This course is part of the Mathematics for Engineers Specialization

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Jeffrey R. Chasnov

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Mathematics for Engineers SpecializationThe Hong Kong University of Science and Technology

About this Course

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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 to be completed.  

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

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Course 4 of 5 in the

Mathematics for Engineers Specialization

Intermediate Level

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

Approx. 41 hours to complete

English

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What you will learn

MATLAB and Scientific Computing
Root Finding and Numerical Matrix Algebra
Quadrature and Interpolation
Numerical Solution of Ordinary and Partial Differential Equations

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Coursera Labs

Includes hands on learning projects.

Learn more about Coursera Labs

Course 4 of 5 in the

Mathematics for Engineers Specialization

Intermediate Level

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

Approx. 41 hours to complete

English

Could your company benefit from training employees on in-demand skills?

Try Coursera for Business

Instructor

Instructor rating

4.88/5 (104 Ratings)

Jeffrey R. Chasnov
Top Instructor

Professor

Department of Mathematics

172,602 Learners

16 Courses

Offered by

The Hong Kong University of Science and Technology

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Syllabus - What you will learn from this course

Content Rating97%(1,514 ratings)

Week1

Week 1

9 hours to complete

Scientific Computing

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.

9 hours to complete

14 videos (Total 104 min), 16 readings, 13 quizzes

14 videos

16 readings

Welcome and Course Information2mCertificate or Audit?1mHow to Write Math in the Discussions Using MathJax1mMATLAB Online5mRounding Binary Numbers10mComputer Numbers10mREALMAX10mREALMIN10mEPS10mLogical Expressions5mLogical Vectors10mQuadratic Equation10mBackground for the Logistic Map20mPeriod-230mReference Solution to "Bifurcation Diagram for the Logistic Map (audit)"1mReference Solution to "Bifurcation Diagram for the Logistic Map"1m

3 practice exercises

Diagnostic Quiz30mWeek One Assessment (audit)Week One Assessment45m

Week2

Week 2

6 hours to complete

Root Finding

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.

6 hours to complete

12 videos (Total 130 min), 9 readings, 6 quizzes

12 videos

9 readings

Estimate the Square-root of Three Using the Bisection Method5mEstimate the Square-root of Three Using Newton's Method5mEstimate the Square-Root of Three Using the Secant Method5mRates of Convergence5mOrder of Convergence of the Secant Method30mThe Four Fourth Roots of Unity5mCompute the Value of m in the Period-Two Cycle1mReference Solution to "Computation of the Feigenbaum Delta (audit)"1mReference Solution to "Computation of the Feigenbaum Delta"1m

2 practice exercises

Week Two Assessment (audit)Week Two Assessment45m

Week3

Week 3

5 hours to complete

Matrix Algebra

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.

5 hours to complete

13 videos (Total 115 min), 11 readings, 7 quizzes

13 videos

Week Three Introduction 1mGaussian Elimination without Pivoting | Lecture 2411mGaussian Elimination with Partial Pivoting | Lecture 255mLU Decomposition with Partial Pivoting | Lecture 2610mOperation Counts | Lecture 279mOperation Counts for Gaussian Elimination | Lecture 288mOperation Counts for Forward and Backward Substitution | Lecture 296mEigenvalue Power Method | Lecture 3011mEigenvalue Power Method (Example) |Lecture 317mMatrix Algebra in MATLAB | Lecture 3212mSystems of Nonlinear Equations | Lecture 3310mSystems of Nonlinear Equations (Example) | Lecture 349mFractals from the Lorenz Equations | Lecture 359m

11 readings

Round-off Errors in Gaussian Elimination10mReduced Round-off Errors in Gaussian Elimination with Partial Pivoting5mThe (PL)U Decomposition of A10mEstimating Computational Time using Operation Counts5mSummation Identities10mOperation Counts for a Lower Triangular System10mConvergence of the Eigenvalue Power Method5mDetermine the Dominant Eigenvalue10mHow to Solve Three Nonlinear equations5mReference Solution to "Fractals from the Lorenz Equations (audit)"1mReference Solution to "Fractals from the Lorenz Equations"1m

2 practice exercises

Week Three Assessment (audit)Week Three Assessment45m

Week4

Week 4

7 hours to complete

Quadrature and Interpolation

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.

