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0x046 Differential Equation

1. Linear ODE

Definition (n-th order ODE) An \(n\)-th order differential equation is said to be linear if it can be written in the form

\[y^{(n)}+p_1(x)y^{(n-1)}+ ... + p_n(x)y = f(x)\]

For convenience, we consider the following form

\[P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+ ... + P_n(x)y = F(x)\]

where the left side can be abbreviated as \(Ly\)

Theorem (initial value problem) Suppose \(Ly=F\) is normal (i.e: \(P_0\) has nozero and \(P_0, ..., P_n, F\) are continuous) on \((a,b)\), let \(x_0\) be a point at \((a,b)\), and \(k_0, ..., k_{n-1}\) be arbitrary number, then the initial value problem

\[Ly=F, y(x_0)=k_0, ..., y^{(n-1)}(x_0) = k_{n-1}\]

has a unique solution on \((a,b)\)

2. Laplace Transform

Definition (Laplace transform) Suppose a function \(f(x)\) is piecewise-continuous, then the Laplace transform of \(f(t)\), denoted \(\mathcal{L}\{ f(t) \}\), is defined as

\[\mathcal{L}\{ f(t) \} = \int_0^{\infty} e^{-st} f(t) dt\]

For the sake of convenience, we also use

\[F(s) = \mathcal{L} \{ f(t) \}\]

Lemma (linearity)

\[\mathcal{L} \{ af(t) + bg(t) \} = aF(s)+bG(s)\]

Lemma (derivatives) Suppose \(f', ..., f^{(n-1)}\) are all continuous function and \(f^{(n)}\) is a piecewise continuous function, then high-order derivatives can be transformed into a linear function

\[\mathcal{L}(f^{(n)}) = s^n F(s) - s^{n-1} f(0) - s^{n-2}f'(0) - ... - f^{(n-1)}(0)\]

Lemma (convolution) Laplace transform changes convolution into multiplication

\[\mathcal{L}(f*g) = F(s)G(s)\]

3. Linear PDE

2nd order linear PDF can be calssified as either elliptic, hyperbolic or parabolic

\[Au_{xx} + 2B u_{xy} + Cu_{yy} + Du_x + Eu_y + Fu + G =00\]

by checking sign of \(B^-AC\)

3.1. Laplace Equation

Laplace is the simplest example of elliptic PDF

\[\nabla^2 u = 0\]

or equivalently

\[\Delta u = 0\]

Its solution is called harmonic functions


A trivial case: suppose \(u\) only depends on \(x\)

\[\frac{d^2u}{dx^2} = 0 \implies u(x) = mx + b\]

\(u\) can be fully determined by boundary conditions (e.g: \((u(0), u(1))\))

An observations that can be generalized to higher dimension here is

  • \(u(x)\) is the average of \(u(x+a)\) and \(u(x-a)\)
  • no local max or min

3.1.1. 2-dimension

In two dimension:

\[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]

The real and imaginary parts of a complex analytic function both satisfy the Laplae equation. (which can be proved easily using Cauchy-Riemann)

Consider a power series expansion of \(f\) inside a circle of radius \(R\)

\[f(z) = \sum_{n=0}^\infty c_n z^n\]

using polar coordinate, we know the real part of \(f(z)\) is

\[a_n r^n \cos(n\theta) - b_n r^n \sin(n\theta)\]

which is the solution to the two dimensional Laplace equation, the coeeficient \(a_n, b_n\) can be determined from the boundary

3.2. Heat Equation

Heat equation is an example of the parabolic equation.

3.3. Wave Equation

Wave equation is an example of the hyperbolic equation.

By a linear change of variables, any equation of the form

\[A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + ... = 0\]


\[B^2 - AC > 0\]

This is called the hyperbolic partial differential equation, which can be transformed to the wave equation, ignoring the lower order terms

4. Reference

[1] Elementar Elementary Differential E ential Equations with Boundar quations with Boundary Value Problems

[2] Duke Math 356 lecture notes

[3] MIT differential equation course