Partial Differential Equation | MA

Brief Information

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 $latex [-\pi , \pi]$
    • $latex \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(nx) + b_n sin(nx)}$
    • where?$latex a_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)cos(nx)}dx$,?$latex a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)}dx$, and
    • $latex b_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)sin(nx)}dx$ for?$latex n=1,2,3,…$
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 $latex (-\pi , \pi]$
  • 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,
      1. f must be absolutely integrable over a period.
      2. 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.
      3. 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 $latex \frac{1}{2}(f(x+)+f(x-))=\frac{1}{2} (\lim_{x \rightarrow x+}f(x)+\lim_{x \rightarrow x-}f(x))$
  • Change of scale from?$latex (-\pi , \pi]$ to?$latex (-L , L]$
  • The Fourier series of a function on $latex [-L , L]$
    • $latex \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(n\pi x/L) + b_n sin(n\pi x/L)}$?where
      • $latex a_n = \frac{1}{L} \int_{-L}^{L}{f(x)cos(n\pi x/L)}dx$,
      • $latex a_0 = \frac{1}{L} \int_{-L}^{L}{f(x)}dx$, and
      • $latex b_n = \frac{1}{L} \int_{-L}^{L}{f(x)sin(n\pi x/L)}dx$ for?$latex n=1,2,3,…$
3.4. Sine and Cosine Expansions
  • Change of scale from $latex (-L , L]$ to $latex [0 , L]$
  • Even functions
  • Odd functions
  • Half-range Fourier expansions
    • The Fourier sine series?of a function $latex f$ defined on?$latex [0 , L]$
      • Assume $latex f$ is an odd function. Then,?$latex a_n$ becomes 0.
      • $latex \sum_{n=1}^{\infty}{b_n sin(n\pi x/L)}$?where
        • $latex b_n = \frac{2}{L} \int_{0}^{L}{f(x)sin(n\pi x/L)}dx$ for $latex n=1,2,3,…$
    • The Fourier cosine series?of a function $latex f$ defined on?$latex [0 , L]$
      • Assume $latex f$ is an even function.?Then,?$latex b_n$ becomes 0.
      • $latex \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(n\pi x/L)}$?where
        • $latex a_n = \frac{2}{L} \int_{0}^{L}{f(x)cos(n\pi x/L)}dx$?for $latex n=0,1,2,3,…$
3.5. The Fourier Integral
  • Change of scale from $latex (-L , L]$ to $latex (-\infty ,\infty )$
  • Preconditions of the Fourier integral representation
    • $latex f$ is piecewise smooth on the interval $latex [-L , L]$ for every $latex L\in \mathbb{R^+}$
    • $latex \int_{-\infty}^{\infty}|f(x)|dx$ converges. Say, $latex f$ is absolutely integrable.
  • The Fourier integral representation of a function f on $latex (-\infty ,\infty )$
    • $latex \int_{0}^{\infty}[A_\omega cos(\omega x) + B_\omega sin(\omega x)]dx$ where
      • $latex A_\omega = \frac{1}{\pi} \int_{-\infty}^{\infty}{f(\xi)cos(\omega \xi)}d\xi$, and
      • $latex B_\omega = \frac{1}{\pi} \int_{-\infty}^{\infty}{f(\xi)sin(\omega \xi)}d\xi$
  • The Fourier sine integral representation of a function f on $latex [0 ,\infty )$
    • $latex \int_{0}^{\infty}{B_\omega sin(\omega x)}dx$ where
      • $latex B_\omega = \frac{2}{\pi} \int_{0}^{\infty}{f(\xi)sin(\omega \xi)}d\xi$
  • The Fourier cosine integral representation of a function f on $latex [0 ,\infty )$
    • $latex \int_{0}^{\infty}{A_\omega cos(\omega x)}dx$ where
      • $latex A_\omega = \frac{2}{\pi} \int_{0}^{\infty}{f(\xi)cos(\omega \xi)}d\xi$
3.6. The Fourier Transform
  • The Fourier transform of a function $latex f$
    • $latex \mathcal{F}[f](\omega)=\hat{f}(\omega)=\int_{-\infty}^{\infty}f(\xi)e^{-i\omega \xi}d\xi$
  • The Fourier transform is linear.
    • $latex \mathcal{F}[\alpha f + \beta g](\omega)=\alpha \mathcal{F}[f](\omega) + \beta\mathcal{F}[g](\omega)$
  • The Fourier transform is a function from a function into another function.
3.7. Convolution
  • The convolution $latex f*g$ of $latex f$ with?$latex g$
    • $latex (f*g)(t)=\int_{-\infty}^{\infty}{f(t-x)g(x)}dx$
  • Convolution is an operator between two functions.
  • Let $latex f$ and $latex g$ be real- or complex- valued functions defined on the real line.
  • $latex f$ has a convolution with $latex g$ if
    • $latex \int_{a}^{b}f(x)dx$ and?$latex \int_{a}^{b}g(x)dx$?exist for every interval $latex [a, b]$
    • $latex \int_{-\infty}^{\infty}|{f(t-x)g(x)}|dx$ converges for every real number $latex t$.
3.8. Fourier Sine and Cosine Transforms
  • Fourier sine transform
    • $latex \mathcal{F}_S[f](\omega)=\hat{f}_S(\omega)=\int_{0}^{\infty}f(\xi) sin(\omega\xi) d\xi$
  • Fourier cosine transform
    • $latex \mathcal{F}_C[f](\omega)=\hat{f}_C(\omega)=\int_{0}^{\infty}f(\xi) cos(\omega\xi) d\xi$

