Об этом курсе
4.7
Оценки: 6
Рецензии: 1

100% онлайн

Начните сейчас и учитесь по собственному графику.

Гибкие сроки

Назначьте сроки сдачи в соответствии со своим графиком.

Начальный уровень

Knowledge of single variable calculus.

Прибл. 11 часа на выполнение

Предполагаемая нагрузка: 5 hours per week...

Английский

Субтитры: Английский

100% онлайн

Начните сейчас и учитесь по собственному графику.

Гибкие сроки

Назначьте сроки сдачи в соответствии со своим графиком.

Начальный уровень

Knowledge of single variable calculus.

Прибл. 11 часа на выполнение

Предполагаемая нагрузка: 5 hours per week...

Английский

Субтитры: Английский

Программа курса: что вы изучите

Неделя
1
6 ч. на завершение

First-Order Differential Equations

Welcome to the first module! We begin by introducing differential equations and classifying them. We then explain the Euler method for numerically solving a first-order ode. Next, we explain the analytical solution methods for separable and linear first-order odes. An explanation of the theory is followed by illustrative solutions of some simple odes. Finally, we present three real-world examples of first-order odes and their solution: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. ...
12 видео ((всего 97 мин.)), 11 материалов для самостоятельного изучения, 6 тестов
12 видео
Course Overview2мин
Introduction to Differential Equations9мин
Week 1 Introduction47
Euler Method9мин
Separable First-order Equations8мин
Separable First-order Equation: Example6мин
Linear First-order Equations13мин
Linear First-order Equation: Example5мин
Application: Compound Interest13мин
Application: Terminal Velocity11мин
Application: RC Circuit11мин
11 материала для самостоятельного изучения
Welcome and Course Information2мин
Get to Know Your Classmates10мин
Practice: Runge-Kutta Methods10мин
Practice: Separable First-order Equations10мин
Practice: Separable First-order Equation Examples10мин
Practice: Linear First-order Equations5мин
A Change of Variables Can Convert a Nonlinear Equation to a Linear equation10мин
Practice: Linear First-order Equation: Examples10мин
Practice: Compound Interest10мин
Practice: Terminal Velocity10мин
Practice: RC Circuit10мин
6 практического упражнения
Diagnostic Quiz15мин
Classify Differential Equations10мин
Separable First-order ODEs15мин
Linear First-order ODEs15мин
Applications20мин
Week One
Неделя
2
8 ч. на завершение

Second-Order Differential Equations

We begin by generalising the Euler numerical method to a second-order equation. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. We make use of an exponential ansatz, and convert the ode to a second-order polynomial equation called the characteristic equation of the ode. The characteristic equation may have real or complex roots and we discuss the solutions for these different cases. We then consider the inhomogeneous ode, and the phenomena of resonance, where the forcing frequency is equal to the natural frequency of the oscillator. Finally, some interesting and important applications are discussed....
22 видео ((всего 218 мин.)), 20 материалов для самостоятельного изучения, 3 тестов
22 видео
Euler Method for Higher-order ODEs9мин
The Principle of Superposition6мин
The Wronskian8мин
Homogeneous Second-order ODE with Constant Coefficients9мин
Case 1: Distinct Real Roots7мин
Case 2: Complex-Conjugate Roots (Part A)7мин
Case 2: Complex-Conjugate Roots (Part B)8мин
Case 3: Repeated Roots (Part A)12мин
Case 3: Repeated Roots (Part B)4мин
Inhomogeneous Second-order ODE9мин
Inhomogeneous Term: Exponential Function11мин
Inhomogeneous Term: Sine or Cosine (Part A)9мин
Inhomogeneous Term: Sine or Cosine (Part B)8мин
Inhomogeneous Term: Polynomials7мин
Resonance13мин
RLC Circuit11мин
Mass on a Spring9мин
Pendulum12мин
Damped Resonance14мин
Complex Numbers17мин
Nondimensionalization17мин
20 материала для самостоятельного изучения
Practice: Second-order Equation as System of First-order Equations10мин
Practice: Second-order Runge-Kutta Method10мин
Practice: Linear Superposition for Inhomogeneous ODEs10мин
Practice: Wronskian of Exponential Function10мин
Do You Know Complex Numbers?
Practice: Roots of the Characteristic Equation10мин
Practice: Distinct Real Roots10мин
Practice: Hyperbolic Sine and Cosine Functions10мин
Practice: Complex-Conjugate Roots10мин
Practice: Sine and Cosine Functions10мин
Practice: Repeated Roots10мин
Practice: Multiple Inhomogeneous Terms10мин
Practice: Exponential Inhomogeneous Term10мин
Practice: Sine or Cosine Inhomogeneous Term10мин
Practice: Polynomial Inhomogeneous Term10мин
When the Inhomogeneous Term is a Solution of the Homogeneous Equation10мин
Do You Know Dimensional Analysis?
Another Nondimensionalization of the RLC Circuit Equation10мин
Another Nondimensionalization of the Mass on a Spring Equation10мин
Find the Amplitude of Oscillation10мин
3 практического упражнения
Homogeneous Equations20мин
Inhomogeneous Equations20мин
Week Two
Неделя
3
6 ч. на завершение

