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 [-\pi , \pi]
    • \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(nx) + b_n sin(nx)}
    • where a_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)cos(nx)}dxa_0 = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)}dx, and
    • b_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)sin(nx)}dx for 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 (-\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 \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 (-\pi , \pi] to (-L , L]
  • The Fourier series of a function on [-L , L]
    • \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(n\pi x/L) + b_n sin(n\pi x/L)} where
      • a_n = \frac{1}{L} \int_{-L}^{L}{f(x)cos(n\pi x/L)}dx,
      • a_0 = \frac{1}{L} \int_{-L}^{L}{f(x)}dx, and
      • b_n = \frac{1}{L} \int_{-L}^{L}{f(x)sin(n\pi x/L)}dx for n=1,2,3,...
3.4. Sine and Cosine Expansions
  • Change of scale from (-L , L] to [0 , L]
  • Even functions
  • Odd functions
  • Half-range Fourier expansions
    • The Fourier sine series of a function f defined on [0 , L]
      • Assume f is an odd function. Then, a_n becomes 0.
      • \sum_{n=1}^{\infty}{b_n sin(n\pi x/L)} where
        • b_n = \frac{2}{L} \int_{0}^{L}{f(x)sin(n\pi x/L)}dx for n=1,2,3,...
    • The Fourier cosine series of a function f defined on [0 , L]
      • Assume f is an even function. Then, b_n becomes 0.
      • \frac{1}{2}a_0 + \sum_{n=1}^{\infty}{a_n cos(n\pi x/L)} where
        • a_n = \frac{2}{L} \int_{0}^{L}{f(x)cos(n\pi x/L)}dx for n=0,1,2,3,...
3.5. The Fourier Integral
  • Change of scale from (-L , L] to (-\infty ,\infty )
  • Preconditions of the Fourier integral representation
    • f is piecewise smooth on the interval [-L , L] for every L\in \mathbb{R^+}
    • \int_{-\infty}^{\infty}|f(x)|dx converges. Say, f is absolutely integrable.
  • The Fourier integral representation of a function f on (-\infty ,\infty )
    • \int_{0}^{\infty}[A_\omega cos(\omega x) + B_\omega sin(\omega x)]dx where
      • A_\omega = \frac{1}{\pi} \int_{-\infty}^{\infty}{f(\xi)cos(\omega \xi)}d\xi, and
      • 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 [0 ,\infty )
    • \int_{0}^{\infty}{B_\omega sin(\omega x)}dx where
      • 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 [0 ,\infty )
    • \int_{0}^{\infty}{A_\omega cos(\omega x)}dx where
      • 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 f
    • \mathcal{F}[f](\omega)=\hat{f}(\omega)=\int_{-\infty}^{\infty}f(\xi)e^{-i\omega \xi}d\xi
  • The Fourier transform is linear.
    • \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 f*g of f with g
    • (f*g)(t)=\int_{-\infty}^{\infty}{f(t-x)g(x)}dx
  • Convolution is an operator between two functions.
  • Let f and g be real- or complex- valued functions defined on the real line.
  • f has a convolution with g if
    • \int_{a}^{b}f(x)dx and \int_{a}^{b}g(x)dx exist for every interval [a, b]
    • \int_{-\infty}^{\infty}|{f(t-x)g(x)}|dx converges for every real number t.
3.8. Fourier Sine and Cosine Transforms
  • Fourier sine transform
    • \mathcal{F}_S[f](\omega)=\hat{f}_S(\omega)=\int_{0}^{\infty}f(\xi) sin(\omega\xi) d\xi
  • Fourier cosine transform
    • \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 f_1 and f_2 defined on I are orthogonal with weight \sigma on I if
    • \int_{I}f_1(x)f_2(x)\sigma(x)dx=0
  • An eigenvalue problem is an equation of the form
    • (Lf)(x)+\lambda\sigma(x)f(x)=0, \ \ a<x<b
    • where \lambda is a real parameter and \sigma is a given function such that \sigma(x)>0 for all x\in (a,b)
    • The numbers \lambda for which the equation has nonzero solution are called eigenvalues. The corresponding nonzero solutions are called eigenfunctions.
  • regular Sturm-Liouville (S-L) eigenvalue problem is an equation of the form
    1. [p(x)f'(x)]'+q(x)f(x)+\lambda\sigma(x)f(x)=0, \ \ a<x<b
    2. with the boundary conditions (BCs)
      \kappa_1f(a) +\kappa_2f'(a)=0,
      \kappa_3f(b) +\kappa_4f'(b)=0,
    3. together with the following conditions.
      • p(x)\in\mathbb{R} is continuous, differentiable, and p(x)>0.
      • q(x)\in\mathbb{R} is continuous.
      • \sigma(x)\in\mathbb{R} is continuous, and \sigma(x)>0.
      • \kappa_1,\kappa_2\in\mathbb{R}. \kappa_1 and \kappa_2 are not zero at the same time.
      • \kappa_3,\kappa_4\in\mathbb{R}. \kappa_3 and \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: f''(x)+af'(x)+bf(x)+\lambda c f(x)=0
    • Transform this equation by multiplying e^{ax} each term.
    • Then, we get [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 f''(x)+af'(x)+bf(x)+\lambda c f(x)=0 is given.
    • Its characteristic equation is s^2+as+(b+\lambda c).
    • Then, s_1=\frac{-a+\sqrt{a^2-4(b+\lambda c)^2}}{2} and s_2=\frac{-a-\sqrt{a^2-4(b+\lambda c)^2}}{2}
    • Then, there are  three cases when a^2-4(b+\lambda c)^2>0, =0, <0.
      1. If a^2-4(b+\lambda c)^2>0, then s_1,s_2 \in \mathbb{R} and s_1 \neq s_2. Then, f(x)=C_1 e^{s_1} + C_2 e^{s_2}
      2. If a^2-4(b+\lambda c)^2=0, then s_1,s_2 \in \mathbb{R} and s_1 = s_2. Then, f(x)=(C_1 + C_2 x)e^{s_1}
      3. If a^2-4(b+\lambda c)^2<0, then s_1,s_2 \in \mathbb{C}-\mathbb{R} and s_1 \neq s_2. So, s_1=\alpha + \beta i, s_2=\alpha - \beta i such that \alpha=\frac{-a}{2},\beta=\frac{\sqrt{4(b+\lambda c)^2-a^2}}{2}, i=\sqrt{-1}. Then, f(x)=e^{\alpha x}[C_1 cos(\beta x) + C_2 sin(\beta x)]
    • If f(x) is not a zero function, then f(x) is an eigenvalue function and there exists the corresponding eigenvalue.
  • The generalized Fourier series expansion for u(x) in the eigenfunctions \{f_n(x)\}_{n=1}^{\infty}
    • 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 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 \{f_n(x)\}_{n=1}^{\infty} has a unique generalized Fourier series expansion, then \{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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