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The Hong Kong University of Science and Technology
Numerical Methods for Engineers
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The Hong Kong University of Science and Technology

Numerical Methods for Engineers

This course is part of Mathematics for Engineers Specialization

Jeffrey R. Chasnov

Instructor: Jeffrey R. Chasnov

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27,416 already enrolled

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6 modules
Gain insight into a topic and learn the fundamentals.
4.9

(392 reviews)

Intermediate level

Recommended experience

Recommended experience

Intermediate level

Knowledge of calculus, matrix algebra, differential equations and a computer programming language

Flexible schedule
Approx. 40 hours
Learn at your own pace
92%
Most learners liked this course

6 modules
Gain insight into a topic and learn the fundamentals.
4.9

(392 reviews)

Intermediate level

Recommended experience

Recommended experience

Intermediate level

Knowledge of calculus, matrix algebra, differential equations and a computer programming language

Flexible schedule
Approx. 40 hours
Learn at your own pace
92%
Most learners liked this course
  • About
  • Outcomes
  • Modules
  • Recommendations
  • Testimonials
  • Reviews

What you'll learn

  • MATLAB and the foundations of scientific computing

  • Root finding methods such as Newton's method, and numerical linear algebra using the LU decomposition

  • Integration methods such as adaptive quadrature, and interpolation algorithms using a cubic spline

  • Numerical methods for solving ODEs, such as Runge-Kutta, and the finite difference method for solving PDEs

Skills you'll gain

  • Matlab
  • Programming Principles
  • Simulation and Simulation Software
  • Mathematical Modeling
  • Engineering Calculations
  • Applied Mathematics
  • Calculus
  • Integral Calculus
  • Scripting
  • Estimation
  • Engineering Analysis
  • Linear Algebra
  • Scientific Visualization
  • Computational Thinking
  • Simulations
  • Differential Equations
  • Algorithms
  • Mathematical Software
  • Plot (Graphics)
  • Numerical Analysis

Details to know

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Assessments

8 assignments

Taught in English

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This course is part of the Mathematics for Engineers Specialization
When you enroll in this course, you'll also be enrolled in this Specialization.
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There are 6 modules in this course

This course covers the most important numerical methods that an engineer should know, including root finding, matrix algebra, integration and interpolation, ordinary and partial differential equations. We learn how to use MATLAB to solve numerical problems, and access to MATLAB online and the MATLAB grader is given to all students who enroll.

We assume students are already familiar with the basics of matrix algebra, differential equations, and vector calculus. They should have a working knowledge of a programming language, and be willing to learn MATLAB. The course contains 74 short lecture videos and MATLAB demonstrations. After each lecture or demonstration, there are problems to solve or programs to write. The course is organized into six weeks, and at the end of each week, there is an assessed quiz and a longer programming project. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/numerical-methods-for-engineers.pdf And watch the promotional video from the link https://youtu.be/qFJGMBDfFMY

MATLAB is a high-level programming language extensively utilized by engineers for numerical computation and visualization. We will learn the basics of MATLAB: how real numbers are represented in double precision; how to perform arithmetic with MATLAB; how to use scripts and functions; how to represent vectors and matrices; how to draw line plots; and how to use logical variables, conditional statements, for loops and while loops. For your programming project, you will write a MATLAB code to compute the bifurcation diagram for the logistic map.

