Modeling with PDE

The introduction made the case that some phenomena cannot be captured without resolving space, and that doing so turns an ordinary differential equation into a partial differential equation (PDE). This page follows that step all the way through on a single physical example — heat transfer. We motivate where space matters, set up a concrete scenario, derive the governing PDE from a basic law of physics, attach the conditions that pin down a unique solution, and finally place the result inside a general classification of second-order PDEs.

Where Space Is Essential

Resolving space is not a niche concern; it is the rule rather than the exception across physics and continuum mechanics. Whenever the quantity of interest varies from place to place inside a body or a region, a spatially resolved model is the only honest description. A few representative fields make the breadth clear:

Fluid and thermo-mechanics. Where does a tornado rise? Is a given vehicle body aerodynamically beneficial? The flow field differs at every point around the object.
Structural mechanics. Can a building withstand mechanical stresses, and where are its weak points? Stress is a field spread over the whole structure.
Process engineering. Where, and how hot, does a nuclear, chemical, or biological reactor become? Safety hinges on the location of the hot spots, not just their average.
Electromagnetism. How high is the electron density inside a transistor? The answer is a density that varies across the device.
Geophysics. Where and when will the next earthquake strike, and how devastating will it be? The phenomenon is inseparable from where it happens.

Each question is about a quantity that lives at every point of a region — exactly the situation a PDE is built to describe.

With or Without Time?

Before committing to a PDE, there is a second question to settle: does the phenomenon also depend on time, or only on position? The answer splits problems into two families.

A phenomenon is stationary when it does not depend on time, only on position — the situation has settled into a steady pattern that no longer changes. Dissolving a soluble substance in water with no external influence tends, after a while, toward such a stationary distribution; so does the temperature in a room that is heated constantly. A useful observation: a stationary phenomenon in one spatial dimension depends on a single variable again, so its model collapses back to an ODE.

A phenomenon is instationary when it depends on both time and location — it is genuinely evolving. The oscillations of a swing-boat, turbulence (chaotic flow over a barrier, or the wake instability behind a plane), and a control loop with continuously adapted boundary conditions are all of this kind. An instationary phenomenon is unavoidably a PDE: the unknown depends on several variables at once.

The Object of Desire: Temperature

Heat transfer is the central problem of thermodynamics, and it is the example we will carry through the rest of this page. Physically, heat acts on the surface of an object, and the aim is to predict how it then propagates and distributes through the interior. The setting shows up everywhere: a hot-wire, a cooking pot on a stove, the cooling water in a nuclear reactor, a room in winter (where should the heating go?), a room in summer (sun heating through the windows).

The quantity we are after is the temperature $T$ . What makes this a spatial problem is that $T$ is not a single number but a field — its arguments depend on how many spatial dimensions the situation demands:

T(x; t) \quad\text{or}\quad T(x, y; t) \quad\text{or}\quad T(x, y, z; t)

The semicolon separates the spatial arguments from time, a reminder that space and time play different roles in the model. Two further ingredients turn this general object into a concrete, solvable problem, and both are crucial:

Boundary and initial conditions — what is happening at the edges of the object and how it started out.
Material properties — above all the thermal conductivity, which sets how readily heat moves through the medium.

The Scenario

To make everything concrete, fix a domain $\Omega \subset \mathbb{R}^3$ with boundary $\partial\Omega$ , observed over a time interval $[0, \tau]$ . The running example is a cylindrical cooking pot filled with water, standing on a circular cooktop:

The bottom is the circular cooktop (the heating element).
The content is water, open to the room at the top (no lid).
The side wall closes the cylinder.

At the beginning everything is cold — cold water, cold hotplate. Then the hotplate is switched on: it quickly reaches and exceeds room temperature, and the water eventually heats up. Two competing effects run at once — strong heating from below and cooling through the side wall and the open top into the surrounding room. The boundary $\partial\Omega$ therefore splits naturally into three parts — the bottom circle, the top circle, and the cylindrical wall between them — and we will give each its own condition once we know what kinds of conditions exist.

A Multi-Piece Model

Building the model is the standard example of a physical derivation, and the result is not one equation but a small assembly of parts. This mirrors the structure already seen for population dynamics: an equation that encodes the physics — how the different quantities are correlated — together with conditions that fix the concrete scenario the physics is playing out in. As with an ODE, the equation alone does not single out one answer; it determines a whole family of solutions, and only the scenario-fixing conditions select the one that actually occurs.

