Resources Used

• Partial Differential Equations: An Introduction, 2nd Ed, by Walter A. Strauss.
• Lecture Notes by Kostyantyn Zheltukhin.
• Elements of Partial Differential Equations by Ian N. Sneddon.

Notation

• 0 \in \N and \N^+ := \N \setminus \{0\}.
• Given a set X, its closure will be denoted with \overline{X} and boundary with \partial X.
• C^n(\Omega) denotes the set (or space) of n-times continuously differentiable real-valued functions on the domain \Omega, and C^\infty refers to the smooth functions which are infinitely differentiable.

Thm. Integrating Factor

Suppose we have the first order differential equation

then we can solve this ODE using the integrating factor

so that, by multiplying both sides by \mu, we reach the following identity:

Def. Curl

Given a vector field \bold{F} = (F_1, F_2, F_3) in three-dimensions, the curl of \bold{F} is defined as

Thm. Green's Theorem

Let \Omega be a domain in \R^2 and P, Q \in C^1(\Omega), then

where \partial\,\Omega is positively oriented, piecewise smooth, simple closed curve whose region is \Omega.

Thm. Characteristic Equation Method

Let us be given a second-order linear ordinary differential equation with constant coefficients:

then recall that we let

so that

which substitution of yields:

and as e^{rx} \neq 0, the polynomial is called the characteristic equation whose roots determine the general solution.

Thm. Hyperbolic Functions

Here are some of useful identities regarding the hyperbolic functions:

So hyperbolic \cosh and \sinh are quite similar to their circular counterparts \cos and \sin.

Def. PDE

A partial differential equation is an equation that involves an unknown function u(x_1,...,x_n) of n \geq 2 variables (otherwise it's called ordinary) and some of its partial derivatives with respect to one or more of x_1,...,x_n. Precisely u is a function such that u: U \to \R where U is an open subset of \R^n called the domain of the PDE.

Notice how the parameters x_1,...,x_n are independent variables and the unknown function u is a dependent variable depending on x_1,...,x_n.

The order of a PDE is the highest order that appears in the equation.

A solution of a PDE is a known function u that satisfies the equation.

Notation. Partial Derivative

From now on we will denote the partial derivative of u with respect to its parameter x as

Moreover, higher-order partial derivatives are simply written in sequence. For example, the partial derivative of u_{x} with respect to y will be denoted as

Recall from calculus that u_{xy} = u_{yx} (when the relevant derivatives are continuous).

Unless otherwise stated we will assume all derivatives exist and are continuous.

Def. Initial and Boundary Conditions

These are not formal definitions but rather conventions regarding PDEs.

Because PDEs typically have infinitely many solutions, we may want to impose auxiliary conditions. These conditions are usually classified into two classes: initial and boundary conditions, which themselves (as we will see later) are further classified.

An initial condition specifies as the name implies initial conditions. For example, for an unknown function u of n variables

is an initial condition where c \in \R is a constant.

A boundary condition specifies the boundary behavior of the solution. For example, defining the domain of the PDE is part of specifying boundary data, and so is prescribing the normal derivative \partial u/\partial n on the boundary. We will see more about boundary conditions and classify them later on.

Def. Well-Posed Problem

A well-posed PDE problem consists of a PDE in a domain with initial and/or boundary conditions (or other auxiliary conditions) that satisfies

1. existence i.e. there exists at least one solution satisfying these conditions
2. uniqueness i.e. there is at most one solution
3. stability i.e. the solution continuously depends on the parameters (so that it's not chaotic)

Ex. Constant Coefficients

Let us be given a first-order homogeneous partial differential equation

with non-zero constant coefficients a,b \in \R.

