1 Partial differential equations – MATH 2031 Lecture Notes (Epiphany)

1.1 What is a partial differential equation (a PDE)

Lets write down a famous PDE, the heat/diffusion equation \[ \ddy{u}{t} = D\left(\pder{u}{x}{2} + \pder{u}{y}{2}\right) \] Here we have a scalar function $u(x,y,t)$, the dependent variable. This function is dependent on two spatial variables, $x$ and $y$, and time $t$. We refer to these as the independent variables. The equation is a balance between various partial derivatives of this function.

A solution to this equation in the above video shows the density $u$ spreading out. It can be used to model heat diffusing in a room or chemical populations diffusing in solution. The parameter $D$ is referred to as the diffusion coefficent and represents the rate of spreading (though this is not immidatley obvious here).

Lets be more formal as to what we mean by a PDE:

Definition 1.4 A partial differential equation is an equation containing an unknown function of two or more variables and its partial derivatives with respect to these variables.

We categorise PDE’s by their highest deirvative

Definition 1.3 The order of a partial differential equation is the order of if highest-order derivative(s).

They are much easier to tackle if they are linear:

Definition 1.2 A partial differential equation is linear/non-linear if it is respectively linear/non-linear in the dependent variable.

For example, consider the following equations:

\[\begin{align} (a) & \ddy{u}{t} + \pder{u}{x}{3} - 6u\ddy{u}{x} = 0,\\ (b) & \pder{u}{t}{2} + \sin(u)\frac{\partial^2{u}}{\partial x \partial y} =0,\\ (c) & \pder{u}{t}{3} + \sin(x)\frac{\partial^2{u}}{\partial x \partial y}=0. \end{align}\]

Tip 1.1: Identify the dependent and independent variables of these PDE’s. State their order and classify them as linear or non-linear.

In case (a) the dependent variable is $u$, the independent variables are $x$ and $t$. It is third order as the highest derivative in any form is $\pder{u}{x}{3}$. It is non-linear as $u\ddy{u}{x}$ is a non-linear product.

For case (b) the dependent variable is $u$, the independent variables are $x$, $y$ and $t$. It is second order, there are two derivatives of that order $\pder{u}{t}{2}$ and $\frac{\partial^2{u}}{\partial x \partial y}$. It is non-linear due to the $\sin(u)$ (and made further so by the fact this is multiplied by a deriavtive).

For case (c), its dependent variable is $u$ and it has two independent variables $x$, $y$ and $t$. It is third order. It is linear, despite the fact that $\sin(x)$ is a non-linear function and mutiplied by $\frac{\partial^2{u}}{\partial x \partial y}$. That is because that non-linearity is in the independent variable not the dependent variable.

Equation (a) is also a famous equation, it is called the Korteweg-DeVries equation. It can be used to model shallow water waves. An example solution is shown below in which a larger faster wave passes through a smaller slower one without them merging

1.2 Solutions to partial differential equations

We have now seen two nice “solutions” to some famous PDE’s. But what do we mean precisely by the term solution?

Definition 1.1 A solution of a PDE in some region $R$ of the space of the independent variables is a function that possesses all of the partial derivatives that are present in the PDE, in some region containing R, and also satisfies the PDE everywhere in $R$.

Lets unpack this with an example. Consider again the heat equation: \[ \ddy{u}{t} = D\left(\pder{u}{x}{2} + \pder{u}{y}{2}\right) \] This equation is in two independent spatial variables $x,y$, so the spatial region $R$ in our definition should be two dimensional. It should also be true for all time $t\in(0,\infty]$. Lets assume it is a square of length $L$, so that $x\in[0,L]$ and $y\in[0,L]$ and $R = [0,L]\times[0,L]\times(0,\infty]$. Then a solution to the heat equation is

\[ u(x,y,t) = \mathrm{e}^{-D\frac{(n^2 + m^2)\pi^2}{L^2} t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right), \tag{1.1}\] for some arbitrary integers $(n,m)$. But also so is \[ u(x,y,t) = \mathrm{e}^{-D\frac{(n^2 + m^2)\pi^2}{L^2} t}\sin\left(\frac{n\pi x}{L}\right)\sin\left(\frac{m\pi y}{L}\right). \tag{1.2}\]

Try it yourself

Confirm this by differentiating both solutions and substituting them into the equation.