The model has two pieces, both of which have to be derived:

The PDE. It describes the interplay between how the temperature changes with respect to space (here, in three dimensions) and how it changes with respect to time. One equation suffices for each wanted function, and that equation determines the family of solutions.
The initial and boundary conditions. These pick out the unique solution. The initial condition gives the temperature field over the entire domain at the start. The boundary conditions give the temperature values and changes at the boundary, at all times.

The split between initial and boundary conditions is worth pausing on, because the two carve up the space–time picture along different axes.

The space–time picture — why one condition is “at the start,” the other “at the edge”

Think of the full problem as living on a space–time cylinder: the spatial domain $\Omega$ stacked up along the time axis $[0, \tau]$ . The solution $T$ is defined at every $(\text{point in }\Omega,\ \text{instant in }[0,\tau])$ pair inside that stack.

The initial condition is data on the bottom lid of the stack: the single instant $t = 0$ , but the whole spatial domain $\Omega$ , interior and boundary alike. It answers “what did the temperature field look like the moment we started?”
The boundary conditions are data on the side wall of the stack: only the spatial boundary $\partial\Omega$ , but for all times $t \in [0, \tau]$ . They answer “what is being done to the edges of the object throughout the whole experiment?”

So the two are not redundant and not in competition — they cover complementary faces of the same space–time region. A time-dependent heat problem genuinely needs both: the start determines where the evolution begins, the edges steer it the whole way through.

Deriving the Heat Equation

The equation we are after

The derivation will produce the following PDE, which is worth stating up front so the steps have a target.

The heat equation relates the change of a temperature field $T$ in time to the sum of its pure second derivatives in space:

\kappa\,(T_{xx} + T_{yy} + T_{zz}) = \kappa \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) = \kappa\,\Delta T = \frac{\partial T}{\partial t} = T_t

or, written shortly,

\kappa\,\Delta T = T_t.

Here $\kappa > 0$ is a material constant (the thermal diffusion coefficient), and $\Delta$ is the Laplace operator, defined by

\Delta T := T_{xx} + T_{yy} + T_{zz},

the sum of the pure second-order partial derivatives in the three spatial directions.

Before deriving anything, it pays to read what the equation already tells us. The Laplace operator $\Delta T$ measures how the temperature at a point compares with the average of its surroundings — it is the part of the model that drives diffusion, the smoothing-out of differences. The equation says the rate of change in time, $T_t$ , is proportional to that imbalance: where a point is colder than its neighborhood, $\Delta T > 0$ and the point warms; where it is hotter, it cools. Heat flows so as to flatten differences.

A particularly clean reading comes from asking what happens in a stationary situation — once the temperature has stopped changing, so that $T_t = 0$ . The equation then forces $\Delta T = 0$ : the sum of the second derivatives in the spatial directions vanishes everywhere. The steady temperature field is one with no remaining imbalance for diffusion to act on — a fact we can read straight off the equation before solving anything.

The Laplace equation (also called the potential equation) is the stationary case of the heat equation, where the temperature no longer changes in time:

\Delta T = 0.

Its solutions are the steady-state temperature fields — those in perfect diffusive balance at every point.

A balance equation from energy conservation

Derivations like this one almost always start from a basic law of physics. Here the law is the conservation of energy: energy is neither created nor destroyed, only transferred. The first task is to turn that law into a balance equation — a bookkeeping statement of the form “if our quantity of interest changes, what are the possible causes?”

Pick a small subdomain $D \subseteq \Omega$ — picture a one-cubic-centimeter cube of water inside the pot. The left-hand side of the balance will say the heat content of $D$ changes over time; the right-hand side will list the only two things that can cause that change:

Sources or sinks inside $D$ . Something in the interior of the cube produces heat (a source) or absorbs it (a sink) — imagine a tiny heater, or its opposite, sitting inside the cube.
Heat flowing across the boundary of $D$ . Heat enters or leaves the cube through its surface — the more realistic effect in our pot, where warmth seeps in from the hot region below and out through the cooler sides.

Writing this out as integrals gives the balance equation:

\frac{\partial}{\partial t} \int_D \rho\, c\, T \, dV = \int_D q \, dV + \int_{\partial D} k\, \nabla T \cdot d\mathbf{S}

Reading it term by term:

Left-hand side — the effect. To get the total heat stored in $D$ we sum the temperature over the whole cube, which is a volume integral over $D$ (with $\rho$ the density and $c$ the specific heat converting temperature into stored energy, and $dV$ the volume element). That stored heat changes in time, so we take its time derivative. This is “how the energy held in $D$ changes per unit time.”
First term on the right — internal sources and sinks. We again sample the whole interior, a volume integral over $D$ , of the source term $q$ (positive for a source, negative for a sink).
Second term on the right — flow through the surface. Heat flow across the boundary is a surface integral over $\partial D$ , sweeping every patch of the cube’s surface. The local heat flow is driven by the temperature’s spatial change, i.e. its gradient $\nabla T$ ; since the gradient is a vector, we take its inner product with the outward surface element $d\mathbf{S}$ to extract the component actually crossing the boundary, scaled by the heat conductance $k$ . The result measures how much heat leaves $D$ through its surface.