Ex. Variable Coefficient

Now, let us be given a first-order homogeneous partial differential equation

Def. First Order (Quasi-)Linear Equation

A linear first order partial differential equation is defined as

where z = z(x,y). More generaly, a quasilinear equation is defined as

and both are are solved in the same way which is

todo add more from handwritten notes

Thm. Quasilinear Cauchy Problem

Consider the first order quasilinear equation

where for some domain \Omega we have x,y,z \in \Omega and A,B,C \in C^1(\Omega) and A^2 + B^2 + C^2 \neq 0 in \Omega.

Passing through the curve

where s \in (a,b) \subseteq \Omega and \alpha, \beta, \gamma \in C^1 (\Omega).

The Cauch Problem above has an unique solution in some neighborhood of \Gamma if

is non-zero on \Gamma.

Moreover, if \Delta |_\Gamma = 0 for all s \in (a,b), then either

• We have infinitely many solns if \Gamma is characteristic curve, or
• We have no solutions if \Gamma is not characteristic curve.

Def. Conservative Vector Field

A vector field \bold{\phi}: \Omega \to \R^n where \Omega is an open subset of \R^n is said to be conservative if there exists a C^1 scalar field \varphi such that

Thm. Solution of Pfaffian PDEs

Let z = z(x,y) and consider the Pfaffian PDE:

where P^2 + Q^2 + R^2 \neq 0 in \Omega \subseteq \R^3, then it has a solution if and only if, for \vec{G} = (P,Q,R):

Moreover,

Also recall that always z_{xy} = z_{yx}.

Def. Simple 2nd Order PDE

The homogeneous linear equation of order two in two variables is given by

where \cdots denotes terms of order one or lower and a_{ij} are real coefficients.

The coefficient 2 is introduced for convenience.

Thm. Canonical Forms

By a linear transformation of the independent variables, a 2nd order linear PDE in two variables can be reduced to one of the three forms. Let \Delta = a_{12}^2 - a_{11}a_{22}, then it is of the form

1. hyperbolic if \Delta > 0 so that

2. parabolic if \Delta = 0 so that

3. elliptic if \Delta < 0 so that

where \cdots denotes terms of order one or lower. Assuming a_{11} \neq 0, we generally have

This identity is correct even if a_{ij}'s are not coefficients but rather functions.

In particular...

Def. Wave Equation

The (one-dimensional) wave equation (on the line) is defined as the (hyperbolic) equation:

where x \in (-\infty, \infty) and t > 0. The general solution is easy to solve since operators factor out nicely:

so that

where f and g are twice-differentiable arbitrary functions of single variable.

Thm. Cauchy Problem for the Wave Equation

Assume we are given an initial value problem for the wave equation so that

where g and h are the given initial displacement and velocity. There is one and only one solution of this problem which is

called the d'Alembert's formula.

Moreover, this problem (generally inhomogenous wave equation) is well-posed since unique solution exists and it's stable.

Def. Inhomogeneous Wave Equation

Let f \in C^2(\R^2, \R), then the inhomogeneous wave equation is defined as

and unless otherwise stated assume t > 0 and x \in (-\infty, \infty).

Thm. Simple Inhomogenous Wave Equation

Assume we are given a inhomogenous wave equation with the initial conditions

then the solution u(x,t) is given by applying the Green's Theorem on the domain of dependence so that

where D is the characteristic triangle.

where

Thm. Generalized d'Alembert's Formula

Let

be an inhomogeneous wave equation with the initial conditions

then the solution is given by the formula

notice that the double integral is over the characteristic triangle \Delta.

Moreover, if this problem on the half line i.e. with extra boundary condition

then the solution is the same as above for x > ct > 0, but for 0 < x < ct, we have

where t - x/c is the reflection point and D is the corresponding shaded region.

Thm. Causality

Effect of initial position g(x) is a pair of waves traveling in either direction at speed c and at half the original amplitude. The effect of an initial velocity h(x) is a wave spreading out at speed \le c in both directions, so part of the wave may lag behind (if there is an initial velocity) but no part goes faster than speed c. This is called the principle of causality.