What's included

14 videos14 readings2 assignments9 app items

14 videos•Total 104 minutes
  • Course Overview•3 minutes•Preview module
  • Week One Introduction•2 minutes
  • Binary Numbers | Lecture 1•11 minutes
  • Double Precision | Lecture 2•13 minutes
  • MATLAB as a Calculator | Lecture 3•5 minutes
  • Scripts and Functions | Lecture 4•7 minutes
  • Vectors | Lecture 5•5 minutes
  • Line Plots | Lecture 6•7 minutes
  • Matrices | Lecture 7•8 minutes
  • Logicals | Lecture 8•4 minutes
  • Conditionals | Lecture 9•3 minutes
  • Loops | Lecture 10•5 minutes
  • Logistic Map (Part A) | Lecture 11•16 minutes
  • Logistic Map (Part B) | Lecture 12•8 minutes
14 readings•Total 133 minutes
  • Welcome and Course Information•1 minute
  • How to Write Math in the Discussions Using MathJax•1 minute
  • MATLAB Online•5 minutes
  • Rounding Binary Numbers•10 minutes
  • Computer Numbers•10 minutes
  • REALMAX•10 minutes
  • REALMIN•10 minutes
  • EPS•10 minutes
  • Logical Expressions•5 minutes
  • Logical Vectors•10 minutes
  • Quadratic Equation•10 minutes
  • Background for the Logistic Map•20 minutes
  • Period-2•30 minutes
  • Reference Solution to "Bifurcation Diagram for the Logistic Map"•1 minute
2 assignments•Total 75 minutes
  • Diagnostic Quiz•30 minutes
  • Week One Assessment•45 minutes
9 app items•Total 225 minutes
  • MATLAB as a Calculator•15 minutes
  • Binet's Formula for the Fibonacci Numbers•15 minutes
  • Table of Sines and Cosines•15 minutes
  • Logarithmic Spiral•15 minutes
  • Lemniscate•15 minutes
  • Manipulating Matrices•15 minutes
  • Banded Matrices•15 minutes
  • Recursion Definition for the Fibonacci Numbers•60 minutes
  • Bifurcation Diagram for the Logistic Map•60 minutes

Root finding is a numerical technique used to determine the roots, or zeros, of a given function. We will explore several root-finding methods, including the Bisection method, Newton's method, and the Secant method. We will also derive the order of convergence for these methods. Additionally, we will demonstrate how to compute the Newton fractal using Newton's method in MATLAB, and discuss MATLAB functions that can be used to find roots. For your programming project, you will write a MATLAB code using Newton's method to compute the Feigenbaum delta from the bifurcation diagram for the logistic map.

What's included

12 videos8 readings1 assignment3 app items1 plugin

12 videos•Total 129 minutes
  • Week Two Introduction •1 minute•Preview module
  • Bisection Method | Lecture 13•9 minutes
  • Newton's Method | Lecture 14•10 minutes
  • Secant Method | Lecture 15•9 minutes
  • Order of Convergence| Lecture 16•5 minutes
  • Convergence of Newton's Method | Lecture 17•11 minutes
  • Fractals from Newton's Method | Lecture 18•8 minutes
  • Coding the Newton Fractal | Lecture 19 •21 minutes
  • Root-Finding in MATLAB | Lecture 20•9 minutes
  • Feigenbaum Delta (Part A) | Lecture 21•16 minutes
  • Feigenbaum Delta (Part B) | Lecture 22•17 minutes
  • Feigenbaum Delta (Part C) | Lecture 23•8 minutes
8 readings•Total 57 minutes
  • Estimate the Square-root of Three Using the Bisection Method•5 minutes
  • Estimate the Square-root of Three Using Newton's Method•5 minutes
  • Estimate the Square-Root of Three Using the Secant Method•5 minutes
  • Rates of Convergence•5 minutes
  • Order of Convergence of the Secant Method•30 minutes
  • The Four Fourth Roots of Unity•5 minutes
  • Compute the Value of m in the Period-Two Cycle•1 minute
  • Reference Solution to "Computation of the Feigenbaum Delta"•1 minute
1 assignment•Total 45 minutes
  • Week Two Assessment•45 minutes
3 app items•Total 95 minutes
  • Fractals from the Four Fourth Roots of Unity•15 minutes
  • Elliptical Planetary Orbits•20 minutes
  • Computation of the Feigenbaum Delta•60 minutes
1 plugin•Total 23 minutes
  • Deep Dive into The Newton Fractal•23 minutes

Numerical linear algebra is the term used for matrix algebra performed on a computer. When conducting Gaussian elimination with large matrices, round-off errors may compromise the computation. These errors can be mitigated using the method of partial pivoting, which involves row interchanges before each elimination step. The LU decomposition algorithm must then incorporate permutation matrices. We will also discuss operation counts and the big-Oh notation for predicting the increase in computational time with larger problem sizes. We will show how to count the number of required operations for Gaussian elimination, forward substitution, and backward substitution. We will explain the power method for computing the largest eigenvalue of a matrix. Finally, we will show how to use Gaussian elimination to solve a system of nonlinear differential equations using Newton's method. For your programming project, you will write a MATLAB code that applies Newton's method to the Lorenz equations.