The same bookkeeping for a population — births, deaths, and migration

The structure of this balance is not special to heat; it is how any conserved (or accounted-for) quantity is tracked. Replace “heat in the cube $D$ ” with “people in a country.” The left-hand side becomes the rate of change of the country’s population. The right-hand side splits into exactly the same two causes:

an interior term — births minus deaths, the local creation and removal of people inside the borders (the analog of sources and sinks $q$ );
a boundary term — immigration minus emigration, the net flow of people across the border (the analog of the surface flux $k\,\nabla T \cdot d\mathbf{S}$ ).

Same equation, different physics. Recognizing this shape is what lets one derivation technique serve dozens of modeling problems.

From the balance to the PDE

The balance equation is awkward as written, because it mixes two kinds of integrals — a volume integral on the left and over the source term, but a surface integral for the flux. The remedy is Gauss’s divergence theorem, which converts the surface flux into a volume integral over $D$ . After moving everything to one side, the whole balance becomes a single volume integral:

\int_D \left( \rho\, c\, T_t - q - k\,\Delta T \right) dV = 0

The conversion is exactly where the gradient in the flux term turns into the Laplace operator: the divergence theorem trades $\int_{\partial D} k\,\nabla T \cdot d\mathbf{S}$ for $\int_D k\,\nabla \cdot \nabla T \, dV = \int_D k\,\Delta T \, dV$ , since the divergence of the gradient is precisely $\Delta T$ .

Now comes the decisive step. This integral vanishes — but not just for one particular cube. The subdomain $D$ was an arbitrary small piece of the pot, so the integral is zero for every choice of $D$ . When a continuous integrand integrates to zero over every subdomain, the integrand itself must be zero everywhere.

Why “integral zero” forces “integrand zero” here, but not in general

In general the implication fails: $\int_0^{2\pi} \sin x \, dx = 0$ , yet $\sin x$ is certainly not the zero function — the positive and negative parts simply cancel. A single integral being zero says nothing about the integrand.

What rescues us is the word arbitrary. We do not know merely that the integral over the whole pot is zero; we know it is zero over every subdomain $D$ , no matter how small or where placed. Suppose the continuous integrand were positive at some point. By continuity it would stay positive on a small ball around that point, and the integral over that ball would come out strictly positive — contradicting the assumption that every subdomain integrates to zero. The same argument rules out a negative value. So the integrand has nowhere left to be nonzero. (Continuity is essential: it is what lets a nonzero value at a single point spread into a whole neighborhood of nonzero values.)

Setting the integrand to zero and solving for $T_t$ gives the heat equation in its most general form, with sources:

T_t = \kappa\, \Delta T + \frac{q}{\rho\, c}, \qquad \kappa = \frac{k}{\rho\, c}

The constant $\kappa > 0$ is the thermal diffusion coefficient (or thermal diffusivity) — it bundles the material properties $k$ , $\rho$ , and $c$ into the single number that sets how fast the body equilibrates. Its presence in front of the Laplace operator is what makes $\Delta T$ describe a diffusion or balancing process.

Two important special cases follow immediately:

No external influence ( $q = 0$ ). With no internal sources or sinks, the source term drops and we recover the heat equation exactly: $T_t = \kappa\, \Delta T$ .
Stationary case ( $T_t = 0$ ). Once the field stops changing in time, we are left with the Laplace equation $\Delta T = 0$ .

The general recipe

This derivation is one instance of a recipe that recurs throughout physical modeling, and it is worth extracting in the abstract:

Start from a basic physical rule. The famous apple (gravity), conservation of mass, conservation of energy — some fundamental law that is taken as given.
Obtain a balance equation. Express the rule as a summary relationship, typically involving integrals over a region and its boundary.
Simplify using analysis. The divergence theorem is the usual suspect, turning surface terms into volume terms; then “the integral vanishes for every subdomain” upgrades to “the integrand is zero.”
Simplify by physical restrictions. Drop terms the situation rules out — for us, the absence of source terms ( $q = 0$ ).
Read off the differential equation. What remains is the desired PDE.

Recognizing this skeleton means that deriving the next physical model — fluid flow, electromagnetism, elasticity — is less a fresh invention than a re-run of these five steps with different ingredients.