Since both sides of the equation are equal the equation is satisfied on $R$. Clearly the derivatives exist everywhere in the domain so it satisfies our definition. But are both valid? Well that depends on the boundary conditions. In this case we must make some statement on what values $u$ takes on the edges $x=0,L$ and $y=0,L$. We consider two cases:

\[\begin{align} (a)\, & u(0,y,t) = u(L,y,t) = u(x,0,t) = u(x,L,t)=0.\\ (b)\, & \ddy{u}{x}(0,y,t) = \ddy{u}{x}(L,y,t) = \ddy{u}{y}(x,0,t) = \ddy{u}{y}(x,L,t)=0. \end{align}\]

The first set of boundary conditions (a) are an example of Dirichlet boundary conditions, conditions placed on the value of the function on the boundary. The second, (b) are often called Neumann boundary conditions, conditions placed on the derivatives of the function on the boundary.

We see that, since $\sin(0) = \sin(n\pi) = 0$, Equation 1.2 satisfies the Dirchlet boundary conditions (a). However, as $\cos(0)=1\neq 0$ the solution Equation 1.1 does not.

Try it yourself

By differentiating the solutions, confirm that Equation 1.2 cannot satisfy boundary conditions (b) and that Equation 1.1 can.

Solutions to partial differential equations (and indeed ordinary differential equations) are not fully defined without their boundary conditions.

We have seen that boundary conditions can change the solution’s nature. In this case the only obvious change is that there is a non-zero spatially uniform behaviour $n=m=0$ in the cosine case (no $x$ or $y$ dependence), which must be zero in the case of the sin solutions. You will see later this can have significant effects on the behaviour of the system. Throughout this course, when we discuss more general results about differential equations (operators as we shall come to call them), one must always consider the solution and boundary conditions as a single definition.

1.3 Linear equations and the principle of superposition.

A fundamental property of linear PDE’s is that we can add solutions and they will still solve the equation. For example, with Neumann boundary conditions (b) the heat equation has cosine solutions (Equation 1.1) which depended on the choice of arbitrary integers $n$ and $m$. We can sum over all the possible values of $n$ and $m$ as follows:

\[ u(x,y,t) = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_n\mathrm{e}^{-D\frac{(n^2+ m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right) \] To see that it is a solution we differentiate:

\[\begin{align} \pder{u}{x}{2} &= -\frac{n^2\pi^2}{L^2}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{nm}\mathrm{e}^{-D\frac{(n^2 + m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right) = -\frac{n^2\pi^2}{L^2}u,\\ \pder{u}{y}{2} &= -\frac{m^2\pi^2}{L^2}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{nm}\mathrm{e}^{-D\frac{(n^2+ m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right) = -\frac{m^2\pi^2}{L^2}u,\\ \ddy{u}{t} &= -D\frac{(n^2+ m^2)\pi^2}{L^2}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{nm}\mathrm{e}^{-D\frac{(n^2 + m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right)=-D\frac{(n^2 + m^2)\pi^2}{L^2}u. \end{align}\]

Inserting this into the heat equation gives: \[ \ddy{u}{t} = D\left(\pder{u}{x}{2} + \pder{u}{y}{2}\right) \Rightarrow -D\frac{(n^2+ m^2)\pi^2}{L^2}u = -D\frac{(n^2+ m^2)\pi^2}{L^2}u \]

We don’t need to consider negative $n,m$ due to the symmetry of the sin and cos functions and the fact we have free constants $C_n$.

There are a number of points to raise about this sum which we shall repeatedly come back to over the term:

At $t=0$ this is just a Fourier series (think back to second term of calculus I). We know these series can represent most functions we are likely to encounter in this course. So the solution can have almost any form initially (at $t=0$).
The terms of a Fourier series are linearly independent: it can only be zero if all terms are individually zero (all $C_n=0$). Each “mode” of the system (each component of the sum for $(n,m)$) has distinct behaviour independent of the others.

The first point means we should expect lots of possible complexity in PDE’s (even a linear one like this)! The second means we can break that complex behaviour down (in the linear case) into the behaviour of simple functions.

And example of this superposition is shown in Figure 1.2 where we see three regular modes superimposed to produce a complex and irregular function.

This second observation is fundamental to the mathematics we will explore in this course:

One of the main aims of this term is to show that a large class of equations, crucial in physical and biological modelling, have solutions $u(x,t)$ in the following “generalised Fourier series” form (here in 1D to keep it simple): \[ u(x,t) = \sum_{n=0}^{\infty}T_n(t)S_n(x). \] That is some spatial function $S_n(x)$ which describes the individual spatial shapes the solutions are made from, and, for each spatial function $S_n$, some function $T_n(t)$ which detemines how it varies in time.