What's included

13 videos10 readings1 assignment4 app items

13 videos•Total 114 minutes
  • Week Three Introduction •1 minute•Preview module
  • Gaussian Elimination without Pivoting | Lecture 24•11 minutes
  • Gaussian Elimination with Partial Pivoting | Lecture 25•5 minutes
  • LU Decomposition with Partial Pivoting | Lecture 26•10 minutes
  • Operation Counts | Lecture 27•9 minutes
  • Operation Counts for Gaussian Elimination | Lecture 28•8 minutes
  • Operation Counts for Forward and Backward Substitution | Lecture 29•6 minutes
  • Eigenvalue Power Method | Lecture 30•11 minutes
  • Eigenvalue Power Method (Example) |Lecture 31•7 minutes
  • Matrix Algebra in MATLAB | Lecture 32•12 minutes
  • Systems of Nonlinear Equations | Lecture 33•10 minutes
  • Systems of Nonlinear Equations (Example) | Lecture 34•9 minutes
  • Fractals from the Lorenz Equations | Lecture 35•9 minutes
10 readings•Total 71 minutes
  • Round-off Errors in Gaussian Elimination•10 minutes
  • Reduced Round-off Errors in Gaussian Elimination with Partial Pivoting•5 minutes
  • The (PL)U Decomposition of A•10 minutes
  • Estimating Computational Time using Operation Counts•5 minutes
  • Summation Identities•10 minutes
  • Operation Counts for a Lower Triangular System•10 minutes
  • Convergence of the Eigenvalue Power Method•5 minutes
  • Determine the Dominant Eigenvalue•10 minutes
  • How to Solve Three Nonlinear equations•5 minutes
  • Reference Solution to "Fractals from the Lorenz Equations"•1 minute
1 assignment•Total 45 minutes
  • Week Three Assessment•45 minutes
4 app items•Total 90 minutes
  • The LU Decomposition of a Matrix•10 minutes
  • Eigenvalues and Eigenvectors•10 minutes
  • Fixed-Point Solutions of the Lorenz Equations•10 minutes
  • Fractals from the Lorenz Equations•60 minutes

The computation of definite integrals is known as quadrature. We will explore the fundamentals of quadrature, including elementary formulas for the Trapezoidal rule and Simpson’s rule; development of composite integration rules; an introduction to Gaussian quadrature; construction of an adaptive quadrature routine where the software determines the appropriate integration step size; and the usage of the MATLAB function integral.m. Additionally, we will learn about interpolation. A good interpolation routine can estimate function values at intermediate sample points. We will learn about linear interpolation, commonly employed for plotting data with numerous points; and cubic spline interpolation, used when data points are sparse. For your programming project, you will write a MATLAB code to compute the zeros of a Bessel function. This task requires the combination of both quadrature and root-finding routines.