See and add principle of causality figure p. 39

Thm. Domain of Influence and Dependence

An initial condition (position or velocity or both) at the point (x_0, 0) can affect the solution for t > 0 only in the shaded sector, which is called the domain of influence of the point (x_0, 0). The domain of influence corresponds to the shaded are for the (upwards infinite) triangle defined by two lines x+ct = x_0, x - ct = x_0 and the point (x_0,0) where x_0 \in [x-ct, x+ct].

The domain of dependence or the past history of the point (x_0,t_0) corresponds to the shaded area for the (downwards) triangle (x_0-ct_0, 0), (x_0+ct_0,0) and (x_0,t_0) called the characteristic triangle.

Def. Energy

Consider a homogenous wave equation, then we define the energy integral as

which is constant (exercise).

Thm. Reflection of Waves

Assume we are given a Dirichlet problem on the half-line (0,\infty) for the wave equation so that

where x \in (0,\infty) and t>0. Then, the solution for x>ct is given by

and for 0<x<ct we have

See Figure 2 in p. 62

Thm. Duhamel's Principle

TODOs

• Add semi infinite: free/fixed end wave equations and their domains.
• Solution of non-homogenous wave equations via u = v+w, see notes p52.

Def. Heat Equation

The (one-dimensional) heat equation is defined as the (parabolic) equation:

where unless otherwise stated

for non-zero \ell, k \in \R and t > 0 with x \in (0, \ell).

It is also known as the heat equation especially in natural sciences. The heat equation is harder to solve than the wave equation, so we will postpone the general solution.

Similar to the wave equation, the heat equation also has an unique stable solution.

Def. Energy

Consider we are given an homogenous diffusion equation, then we define it's energy integral as

which is non-increasing (exercise) i.e. E(t) < E(0).

Thm. (Weak) Maximum Principle

We first define the open domain \Omega_T in \R as

and

then, if u_t - k^2 u_{xx} \leq 0 in \Omega_T, we have

So, if u(x,t) satisfies the diffusion equation in a rectangle, 0 \leq x \leq \ell and 0 \leq t \leq T in space-time, then the maximum value of u(x,t) is attained either initially at t = 0 or on the lateral sides x = 0 or x = \ell.

Indeed there is a stronger version of the maximum principle called the strong maximum principle which asserts the maximum cannot be attained anywhere inside the rectangle but only on the bottom or the lateral sides (unless u is constant). The corners are allowed.

The minimum value has the same property so that it too can be attained only on the bottom or the lateral sides.

In particular assume that u_t - k u_{xx} = 0 in \Omega_T, then

and

hence

Thm. Invariance Properties

The diffusion equation (1) has some basic invariance properties, namely

1. The translation u(x - y, t) of any solution u(x,t) is another solution.
2. Any derivative u_x, u_t, or u_xx etc. of a solution is another solution.
3. A linear combination of solutions is again a solution — linearity.
4. An integral of a solution is again a solution.

Thm. Diffusion on the Whole Line

Let us be given the following problem

where x \in (-\infty, \infty) and t \in (0, \infty). Assuming \phi(\infty) = \phi(-\infty) = 0, we have

for t > 0. This is called the source function, so that for the solution we have

The source function S(x,t) is defined for all real x and positive t. Moreover, S(x,t) is positive and is even in x so that S(-x,t) = S(x,t).

Note that the general solution integral is not usually expressible with the elementary functions, therefore recall that

\text{erf}(x) := \dfrac{2}{\sqrt{\pi}} \int_0^x e^{-p^2} dp

Thm. Comparison of Wave and Diffusion

*along characteristics with speed c.