We will come to refer to the $S_n$ as eigenmodes, they differ dependent on the equation, so in some sense each equation comes equipped with its own personalised set of individual behaviours. In the models these behavious often have some clear meaning. For example many models of stripe and spot patterns on animals and other organisms have the $\cos$ and $\sin$ modes we have seen above. These represent either spotted or striped patterns. The video above is an example of such a model where cos and sin modes are fundamental to the formation of a spotted pattern.

1.4 The linearity property and operators

A common object throughout this course will be a linear operator $L$, which acts on (scalar) functions $u$: e.g.: \[ L =\ddy{}{t} - D\pder{}{x}{2} \] so that we can write the heat equation as: \[ Lu = \ddy{u}{t} - D\pder{u}{x}{2} = 0. \] The linearity implies the following property: \[ L\left(\sum_{n=0}^{\infty}u_n\right) = \sum_{n=0}^{\infty}Lu_n. \] So that if $L u_n=0$ for all $n$ then $L u=0$, this is the superposition principle we discussed above. The point of this, is that, like in the case of the heat equation, the $u_n$ are often simple functions, i.e. the equation $L u_n=0$ is often tractable or relatively simple. It is this principle which we shall use in following sections and the linearity property is crucial to it.

1.5 Elliptic, parabolic and hyperbolic equations.

Linear PDE’s also give us our first insight into a fundamental classification of PDE’s. Consider the general form of a second order $n$-dimensional linear equation: \[ \sum_{i,j=1}^{n}A_{ij}\partial_i \partial_j u + \sum_{i=1}^{n}B_i \partial_i u +C u + D=0. \] where $A,B,C,D$ are functions which may depend on the spatial variables $x_i,i = 1,2\dots n$. As we have seen previously if we assume sufficient regualarity then partial derivatives commute: \[ \partial_i\partial_j u = \partial_j\partial_i u \] and we can assume the matrix $A_{ij}$ is symmetric. It transpires that key properties of these equations arise from the properties of the matrix $A$, specifically its eigenvalues. Since $A$ is symmetric, its $n$ eigenvalues are real. Let $a$ be the number of zero eigenvalues and $b$ be the number of positive eigenvalues. Then we split equations into three types given the numbers $a$ and $b$:

Elliptic equations for which $a=0$ and either $b=0$ or $b=n$.
Parabolic equations for which $a>0$ and $b \leq n-1$ (which implies $\det(A)=0$).
Hyperbolic equations for which $a=0$ and $b=1,n-1$.

A stricter parabolic definition.

Many authors (including the wikipedia page) have the following stricter definition of parabolic equations: those for which $a>0$ and all non-zero eigenvlaues are either all negative or all positive. This essentially means the “part” of the equation (of the matrix $A$) which has a non zero determinant is elliptic. In this case there are a number of strong theorems one can prove about the equation. If you do PDE’s in third year you will encounter some of them.

On this course the definition I have given you above Parabolic equations for which $a>0$ and $b<n-1$ (which implies $\det(A)=0$). is the one you should use to answer questions.

Tip 1.2: Classify the following PDE \[ 3\pder{u}{x}{2} + \pder{u}{y}{2} + 4 \pder{u}{z}{2} + 4\frac{\partial^2 u}{\partial y \partial z} =0 \]

The associated matrix $A$ is \[ A = \left( \begin{array}{ccc} 3 & 0 & 0\\ 0 & 1 & 2\\ 0 & 2 & 4 \end{array} \right) \] To get the eigenvalues we solve $\det(A-\lambda I) = -\lambda(\lambda-3)(\lambda-5)=0$. So $a=1$, thus it is parabolic.

The reasons for and meaning of this classification are quite deep and beyond the scope of this course (if you’re interested then PDE’s in third year is for you). For what interests us here, these three equation types have distinct characteristics which manifest in the types of solution methods we will encounter in this course.

Try it yourself

Questions 1 and 2 of tutorial sheet one have a bunch of examples for you to try yourself.

1.5.1 Elliptic PDE’s

The classic elliptic PDE is Poisson’s equation: \[ \nabla^2 u = f({\bf x}). \] Here $\nabla^2 u = \nabla\cdot\nabla u$ (a scalar) is the Laplacian (you should confirm it is elliptic via our scheme above). The video above is a soluton of Poisson’s equation with $f = Q\delta(x,y) +Q\delta(x-x_c,x-y_c)$.

This “function” $\delta$, the delta function, is one we will come back to in chapters 3 and 4 at great length.