What's included

13 videos11 readings1 assignment3 app items

13 videos•Total 110 minutes
  • Week Four Introduction •1 minute•Preview module
  • Midpoint Rule | Lecture 36•8 minutes
  • Trapezoidal Rule | Lecture 37•8 minutes
  • Simpson's Rule | Lecture 38•6 minutes
  • Composite Quadrature Rules | Lecture 39•12 minutes
  • Gaussian Quadrature | Lecture 40•8 minutes
  • Adaptive Quadrature | Lecture 41•11 minutes
  • Quadrature in MATLAB | Lecture 42•3 minutes
  • Interpolation | Lecture 43•10 minutes
  • Cubic Spline Interpolation (Part A) | Lecture 44•15 minutes
  • Cubic Spline Interpolation (Part B) | Lecture 45•10 minutes
  • Interpolation in MATLAB | Lecture 46•5 minutes
  • Bessel Functions and their Zeros | Lecture 47•6 minutes
11 readings•Total 106 minutes
  • The Midpoint Rule is the Area of a Rectangle•5 minutes
  • Midpoint Rule for a Quadratic Function•10 minutes
  • Derive the Trapezoidal Rule•10 minutes
  • Derive Simpson's Rule•15 minutes
  • Simpson's 3/8 Rule•10 minutes
  • Three-point Legendre-Gauss Quadrature•10 minutes
  • Computing the Error in an Adaptive Quadrature•10 minutes
  • Linear and Quadratic Interpolation•10 minutes
  • Cubic Spline Interpolation with Endpoint Slopes Known•10 minutes
  • Cubic Spline Interpolation with the Not-a-Knot Condition•15 minutes
  • Reference Solution to "Bessel Function Zeros"•1 minute
1 assignment•Total 45 minutes
  • Week Four Assessment•45 minutes
3 app items•Total 105 minutes
  • Cornu Spiral•30 minutes
  • Interpolate Two Data Files•15 minutes
  • Bessel Function Zeros•60 minutes

We will learn about the numerical integration of ordinary differential equations (ODEs). We will introduce the Euler method, a single-step, first-order method, and the Runge-Kutta methods, which extend the Euler method to multiple steps and higher order, allowing for larger time steps. We will show how to construct a family of second-order Runge-Kutta methods, discuss the widely-used fourth-order Runge-Kutta method, and adopt these methods for solving systems of ODEs. We will show how to use the MATLAB function ode45.m, and how to solve a two-point boundary value ODE using the shooting method. For your programming project, you will conduct a numerical simulation of the gravitational two-body problem.

What's included

13 videos9 readings1 assignment3 app items

13 videos•Total 124 minutes
  • Week Five Introduction •1 minute•Preview module
  • Euler Method | Lecture 48•7 minutes
  • Modified Euler Method | Lecture 49•9 minutes
  • Runge-Kutta Methods | Lecture 50•12 minutes
  • Second-Order Runge-Kutta Methods | Lecture 51•7 minutes
  • Higher-Order Runge-Kutta Methods | Lecture 52•10 minutes
  • Higher-Order ODEs and Systems | Lecture 53•7 minutes
  • Adaptive Runge-Kutta Method | Lecture 54•13 minutes
  • Integrating ODEs in MATLAB (Part A) | Lecture 55•15 minutes
  • Integrating ODEs in MATLAB (Part B) | Lecture 56•7 minutes
  • Shooting Method for Boundary Value Problems | Lecture 57•11 minutes
  • The Two-Body Problem (Part A) | Lecture 58•9 minutes
  • The Two-Body Problem (Part B) | Lecture 59•10 minutes
9 readings•Total 76 minutes
  • When the Euler Method is Exact•10 minutes
  • When the Modified Euler Method is Exact•10 minutes
  • Ralston's Method•5 minutes
  • Runge-Kutta Methods and Quadrature Formulas•10 minutes
  • Fourth-Order Runge-Kutta Method and Simpson's Rule•10 minutes
  • Systems of ODEs•10 minutes
  • Example of Adaptive Integration•10 minutes
  • Circular orbits•10 minutes
  • Reference Solution to "Two-Body Problem"•1 minute
1 assignment•Total 45 minutes
  • Week Five Assessment•45 minutes
3 app items•Total 120 minutes
  • The Lorenz Equations•30 minutes
  • Swing a Pendulum to the Top•30 minutes
  • Two-Body Problem•60 minutes

We will learn how to solve partial differential equations (PDEs). While this is a vast topic with various specialized solution methods, such as those found in computational fluid dynamics, we will provide a basic introduction to the subject. We will categorize PDE solutions into boundary value problems and initial value problems. We will then apply the finite difference method for solving PDEs. We will solve the Laplace equation, a boundary value problem, using two methods: a direct method via Gaussian elimination; and an iterative method, where the solution is approached asymptotically. We will next solve the one-dimensional diffusion equation, an initial value problem, using the Crank-Nicolson method. We will also employ the Von Neumann stability analysis to determine the stability of time-integration schemes. For your programming project, you will solve the two-dimensional diffusion equation using the Crank-Nicolson method.