Thm. Diffusion on the Half-Line

Let's take the domain for the diffusion equation to be the half-line (0, \infty) and take the Dirichlet boundary conditions at the single point x = 0, so that the problem becomes

where x \in (0, \infty) and t \in (0, \infty). The general solution for this problem is of the form

Now consider the Neumann problem, so that it becomes

where x \in (0, \infty) and t \in (0, \infty), then the general solution for this problem is of the form

Thm. Diffusion with a Source

Let's now consider the diffusion equation with a source f(x,t), i.e.

then the general solution is of the form

Introduction

Consider the space bounded heat equation

and assume u = X(x)T(t) then by substitution we have

and

They must be equal to a constant since left side depends on x and right side depends on t only. Negativity is for convenience.

for some constant \lambda. This implies two simple ODEs

This can be considered as an eigenvalue problem. In this case, \lambda \leq 0 implies trivial solutions and for \lambda > 0 we have the set of solutions

where \lambda_n corresponds to eigenvalues and x_n corresponds to eigenfunctions. Similarly, solving the eigenvalue problem for T yields

hence

so that

is a solution (assuming that the series convergent and can be differentiated term by term).

Def. Hilbert Space

Consider a linear space \cal{H} of piecewise continuous functions on [-\ell, \ell] on space \cal{H}, and define an inner product

then (\cal{H}, \Braket{\cdot, \cdot}) is called a pre-Hilbert space and if \cal{H} is complete it is called a Hilbert Space.

Recall that inner product induces a norm so that

Thm. An Orthogonal Basis for \cal{H}

The sequence

is an orthogonal basis for the pre-Hilbert space \cal{H} and any f \in \cal{H} can be written as

so that the series converge to f in the norm \| \cdot \|_2.

Thm. Parseval's Equality

Using the basis given above and the norm \| \cdot \|_2, we reach the following identity called the Parseval's Equality:

Note that \|1\|_2 = \sqrt{2\ell} and \|\cos\left(\dfrac{n \pi}{\ell}x\right)\|_2 = \|\sin\left(\dfrac{n \pi}{\ell}x\right)\|_2 = \sqrt{\ell}.

Thm. Fourier Sine Series

Suppose f_\text{odd} piecewise continuous on [-\ell, \ell] and is an odd function, then

in the interval (-\ell, \ell) and is called the Fourier Sine Series for the function f_\text{odd}.

These series, as we saw earlier, are used for wave and diffusion equations with Dirichlet boundary conditions.

Moreover, the coefficients b_n are given by:

This is equivalent to the Fourier Series of f_\text{odd} since the even cosine terms cancel out in the full Fourier Series.

Thm. Fourier Cosine Series

Suppose f_\text{even} piecewise continuous on [-\ell, \ell] and is even, then

in the interval (-\ell, \ell) and is called the Fourier Cosine Series for the function f_\text{even}.

These series, as we saw earlier, are used for wave and diffusion equations with Neumann boundary conditions on (0, l).

Moreover, the coefficients a_n are given by:

This is equivalent to the Fourier Series of f_\text{even} since the odd sine terms cancel out in the full Fourier Series.

Thm. Fourier Convergence Theorem

Let f be a function on [-\ell, \ell], and extend it periodically with period 2\ell, so that outside [-\ell, \ell] we have

Suppose f is piecewise smooth on [-\ell, \ell], then the fourier series of f converges at each point x to

This implies geometrically, for the endpoints of the periodically extended f, the Fourier series converge to their midpoint.

Moreover if f is continuous on [-\ell, \ell] and the endpoint values match periodically i.e.

then there is no jump at the endpoints.

Ex. \cal{F}(1)

For some problems it is often useful to consider the Fourier series of the constant 1, which equals to

where

Def. Laplace Operator

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient of f \in C^2(\Omega). Therefore, if f is a twice-differentiable real-valued function, then the Laplacian of f is the real valued function

Moreover, f is called harmonic if \Delta f = 0.

In this section we will consider the Laplace Equation defined by \Delta u = 0, and then the Poisson Equation defined by \Delta u = F(x) with Dirichlet boundary condition:

and Neumann boundary condition:

where \vec{n} is the unit outward normal to \partial \Omega.

