applied Mathematical Problems with Solving linear programming Problems .. Solving quadratic programming Problems .. Solving general nonlinear programming Problems .

9 Mixed Integer Programming Problems .. Solving mixed integer programming Problems .. Solving binary programming Problems .. Linear Matrix Inequalities .. A general introduction to LMIs .. Lyapunov inequalities .. Classification of LMI Problems .. LMI problem solutions with MATLAB .. Optimization of LMI Problems by YALMIP Toolbox .201 Exercises ..2037 Differential Equation Analytical Solution Methods for Special Classes of ODEs.. Mathematical descriptions .. Analytical solution methods .. Applications of Laplace transforms .. Analytical solutions to LTI state-space equations .. Analytical solutions to special nonlinear differentialequations .. Numerical Solutions to ODEs .. Overview of numerical solution algorithms .. Fixed-step Runge-Kutta algorithm and its MATLAB implementation .. Numerical solution to first-order vector ODEs.

10 Transforms to standard ODEs .. Validation of numerical solutions to ODEs .. Numerical Solutions to Special Ordinary Differential Equations Solutions of stiff ODEs .. Solutions of implicit differential equations .. Solutions to differential algebraic equations .. Solutions to delay differential equations .. Solving Boundary Value Problems .. Solutions to two-point boundary value Problems .. Solutions to general boundary value Problems .. Introduction to Partial Differential Equations .. Solving a set of 1D PDEs .. Mathematical description to 2D PDEs .. The GUI for the PDE Toolbox an introduction .. Solving ODEs with Block Diagrams in Simulink.. A brief introduction to Simulink .. Simulink relevant blocks .. Using Simulink for modeling and simulation of ODEs .260 Exercises ..263 Contentsix8 Data Interpolation and Functional Approximation Interpolation and Data Fitting.