For now pretend $\delta(x-x_c,x-y_c)$ is a function which is $1$ at $x=x_c$, $y=y_c$ and $0$ for all other $(x,y)$ i.e. it is only non-zero at a single point. Similarly $\delta(x,x)$ is $1$ at $(0,0)$ and $0$ otherwise. So this is an equation of interacting sources of charge (of value $Q$).

We see in the video, where the sources are the centre of the red and blue circles, the location of $(x_c,y_c)$ is moved around the field lines (lines of constant $u$) initally looking like a classic dipole change shape. The crucial thing to note is that the curves are typically nice and smooth, and crucially change globally (everywhere) under a local change in the location of the source $(x_c,y_c)$. Also note it has no time dependence (it is what we call a steady state), this is another common property of these equations.

Elliptic PDE properties

These three properties

Smoothness
Global response
Time independence

are common to most elliptic PDE’s

The solution to Poisson’s equation, if we assume a domain $R = \mathbb{R}^2$ with $u(x,y) =0$ on the boundary $\partial R$ takes the form:

\[ u(x,y) = \int_{R}G(x,y,x',y')f(x,y)\rmd x'\rmd y', \tag{1.3}\] where \[ G(x,y,x,y') = \frac{1}{2\pi}\mathrm{ln}({\bf r}),\quad {\bf r} = \sqrt{(x-x')^2+(y-y')^2}. \]

The function $G(x,y,x,y')$ is called the Green’s function, in chapter 3 we will see this is the solution to the equation \[ \nabla^2 u = \delta. \] and we will see the solution to the Poisson equation with the right hand side $f$ is a version of the superposition theorem.

We see an illustration of the global nature of elliptic problems in the solution form Equation 1.3. For each point $(x,y)$, the solution is integrated over all values of $f(x',y')$ for $(x',y')\in R$.

1.5.2 Parabolic PDE’s

The prototype of a parabolic PDE is the heat equation

\[ \ddy{u}{t} = D\nabla^2 u. \] Again, I challenge you to show it is parabolic ¹ Note that the operator $\nabla^2 u$ also featured in the elliptic Poisson equation, the heat equation adds time to the picture. Indeed the heat equation with a source, \[ \ddy{u}{t} = D\nabla^2 u + f(x,y) \] is also a parabolic equation.

Three common properties of parabolic equations are:

Parabolic PDE properties

Smoothness
Global repsonse
Time dependence

We have seen above that one class of solutions to the heat equation is the sum

\[ u(x,y,t) = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_n\mathrm{e}^{-\frac{D(n^2+ m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right) \] Note that, for all modes except $n=m=0$ we see interesting local variations in behaviour are rapidly killed off by the exponential decay of the time-dependent function in the solution. In other parabolic PDE’s its not always quite so simple, but there is a general tendency for them to lead to a simplification (smoothing) of the initial condition.

Just for fun here are some more examples:

The video above is the advection diffusion equation \[ \ddy{u}{t} = D \nabla^2 u -{\bf v}\cdot\nabla u,\quad {\bf v} = V(\cos(\theta),\sin(\theta)). \] The vector field ${\bf v}$ acts to move (advect) some density $u$. Imagine a dissolved drug being spun around in a spinning glass of water, this is advection, represented by the second term on the right hand side. The first term is diffusion, which we have already seen spreads the density out. This is why we see the complex swirl pattern I create in the video smooth out to a nearly uniform rotating ring.

The video above is a solution of the following coupled PDE’s:

\[\begin{align} \ddy{u}{t} &= D_u\nabla^2 u +au + (u+cv)(u^2+v^2)\\ \ddy{v}{t} &= D_v\nabla^2 v +av + (cu-v)(u^2+v^2) \end{align}\]

Models like these are used to model electrocardiogram activity amongst other applications.

1.5.3 Hyperbolic PDE’s

The prototype PDE for the hyperbolic case is the wave equation: \[ \pder{u}{t}{2}= c^2\nabla^2 u. \] Remarkably for any functions $f(z),g(z)$, the solutions to this equation (in one spatial dimension) take the form: \[ u(x,t) = f(x-ct) + g(x+ct). \] This can be seen from the chain rule (with $z=x-ct$ and $z^*=x+ct$):

\[\begin{align} \ddy{u}{t} &= \dds{f}{z}\ddy{z}{t} + \dds{g}{z^*}\ddy{z^*}{t} = c \left(-\dds{f}{z} + \dds{g}{z^*}\right),\\ \ddy{u}{x} &= \dds{f}{z}\ddy{z}{x} + \dds{g}{z^*}\ddy{z}{x} = \dds{f}{z} + \dds{g}{z^*},\\ \pder{u}{t}{2} &= c^2\left(\deriv{f}{z}{2} + \deriv{g}{z^*}{2}\right),\\ \pder{u}{x}{2} &= \deriv{f}{z}{2} + \deriv{g}{z^*}{2}. \end{align}\]