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 $latex f_1$ and $latex f_2$ defined on $latex I$ are orthogonal with weight $latex \sigma$ on $latex I$ if
    • $latex \int_{I}f_1(x)f_2(x)\sigma(x)dx=0$
  • An eigenvalue problem is an equation of the form
    • $latex (Lf)(x)+\lambda\sigma(x)f(x)=0, \ \ a<x<b$
    • where $latex \lambda$ is a real parameter and $latex \sigma$ is a given function such that $latex \sigma(x)>0$ for all $latex x\in (a,b)$
    • The numbers $latex \lambda$ 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
    1. $latex [p(x)f'(x)]’+q(x)f(x)+\lambda\sigma(x)f(x)=0, \ \ a<x<b$
    2. with the boundary conditions (BCs)
      $latex \kappa_1f(a) +\kappa_2f'(a)=0$,
      $latex \kappa_3f(b) +\kappa_4f'(b)=0$,
    3. together with the following conditions.
      • $latex p(x)\in\mathbb{R}$ is continuous, differentiable, and $latex p(x)>0$.
      • $latex q(x)\in\mathbb{R}$ is continuous.
      • $latex \sigma(x)\in\mathbb{R}$ is continuous, and?$latex \sigma(x)>0$.
      • $latex \kappa_1,\kappa_2\in\mathbb{R}$. $latex \kappa_1$ and $latex \kappa_2$ are not zero at the same time.
      • $latex \kappa_3,\kappa_4\in\mathbb{R}$. $latex \kappa_3$ and $latex \kappa_4$ are not zero at the same time.
  • How to transform a second-order ODE to?a regular Sturm-Liouville problem
    • A?second-order ODE: $latex f”(x)+af'(x)+bf(x)+\lambda c f(x)=0$
    • Transform this equation by multiplying $latex e^{ax}$ each term.
    • Then, we get $latex [e^{ax}f'(x)]’+be^{ax}f(x)+\lambda ce^{ax} f(x)=0$, which is the form of a regular Sturm-Liouville problem.
  • How to solve a regular Sturm-Liouville problem
    • Generally, a?second-order ODE $latex f”(x)+af'(x)+bf(x)+\lambda c f(x)=0$ is given.
    • Its characteristic equation is $latex s^2+as+(b+\lambda c)$.
    • Then, $latex s_1=\frac{-a+\sqrt{a^2-4(b+\lambda c)^2}}{2}$ and?$latex s_2=\frac{-a-\sqrt{a^2-4(b+\lambda c)^2}}{2}$
    • Then, there are ?three cases when?$latex a^2-4(b+\lambda c)^2>0, =0, <0$.
      1. If?$latex a^2-4(b+\lambda c)^2>0$, then $latex s_1,s_2 \in \mathbb{R}$ and?$latex s_1 \neq s_2$. Then, $latex f(x)=C_1 e^{s_1} + C_2 e^{s_2}$
      2. If?$latex a^2-4(b+\lambda c)^2=0$,?then $latex s_1,s_2 \in \mathbb{R}$ and?$latex s_1 = s_2$. Then, $latex f(x)=(C_1 + C_2 x)e^{s_1}$
      3. If?$latex a^2-4(b+\lambda c)^2<0$,?then?$latex s_1,s_2 \in \mathbb{C}-\mathbb{R}$ and?$latex s_1 \neq s_2$. So, $latex s_1=\alpha + \beta i, s_2=\alpha – \beta i$ such that $latex \alpha=\frac{-a}{2},\beta=\frac{\sqrt{4(b+\lambda c)^2-a^2}}{2}, i=\sqrt{-1}$. Then, $latex f(x)=e^{\alpha x}[C_1 cos(\beta x) + C_2 sin(\beta x)]$
    • If $latex f(x)$ is not a zero function, then $latex f(x)$ is an eigenvalue function and there exists the corresponding eigenvalue.
  • The generalized Fourier series expansion?for $latex u(x)$ in the eigenfunctions $latex \{f_n(x)\}_{n=1}^{\infty}$
    • $latex u(x)\approx \sum_{n=1}^{\infty}{c_n f_n(x)}=\frac{1}{2}\{u(x+)+u(x-)\},\ a\leq x \leq b$
    • where $latex c_n=\frac{\int_{a}^{b}{u(x)f_n(x)\sigma(x)}dx}{\int_{a}^{b}{f_n^2(x)\sigma(x)}dx},\ for \ n=1,2,3,…$
    • If?$latex \{f_n(x)\}_{n=1}^{\infty}$ has a unique generalized Fourier series expansion, then?$latex \{f_n(x)\}_{n=1}^{\infty}$ 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
  1. Rod with zero temperature at the endpoints.

    • The Fourier sine series
  2. Rod with insulated endpoints.

    • The Fourier cosine series
  3. Rod with mixed homogeneous boundary conditions.

    • The generalized Fourier series
  4. Rod with an endpoint with a zero-temperature medium
  5. Heat conduction in a thin uniform circular ring.
5.2. The Wave Equation
  1. String with fixed endpoints.

  2. Vibrating string with free endpoints.
5.3. The Laplace Equation
  1. The Laplace equation in a rectangle.
  2. The Laplace equation in a circular disk.
5.4. Other Equations
  1. The diffusion-convection IBVP
  2. The dissipative wave propagation problem
  3. 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

 

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