Boundary and Initial Conditions

The PDE is only half the story. By itself it admits a whole family of solutions; to fix the one concrete scenario we need conditions, just as an ODE needs initial data. For a time-dependent problem in space, those conditions come in two flavors that work together: conditions in space (on the boundary) and a condition in time (at the start). Having both at once is not a contradiction — it is exactly what a problem with both spatial and temporal derivatives requires.

Types of boundary conditions

A boundary condition fixes what the solution does on $\partial\Omega$ . There are different classes of them, distinguished by what they prescribe.

A Dirichlet boundary condition prescribes the temperature value itself on the boundary, for all times:

T(t; x, y, z) = \psi \quad \text{on } \partial\Omega.

It is the simplest way to pin a function down on the boundary — you simply state what it equals there (heating or cooling held at a known value). This is given knowledge, not something the simulation computes.

It is the multidimensional form of the Dirichlet condition for an ODE boundary value problem: the same prescribe-the-value idea, now imposed across the whole surface $\partial\Omega$ rather than at the two endpoints of an interval.

A Dirichlet condition is reasonable whenever the boundary value is genuinely known — a modern cooktop whose temperature you dial in, for instance. But often the absolute value at the boundary is not something you can specify; you only know how heat moves across it. For that, a different kind of condition is needed.

A Neumann boundary condition prescribes not the value of $T$ but its normal derivative on the boundary — the directional derivative along the outward unit normal $\mathbf{n}$ :

\frac{\partial T}{\partial n}(t; x, y, z) = \nabla T(t; x, y, z) \cdot \mathbf{n} = \varphi(t; x, y, z) \quad \text{on } \partial\Omega.

This prescribes the heat flow through the boundary (or parts of it). The normal derivative is computed, as any directional derivative is, by taking the gradient and projecting it onto the normal direction $\mathbf{n}$ .

It generalizes the Neumann condition for an ODE boundary value problem, prescribing a derivative rather than a value, from the endpoints of an interval to the outward normal direction on the surface $\partial\Omega$ .

The Neumann condition has a vivid special case. When $\varphi = 0$ , there is no temperature difference driving flow across the boundary, so no heat is transported through it at all — the boundary is completely isolated. Physically: a pot whose surface obeys a zero-flux condition would be safe to touch, because no heat escapes through it.

Which condition makes physical sense is often not trivial, and answering it is part of the modeling work.

Which combinations are admissible

Not every assignment of conditions to the boundary yields a well-posed problem. The possibilities sort out as follows:

Pure Dirichlet — the temperature is given on the whole boundary. This works.
A mix of Dirichlet and Neumann — temperature prescribed on some parts of the boundary, flow prescribed on others. This works, and is what our pot will need.
Pure Neumann — flow prescribed on the entire boundary. This does not suffice for a unique solution. If $T$ is a solution, then $T + \text{const}$ has exactly the same derivatives, so it satisfies every Neumann condition just as well — yet it is a different temperature field. Physically this is the right verdict: knowing only the flow across the boundary cannot determine the absolute temperature level inside.

Alongside the boundary conditions sits the condition in time.

An initial condition prescribes the temperature over the whole domain at the start time — interior and boundary together, but only at the single instant $t = 0$ :

T(0; x, y, z) = \phi(x, y, z) \quad \text{in } \Omega.

This is the complement of a boundary condition: the boundary conditions constrain the spatial edge $\partial\Omega$ across all times, while the initial condition constrains all of $\Omega$ at the single starting time. Together they wrap the space–time region on the faces where data is needed.

Back to the scenario

With the vocabulary in place, the cooking pot’s conditions can be stated. The domain is $\Omega \subset \mathbb{R}^3$ over the time interval $[0, \tau]$ :

Initial condition. The pot starts at a uniform temperature: $T = \text{const}$ throughout the entire pot at $t = 0$ .
Boundary conditions, one per piece of $\partial\Omega$ $\partial Ω$ :
- Hotplate (bottom). It first rises in temperature, then is held constantly hot — a known value on the boundary, hence a Dirichlet condition.
- Side wall and top. These lose heat steadily to the surroundings — a prescribed flow across the boundary, hence a Neumann condition.

This is exactly the admissible Dirichlet–Neumann mix from above: a value where we control it, a flux where we only know the exchange.

How Many Spatial Dimensions?

A good model is as simple as the question allows — every extra dimension multiplies the cost of solving it, so each one has to earn its place. For the pot, this means asking honestly how many spatial dimensions the temperature field really depends on. The answer is built up by successive challenges.