Recall that

We will not prove it however Poission equation has an unique solution in both boundary conditions, and stable making it well-posed.

Thm. Weak Maximum Principle

Let \Omega be a bounded domain in \R^n and u \in C^2(\Omega) \cap C(\overline{\Omega}) and suppose \Delta u \geq 0 in \Omega, then

and as a result

Moreover, if \Delta u = 0 in \Omega, then

Thm. Laplace Dirichlet Problem in a Rectangle

Consider the rectange R \subseteq \R^2 defined as 0 < x < a and 0 < y < b, and the Laplace Equation

we can write u = u_1 + u_2 + u_3 + u_4 where \Delta u_i = 0 in R where each u_i corresponds to an edge of R.

For simplicity, let us consider just w=u_1 so that we have

and set w = X(x)Y(y) for seperation of variables, which results in

which can be used to construct a formal solution

This is not really a good solution, maybe check out p. 163 of Strauss.

Thm. Poisson's Formula

Let's consider the Dirichlet Problem for a circle which is

within polar coordinates, we have

and

so that

Thm. Mean Value Property

Let u be a harmonic function in a disk D, continuous in its closure \overline{D}, then the value of u at the center of D equals to the average of u on its circumference:

Thm. Divergence Theorem

Let V \subseteq \R^3 be a compact with piecewise-smooth boundary and F a smooth vector field, then

where \vec{n} is the outward pointing unit normal on the boundary \partial V.

Thm. Green's First Identity

Let \Omega \subseteq \R^n be a domain where the divergence theorem holds and u,v \in C^2(\Omega) \cap C^1(\overline{\Omega}), then

and by the divergence theorem we get the Green's First Identity:

Notice that for v = 1, we have

Recall that

\dfrac{\partial u}{\partial \vec{n}} = \nabla u \cdot \vec{n}

Notice how the last identity implies the necessary condition for the existence of solution to the Laplace equations:

Def. Uniform Convergence

We say a functional series \sum_{n=1}^\infty f_n(x) converges uniformly on [a,b] if

Thm. in the Wave Equation

Assume we are given a homogeneous wave equation with Dirichlet conditions so that

where x \in (0, \ell). A separated solution is a solution of the form

So if we plug this equation into our PDE, we get

This defines a separation constant \lambda. For the Dirichlet eigenvalue problem one gets \lambda>0, so write \lambda=\beta^2 with \beta>0. Then the equations above are a pair of separate ODEs:

These ODEs are easy to solve and have solutions of the form

where A,B,C,D are constants. The boundary conditions require X(0)=0=X(\ell). From X(0)=0 we get C=0, hence X(x)=D\sin(\beta x), and then

We are not interested in the trivial solution X\equiv 0 (equivalently C=D=0).

therefore \beta = \dfrac{n\pi}{\ell} and there are infinitely many separated solutions of the form

where n = 1,2,\dots and A_n and B_n are arbitrary constants. Noting a sum of solutions is also a solution, we have

The coefficients n \pi c / \ell before the variable t are sometimes called the frequencies.

For the initial condition, notice that

and

The numbers \lambda_n = (n \pi / \ell)^2 are called eigenvalues and the functions X_n(x) are called eigenfunctions where n = 1,2,3,....

Thm. in the Diffusion Equation

Analogously assume we are given a homogeneous diffusion equation with Dirichlet conditions so that

where x \in (0, l), then

so that

and

Thm. in the Wave Equation

Consider the (homogenous) wave equation with the Neumann BC, then it has a solution of the form

where x \in (0, l). A separated solution for this Neumann BC is a solution of the form

Thm. in the Diffusion Equation

Now assume we are given a homogeneous diffusion (heat) equation with Neumann conditions so that

then it has a solution of the form

where

Also, notice that we have the eigenvalues

and the eigenfunctions

for n=0,1,2,\dots (with X_0\equiv 1).

Notice how, unlike Dirichlet condition, n starts from 0 instead of 1.