###### Brief Information

- Name : Partial Differential Equation 편미분방정식론
- Lecturer : Jong-woo Lee 이종우
- Semester : 2016 Fall
- Major : BS, Mathematics
- Textbook
- Syllabus : Syllabus_2016-5-2__Partial Differential Equation.pdf
- In short
- Study partial differential equations.
- Linear first-order PDEs,

### Summary of Lectures

#### 1. First-Order Equations

#### 2. Linear Second-Order Equations

#### 3. Elements of Fourier Analysis

##### 3.1. Why Fourier Series?

- Almost any function could be expanded as a Fourier series.
- Fourier series initiated a century of intense investigation of trigonometric series.

##### 3.2. The Fourier Series of a Function

- The
**Fourier series**of a function on- where , , and
- for

##### 3.3. Convergence of Fourier Series

- The
**periodic extension**of a function to the entire real line- The periodic extension is simply a device to reconcile the fact that we want to represent a function g(x) as a Fourier series.
- g(x) is a function defined on

- Theorem (
**Convergence of Fourier Series**,**Dirichlet’s theorem**)- The following conditions are called the ‘
**Dirichlet’s conditions**.’ –*Dirichlet conditions, Wikipedia* - For a real-valued, periodic, continuous function f,
- f must be absolutely integrable over a period.
- f must have a finite number of extrema in any given bounded interval, i.e. there must be a finite number of maxima and minima in the interval.
- f must have a finite number of discontinuities in any given bounded interval, however the discontinuity cannot be infinite.

- Then, its Fourier series at each point x is equal to the limt

- The following conditions are called the ‘
**Change of scale**from to- The
**Fourier series**of a function on- where
- ,
- , and
- for

- where

##### 3.4. Sine and Cosine Expansions

**Change of scale**from to- Even functions
- Odd functions
- Half-range Fourier expansions
- The
**Fourier sine series**of a function defined on- Assume is an odd function. Then, becomes 0.
- where
- for

- The
**Fourier cosine series**of a function defined on- Assume is an even function. Then, becomes 0.
- where
- for

- The

##### 3.5. The Fourier Integral

- Change of scale from to
- Preconditions of the Fourier integral representation
- is piecewise smooth on the interval for every
- converges. Say, is absolutely integrable.

- The
**Fourier integral representation**of a function f on- where
- , and

- where
- The Fourier
**sine**integral representation of a function f on- where

- where
- The Fourier
**cosine**integral representation of a function f on- where

- where

##### 3.6. The Fourier Transform

- The
**Fourier transform**of a function - The Fourier transform is linear.
- The Fourier transform is a function from a function into another function.

##### 3.7. Convolution

- The
**convolution**of with - Convolution is an operator between two functions.
- Let and be real- or complex- valued functions defined on the real line.
- has a convolution with if
- and exist for every interval
- converges for every real number .

##### 3.8. Fourier Sine and Cosine Transforms

- Fourier sine transform
- Fourier cosine transform

#### 4. Sturm-Liouville Problems

- The Sturm-Liouville problems
- regular Sturm-Liouville problems
- periodic Sturm-Liouville problems
- singular Sturm-Liouville problems

##### 4.1. Regular Sturm-Liouville Problems

- Two function and defined on are
**orthogonal**with weight on if - An
**eigenvalue problem**is an equation of the form- where is a real parameter and is a given function such that for all
- The numbers for which the equation has nonzero solution are called
**eigenvalues**. The corresponding nonzero solutions are called**eigenfunctions**.

- A
**regular Sturm-Liouville (S-L) eigenvalue problem**is an equation of the form- with the boundary conditions (BCs)

,

, - together with the following conditions.
- is continuous, differentiable, and .
- is continuous.
- is continuous, and .
- . and are not zero at the same time.
- . and are not zero at the same time.

- with the boundary conditions (BCs)
**How to transform a second-order ODE to a regular Sturm-Liouville problem**- A second-order ODE:
- Transform this equation by multiplying each term.
- Then, we get , which is the form of a regular Sturm-Liouville problem.

**How to solve a regular Sturm-Liouville problem**- Generally, a second-order ODE is given.
- Its characteristic equation is .
- Then, and
- Then, there are three cases when .
- If , then and . Then,
- If , then and . Then,
- If , then and . So, such that . Then,

- If is not a zero function, then is an eigenvalue function and there exists the corresponding eigenvalue.

- The
**generalized Fourier series expansion**for in the eigenfunctions- where
- If has a unique generalized Fourier series expansion, then is called
**complete**.

##### 4.2. Other Problems

- periodic Sturm-Liouville problems
- singular Sturm-Liouville problems

#### 5. The Method of Separation of Variables

##### 5.1. The Heat Equation

- Rod with zero temperature at the endpoints.

- The Fourier sine series

- Rod with insulated endpoints.

- The Fourier cosine series

- Rod with mixed homogeneous boundary conditions.

- The generalized Fourier series

- Rod with an endpoint with a zero-temperature medium

- Heat conduction in a thin uniform circular ring.

##### 5.2. The Wave Equation

##### 5.3. The Laplace Equation

##### 5.4. Other Equations

- The diffusion-convection IBVP

- The dissipative wave propagation problem

- The two-dimensional steady state diffusion-convection problem

#### 6. Linear Homogeneous Problems

#### 7. The Method of Eigenvalue Expansion

### Summary of Concepts

##### PDEs to Solve

- Linear first-order PDEs
- Separable PDEs – by separation of variables
- Non-separable PDEs – by transition

- Quasilinear first-order PDEs
- Linear second-order PDEs
- The wave equation
- The heat equation
- The Laplace equation

#### How to Solve the PDEs

- The method of separation
- The method of characteristics