The Laplace Transform and Series Solution Methods

We present two new analytical solution methods for solving linear odes. The first is the Laplace transform method, which is used to solve the constant-coefficient ode with a discontinuous or impulsive inhomogeneous term. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. We also introduce the solution of a linear ode by series solution. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses. ...
11 видео ((всего 123 мин.)), 10 материалов для самостоятельного изучения, 4 тестов
11 видео
Definition of the Laplace Transform13мин
Laplace Transform of a Constant Coefficient ODE11мин
Solution of an Initial Value Problem13мин
The Heaviside Step Function10мин
The Dirac Delta Function12мин
Solution of a Discontinuous Inhomogeneous Term13мин
Solution of an Impulsive Inhomogeneous Term7мин
The Series Solution Method17мин
Series Solution of the Airy's Equation (Part A)14мин
Series Solution of the Airy's Equation (Part B)7мин
10 материала для самостоятельного изучения
Practice: The Laplace Transform of Sine10мин
Practice: Laplace Transform of an ODE10мин
Practice: Solution of an Initial Value Problem10мин
Practice: Heaviside Step Function10мин
Practice: The Dirac Delta Function15мин
Practice: Discontinuous Inhomogeneous Term20мин
Practice: Impulsive Inhomogeneous Term10мин
Practice: Series Solution Method10мин
Practice: Series Solution of a Nonconstant Coefficient ODE1мин
Practice: Solution of the Airy's Equation10мин
4 практического упражнения
The Laplace Transform Method30мин
Discontinuous and Impulsive Inhomogeneous Terms20мин
Series Solutions20мин
Week Three
Неделя
4
8 ч. на завершение

Systems of Differential Equations and Partial Differential Equations

We solve a coupled system of homogeneous linear first-order differential equations with constant coefficients. This system of odes can be written in matrix form, and we explain how to convert these equations into a standard matrix algebra eigenvalue problem. We then discuss the important application of coupled harmonic oscillators and the calculation of normal modes. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency. Next, to prepare for a discussion of partial differential equations, we define the Fourier series of a function. Then we derive the well-known one-dimensional diffusion equation, which is a partial differential equation for the time-evolution of the concentration of a dye over one spatial dimension. We proceed to solve this equation for a dye diffusing length-wise within a finite pipe. ...
19 видео ((всего 177 мин.)), 17 материалов для самостоятельного изучения, 6 тестов
19 видео
Systems of Homogeneous Linear First-order ODEs8мин
Distinct Real Eigenvalues9мин
Complex-Conjugate Eigenvalues12мин
Coupled Oscillators9мин
Normal Modes (Eigenvalues)10мин
Normal Modes (Eigenvectors)9мин
Fourier Series12мин
Fourier Sine and Cosine Series5мин
Fourier Series: Example11мин
The Diffusion Equation9мин
Solution of the Diffusion Equation: Separation of Variables11мин
Solution of the Diffusion Equation: Eigenvalues10мин
Solution of the Diffusion Equation: Fourier Series9мин
Diffusion Equation: Example10мин
Matrices and Determinants13мин
Eigenvalues and Eigenvectors10мин
Partial Derivatives9мин
Concluding Remarks2мин
17 материала для самостоятельного изучения
Do You Know Matrix Algebra?
Practice: Eigenvalues of a Symmetric Matrix10мин
Practice: Distinct Real Eigenvalues10мин
Practice: Complex-Conjugate Eigenvalues10мин
Practice: Coupled Oscillators10мин
Practice: Normal Modes of Coupled Oscillators10мин
Practice: Fourier Series10мин
Practice: Fourier series at x=010мин
Practice: Fourier Series of a Square Wave10мин
Do You Know Partial Derivatives?10мин
Practice: Nondimensionalization of the Diffusion Equation10мин
Practice: Boundary Conditions with Closed Pipe Ends10мин
Practice: ODE Eigenvalue Problems10мин
Practice: Solution of the Diffusion Equation with Closed Pipe Ends10мин
Practice: Concentration of a Dye in a Pipe with Closed Ends10мин
Please Rate this Course5мин
Acknowledgements
6 практического упражнения
Systems of Differential Equations20мин
Normal Modes30мин
Fourier Series30мин
Separable Partial Differential Equations20мин
The Diffusion Equation20мин
Week Four

Преподаватели

Avatar

Jeffrey R. Chasnov

Professor
Department of Mathematics

О Гонконгский университет науки и технологий

HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world....

Часто задаваемые вопросы

  • Зарегистрировавшись на сертификацию, вы получите доступ ко всем видео, тестам и заданиям по программированию (если они предусмотрены). Задания по взаимной оценке сокурсниками можно сдавать и проверять только после начала сессии. Если вы проходите курс без оплаты, некоторые задания могут быть недоступны.

  • Оплатив сертификацию, вы получите доступ ко всем материалам курса, включая оцениваемые задания. После успешного прохождения курса на странице ваших достижений появится электронный сертификат. Оттуда его можно распечатать или прикрепить к профилю LinkedIn. Просто ознакомиться с содержанием курса можно бесплатно.

Остались вопросы? Посетите Центр поддержки учащихся.