What's included

17 videos15 readings2 assignments4 app items

17 videos•Total 170 minutes
  • Week Six Introduction •2 minutes•Preview module
  • Boundary and Initial Value Problems | Lecture 60•4 minutes
  • Central Difference Approximation | Lecture 61•8 minutes
  • Discrete Laplace Equation | Lecture 62•9 minutes
  • Natural Ordering | Lecture 63•8 minutes
  • Matrix Formulation | Lecture 64•12 minutes
  • MATLAB Solution of the Laplace Equation (Direct Method) | Lecture 65•17 minutes
  • Jacobi, Gauss-Seidel and SOR Methods | Lecture 66•12 minutes
  • Red-Black Ordering | Lecture 67•3 minutes
  • MATLAB Solution of the Laplace Equation (Iterative Method) | Lecture 68•12 minutes
  • Explicit Methods for Solving the Diffusion Equation | Lecture 69•13 minutes
  • Von Neumann Stability Analysis of the FTCS Scheme | Lecture 70•14 minutes
  • Implicit Methods for Solving the Diffusion Equation | Lecture 71•8 minutes
  • Crank-Nicolson Method for the Diffusion Equation | Lecture 72•13 minutes
  • MATLAB Solution of the Diffusion Equation | Lecture 73•11 minutes
  • Two-Dimensional Diffusion Equation | Lecture 74•12 minutes
  • Concluding Remarks •3 minutes
15 readings•Total 107 minutes
  • Higher-order Central Difference Approximation•10 minutes
  • Mean Value Property of the Laplace Equation•10 minutes
  • Coordinates of the four corners•5 minutes
  • The Discrete Laplace Equation on a Four-by-Four Grid•10 minutes
  • Number of Interior and Boundary Points•10 minutes
  • Iterative Solution of a System of Linear Equations•10 minutes
  • Using a Second-Order Time-Stepping Method•10 minutes
  • FTCS Scheme for the Advection Equation•10 minutes
  • Von Neumann Stability Analysis of the FTCS Scheme for the Advection Equation•10 minutes
  • Implicit Discrete Advection Equation•10 minutes
  • Lax Scheme for the Advection Equation•10 minutes
  • Difference Approximations for the Derivative at Boundary Points•1 minute
  • Reference Solution to "Two-Dimensional Diffusion Equation"•1 minute
  • Please Rate this Course•0 minutes
  • Acknowledgements•0 minutes
2 assignments•Total 55 minutes
  • Classify Partial Differential Equations•10 minutes
  • Week Six Assessment•45 minutes
4 app items•Total 150 minutes
  • Direct Solution of the Laplace Equation•30 minutes
  • Iterative Solution of the Laplace Equation•30 minutes
  • The Diffusion Equation with No-Flux Boundary Conditions•30 minutes
  • Two-Dimensional Diffusion Equation•60 minutes

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4.9 (151 ratings)
Jeffrey R. Chasnov

Top Instructor

Jeffrey R. Chasnov
The Hong Kong University of Science and Technology
17 Courses•233,907 learners

Offered by

The Hong Kong University of Science and Technology

Offered by

The Hong Kong University of Science and Technology

HKUST is a world-class research-intensive university that focuses on science, technology, and business as well as humanities and social science. HKUST offers an international campus, and a holistic and interdisciplinary pedagogy to nurture well-rounded graduates with a global vision, a strong entrepreneurial spirit, and innovative thinking.

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4.9

392 reviews

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IU
5

Reviewed on Jan 3, 2023

very Fantastic core course for all engineering and science students to take. Many thanks again to Prof. Jeffrey Chasnov and everyone for making this happen. God bless you.  

M
MU
5

Reviewed on Aug 23, 2021

It's really a privilege for me to be a part of this course. I was able to learn a lot. Thanks Professor for this amazing course.

V
VT
5

Reviewed on Oct 22, 2023

An excellent course on numerical methods with detailed, crystal clear derivations using Taylor's theorem for each numerical method.

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