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The main goals of these lectures are to introduce concepts of numerical methods and introduce MatLAB in an Engineering framework. By this we do not mean that every problem is a “real life” engineering application, but more that the engineering way of thinking is emphasized throughout the discussion.
The philosophy of this book was formed over the course of many years. My father was a Civil Engineer and surveyor, and he introduced me to engineering ideas from an early age. At the University of Kentucky I took most of the basic Engineering courses while getting a Bachelor’s degree in Mathematics. Immediately afterward I completed a M.S. degree in Engineering Mechanics at Kentucky. While working on my Ph.D. in Mathematics at Georgia Tech I taught all of the introductory math courses for engineers. During my education, I observed that incorporation of computation in coursework had been extremely unfocused and poor. For instance during my college career I had to learn 8 different programming and markup languages on 4 different platforms plus numerous other software applications. There was almost no technical help provided in the courses and I wasted innumerable hours figuring out software on my own. A typical, but useless, inclusion of software has been (and still is in most calculus books) to set up a difficult ‘applied’ problem and then add the line “write a program to solve” or “use a computer algebra system to solve”.
In these lecture notes, instruction on using MatLAB is dispersed through the material on numerical methods. In these lectures details about how to use MatLAB are detailed (but not verbose) and explicit. To teach programming, students are usually given examples of working programs and are asked to make modifications.
The lectures are designed to be used in a computer classroom, but could be used in a lecture format with students doing computer exercises afterward. The lectures are divided into four Parts with a summary provided at the end of each Part.
MatLAB and Solving Equations
Vectors, Functions, and Plots in MatLAB
MatLAB Programs
Newton’s Method and Loops
Controlling Error and Conditional Statements
The Bisection Method and Locating Roots
Secant Methods
Symbolic Computations
Linear Algebra
Matrices and Matrix Operations in MatLAB
Introduction to Linear Systems
Some Facts About Linear Systems
Accuracy, Condition Numbers and Pivoting
LU Decomposition
Nonlinear Systems - Newton’s Method
Eigenvalues and Eigenvectors
An Application of Eigenvectors: Vibrational Modes
Numerical Methods for Eigenvalues
The QR Method
Iterative solution of linear systems
Functions and Data
Polynomial and Spline Interpolation
Least Squares Fitting: Noisy Data
Integration: Left, Right and Trapezoid Rules
Integration: Midpoint and Simpson’s Rules
Plotting Functions of Two Variables
Double Integrals for Rectangles
Double Integrals for Non-rectangles
Gaussian Quadrature
Numerical Differentiation
The Main Sources of Error
Differential Equations
Reduction of Higher Order Equations to Systems
Euler Methods
Higher Order Methods
Multi-step Methods
ODE Boundary Value Problems and Finite Differences
Finite Difference Method – Nonlinear ODE
Parabolic PDEs - Explicit Method
Solution Instability for the Explicit Method
Implicit Methods
Insulated Boundary Conditions
Finite Difference Method for Elliptic PDEs
Convection-Diffusion Equations
Finite Elements
Determining Internal Node Values