So we see the wave equation is satisfied whatever $f$ and $g$ are. We see an example in Figure 1.2. These are called travelling waves for the following reason. The functions $f(z)$ and $g(z)$ are what one might call the master shape. Say we take a value $f(-5)$ (as illustrated) then the parameter value $x$ at which $x-ct=-5$ increases linearly with time $t$ (in the figure $c=1,t=10$ so $x-10 =5$, $x=10$), so effectively the shape $f$ is being shifted to the right at a rate $c$ and the the shape $g$ shifted to the left at a rate $c$ in the solution $u(x,t)$.

There are a number of clear properties in this solution which mark hyperbolic behaviour as distinct. First of all the fact solutions are moving locally in space at a characteristic finite speed $c$. Second, rather than being spread or simplified as in the case of the the parabolic heat equation the initial condition is in some sense maintained. Indeed remarkably one can choose for example

\[\begin{align} f(z) &= \left\{\begin{array}{cc} 0 & \mbox{if $x<-a$}\\ 1 & \mbox{if $x\geq-a$ and $x\leq a$}\\ 0 & \mbox{if $x>a$} \end{array}\right.\\ g(z) &= 0. \end{align}\]

A quick insight into chapter 4

We shall justify this later in chapter 4 where we discuss the notion of weakly differentiable solutions, for now it suffices to say it is a solution to a PDE and it is not even differentiable (in the “usual” sense)!!

It desribes a box function moving at a rate $c$.

In summary some of the key properties of hyperbolic equations are:

Solutions are time-dependent.
Solutions travel in characteristic directions at a finite speed (the behaviour is highly localised).
Solutions can be discontinous
Solutions do not (necessarily) make the inital conditions any smoother.

The last point in particular contrasts with the solution to the heat equation: \[ u(x,y,t) = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{nm}\mathrm{e}^{-D\frac{(n^2+ m^2)\pi^2 }{L^2}t}\cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{m\pi y}{L}\right) \] This is a parabolic eqution whose initial conditions should simplify. Indeed we note that all modes $n,m>0$ decay exponetially to zero. The only mode which doesn’t is the constant-in-space $n=0,m=0$ mode. So initially we can have almost any degree of complexity (lots of large $n,m$ behaviour) but the larger more complex modes are killed off quicker leaving the smaller $n,m$ and simpler modes until only a constant state is left.

As we have seen above, in the case of the wave equation, whatever complexity of $f$ and $g$ we put in remains in the system for all time.

A more dramatic simulation of the wave equation shown in the video below should make the wave like behaviour clear:

Note the effect like sunlight rippling on the surface of a lake after a (good number of) stones have been thrown in it!

One more example, again for fun:

This is the hyperbolic Brusselator model, it is a mixture of something like a pattern formation system with wave-like behaviour. Note how it looks like the wave equation initially but then almost start to thicken up trying to form periodic spot like shapes, which are buffeted by the wave motion.

1.6 What about non-linear PDE’s ?

The eagle-eyed student will note that:

We have largely discussed linear methodologies, superposition and the linear classification.
A number of the interesting examples above are non-linear but where classifed as elliptic/parabolic/hyperbolic.

So does any of this apply to non-linear PDE’s, which generally are the more realistic models? Well

The classification of the equations into elliptic, parabolic and hyperbolic types applies to non-linear models, and the general properties outlined also apply e.g. non-locality (parabolic/elliptic) vs locality (hyperbolic).
Most non-linear PDE’s have to be treated using numerical methods. A common class are spectral methods, which use a Fourier series expansion. The equations for the coefficients are more complicated (not just time dependent) but the idea of solving and equation using simpler linearly indpendent functions is an important one. In fact there are lots of spectral methods using lots of different “basis functions”, most of which are derived from the solution to linear equations.

The first point is outside the scope of this course. The second point, however, will arise in chapter 2 when we discuss the Sturm-Lioville theory.

In short the linear methods we mostly focus on in this course are a fundemental start point for the analysis of the more complex non-linear partial differential equations.

hint: the t dependence means there is an independent variable with no second order behaviour, hence and empty row in the matrix A.↩︎