Syllabus
Contents
Syllabus¶
This Applied Math II course based on the book “Advanced Engineering Mathematics”[Oneil17] will help you learn and (hopefully) equip with the basic skill sets in applied math, which are widely used in a variety of scientific research, such as physics, engineering and atmospheric science. There are three main topics in this class including (1) ordinary differential equations (2) Sturm-Liouville Theorem and (3) partial differential equation. Students are expected to learn how to (1) categorize the problem (2) find the solutions (both analytically and numerically) for 1st- and 2nd-order ODEs and even (3) formulate the questions mathematically. We will also walk through some very important examples in atmospheric science, which requires the techniques in this class for solving the problems.
Prerequisites for this class includes Calculus, and Applied Math 1.
Course Outlines¶
Part I: Ordinary Differential Equation
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What is an ODE ? (Different ODE forms)
1st-order ODE
Categories of 1st-order ODE
Separation of variable
Linear ODE
Potential Equation (for implicit ODE solution)
Bernoulli form
Riccati form
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2nd-order ODE
Some basic theorem of 2nd-order ODE
Theorem 1: The Existence of Solutions of an Initial Value Problem
Theorem 2: The Uniqueness of Initial Value Problems and Wronskian Test for Independence
Theorem 3: General and Particular Solutions
Categories of 2nd-order ODE
Homogeneous ODE with constant coefficients (general solution)
Nonhomogeneous ODE (particular solution)
Euler Form
Series Solution: A more general approach for all of the problems above
Week 5 and 6: Laplace Transform
What is a Laplace Transform and how it works? (s space and t or x space)
Initial Value Problem
Heaviside Function and Shifting Theorem
First Shifting Theorem
Second Shifting Theorem
Convolution (Filtering)
Impulse, Dirac Delta Function and Green’s Function approach
Part II: Sturm-Liouville Theorem
Week 7 and 8: Sturm-Liouville Theorem: Eigenfunctions and Fourier Series
Sturm-Liouville Form
Eigenfunction Expansions
Fourier Analysis
Boundary Conditions and Types of Solutions
Power Spectrum, Windows and Gibbs Phenomenon
Week 9 and 10: Special Functions
Special Functions, Series Solutions and Recurrence Relations
Bessel Function
Legendre Polynimial
Hyperbolic Cylinder Function
Part III: Partial Differential Equation
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History and Formula
Forced Solutions
Solutions on a Real String
Heat Diffusion on a 2D Plane
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Problem Setup and Initial Conditions
Wave Solution with Space and Time Structures
Wave Motion in an Unbounded Medium
D’Alembert’s Solutions, Characteristic Lines, and Dispersion Relationship
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Problem Setup and Initial Conditions
Dirichlet Problem for a Rectangle
Forced Solution-Poisson Equation
Green’s Approach in Higher Dimension (Numerical)