7 hours to complete

13 videos (Total 110 min), 12 readings, 6 quizzes

13 videos

12 readings

The Midpoint Rule is the Area of a Rectangle5mMidpoint Rule for a Quadratic Function10mDerive the Trapezoidal Rule10mDerive Simpson's Rule15mSimpson's 3/8 Rule10mThree-point Legendre-Gauss Quadrature10mComputing the Error in an Adaptive Quadrature10mLinear and Quadratic Interpolation10mCubic Spline Interpolation with Endpoint Slopes Known10mCubic Spline Interpolation with the Not-a-Knot Condition15mReference Solution to "Bessel Function Zeros (audit)"1mReference Solution to "Bessel Function Zeros"1m

2 practice exercises

Week Four Assessment (audit)Week Four Assessment45m

Week5

Week 5

6 hours to complete

Ordinary Differential Equations

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.

6 hours to complete

13 videos (Total 125 min), 10 readings, 6 quizzes

13 videos

10 readings

When the Euler Method is Exact10mWhen the Modified Euler Method is Exact10mRalston's Method5mRunge-Kutta Methods and Quadrature Formulas10mFourth-Order Runge-Kutta Method and Simpson's Rule10mSystems of ODEs10mExample of Adaptive Integration10mCircular orbits10mReference Solution to "Two-Body Problem (audit)"1mReference Solution to "Two-Body Problem"1m

2 practice exercises

Week Five Assessment (audit)Week Five Assessment45m

Week6

Week 6

8 hours to complete

Partial Differential Equations

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.

8 hours to complete

17 videos (Total 170 min), 16 readings, 8 quizzes

17 videos

Week Six Introduction 2mBoundary and Initial Value Problems | Lecture 604mCentral Difference Approximation | Lecture 618mDiscrete Laplace Equation | Lecture 629mNatural Ordering | Lecture 638mMatrix Formulation | Lecture 6412mMATLAB Solution of the Laplace Equation (Direct Method) | Lecture 6517mJacobi, Gauss-Seidel and SOR Methods | Lecture 6612mRed-Black Ordering | Lecture 673mMATLAB Solution of the Laplace Equation (Iterative Method) | Lecture 6812mExplicit Methods for Solving the Diffusion Equation | Lecture 6913mVon Neumann Stability Analysis of the FTCS Scheme | Lecture 7014mImplicit Methods for Solving the Diffusion Equation | Lecture 718mCrank-Nicolson Method for the Diffusion Equation | Lecture 7213mMATLAB Solution of the Diffusion Equation | Lecture 7311mTwo-Dimensional Diffusion Equation | Lecture 7412mConcluding Remarks 3m

16 readings

Higher-order Central Difference Approximation10mMean Value Property of the Laplace Equation10mCoordinates of the four corners5mThe Discrete Laplace Equation on a Four-by-Four Grid10mNumber of Interior and Boundary Points10mIterative Solution of a System of Linear Equations10mUsing a Second-Order Time-Stepping Method10mFTCS Scheme for the Advection Equation10mVon Neumann Stability Analysis of the FTCS Scheme for the Advection Equation10mImplicit Discrete Advection Equation10mLax Scheme for the Advection Equation10mDifference Approximations for the Derivative at Boundary Points1mReference Solution to "Two-Dimensional Diffusion Equation (audit)"1mReference Solution to "Two-Dimensional Diffusion Equation"1mPlease Rate this CourseAcknowledgements

3 practice exercises

Classify Partial Differential Equations10mWeek Six Assessment (audit)Week Six Assessment45m

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4.9

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TOP REVIEWS FROM NUMERICAL METHODS FOR ENGINEERS

by EKApr 20, 2021

Very nice introduction to numerical methods.

The MATLAB assignments were challenging but worth the effort.

Many thanks to Prof Chasnov.

by ACJul 12, 2021

An excellent course on Numerical methods. The interactive nature of the course, using MATLAB programming and exercises, made it a lot of fun to learn the subject.

by ILApr 22, 2022

Excellent course material, organization, and presentation. Also a very useful course for scientists and engineers.

by SSNov 29, 2022

It is a great course and help me to have a better understanding of the numerical method!

View all reviews

About the Mathematics for Engineers Specialization

This specialization was developed for engineering students to self-study engineering mathematics. We expect students to already be familiar with single variable calculus and computer programming. Through this specialization, students will learn matrix algebra, differential equations, vector calculus, numerical methods, and MATLAB programming. This will provide them with the tools to effectively apply mathematics to engineering problems and be well-equipped to pursue a degree in engineering. To get a better understanding of what this specialization has to offer, be sure to watch the Promotional Video!

Frequently Asked Questions

When will I have access to the lectures and assignments?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I subscribe to this Specialization?
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
Is financial aid available?
Yes. In select learning programs, you can apply for financial aid or a scholarship if you can’t afford the enrollment fee. If fin aid or scholarship is available for your learning program selection, you’ll find a link to apply on the description page.

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Numerical Methods for Engineers

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