One dimension? Suppose the hotplate were perfect — heating the base perfectly evenly. Then at the bottom ( $z = 0$ ) the temperature is uniform across the plane, and differences appear only as we move up through the water. The temperature would depend on height alone, and the full domain $\Omega \subset \mathbb{R}^3$ could be replaced by just $[0, 1] \times [0, \tau]$ — one spatial coordinate $z$ plus time. The catch: with permanent heating from inside and permanent cooling at the outer surface, the temperature in a horizontal plane is probably not uniform after all — it should be a little cooler near the boundary than in the middle. Homogeneity in the plane is disturbed by the boundary’s cooling.

Two dimensions? A real hotplate heats from its center, so the temperature varies both with height $z$ and with distance $r$ from the center axis. That argues for two spatial dimensions — replacing $\Omega$ with $[0, 1]^2 \times [0, \tau]$ , coordinates $(z, r)$ . But one more doubt remains: does the precise angular position around the axis truly not matter?

Three dimensions? For a broken hotplate — heating from the center but not rotationally symmetric — it does. An old cooktop with a ring of separate gas flames is hotter directly above a flame than between two of them; a modern spiral element is never perfectly uniform either. So there are directions of more intense heating, the angle $\varphi$ matters, and we need three spatial dimensions: $[0, 1]^2 \times [0, 2\pi] \times [0, \tau]$ , coordinates $(z, r, \varphi, t)$ . This $(z, r, \varphi)$ description is exactly cylindrical coordinates — height, radius, and angle. It is no more accurate than the Cartesian $(x, y, z)$ system; it is simply a more convenient, more adequate characterization of a cylindrical pot, since the geometry lines up with the coordinates.

Solving the Model

Heat transfer is a popular demonstrative example precisely because so much can be said about its solutions. General statements about existence and uniqueness are available, and for simple one-dimensional configurations the solution can even be written down explicitly — by separation of variables (Fourier’s method). Such a case is the textbook example of a simple boundary, or boundary-initial-value, problem.

In general, though, the situation is harder. Existence and uniqueness are often still open, and only a numerical solution is possible. Higher-dimensional cases — three spatial dimensions plus time — are frequently extremely laborious to solve. This is why the numerical treatment of PDEs is one of the most important topics in all of numerical simulation, and it is where the next page turns.

Classifying PDEs

Before moving on to numerics, it is worth stepping back to see what kind of equation the heat equation is. It turns out to be a simple representative of a large, well-studied family.

The general second-order linear form

A linear PDE of 2nd order in $d$ dimensions for an unknown function $T(\mathbf{x})$ , $\mathbf{x} \in \mathbb{R}^d$ , has the form

\sum_{i,j=1}^{d} a_{i,j}(\mathbf{x})\, T_{x_i x_j}(\mathbf{x}) + \sum_{i=1}^{d} a_i(\mathbf{x})\, T_{x_i}(\mathbf{x}) + a(\mathbf{x})\, T(\mathbf{x}) = f(\mathbf{x}).

Linear means only linear combinations of the derivatives appear — no products, powers, roots, or nonlinear functions of the derivatives. (Products of a derivative with a known coefficient function $a_{i,j}(\mathbf{x})$ are fine; what is forbidden is multiplying two unknown-derivative factors together.)
2nd order means the highest derivatives present are second derivatives: the equation involves the function itself ( $T$ ), its $d$ first derivatives ( $T_{x_i}$ ), and up to $d^2$ second derivatives ( $T_{x_i x_j}$ ), and nothing higher.

For the heat equation, the relevant dimension is $d = 4$ : the three spatial dimensions and time, $\mathbf{x} = (x, y, z; t)$ . To see it as an instance of the general form, write it with everything on one side,

\kappa\,\Delta T - T_t = \kappa\, T_{xx} + \kappa\, T_{yy} + \kappa\, T_{zz} - T_t = 0,

and match it against the template. The matching is what produces the coefficients.

Because the second-derivative coefficients $a_{i,j}$ carry two indices, they are naturally collected into a matrix $A = (a_{i,j})$ ; the first-derivative coefficients $a_i$ form a vector $b = (a_i)$ . Reading off the heat equation: each pure spatial second derivative $T_{xx}, T_{yy}, T_{zz}$ comes with the constant $\kappa$ , and there are no mixed second derivatives, so $A$ is diagonal with entries $\kappa, \kappa, \kappa$ — and $0$ in the time slot, because there is no $T_{tt}$ term. The only first derivative is $-T_t$ , so $b$ has a single nonzero entry, $-1$ , in the time slot. There is no term proportional to $T$ itself and nothing on the right-hand side, so $a = 0$ and $f = 0$ .

A = \begin{pmatrix} \kappa & 0 & 0 & 0 \\ 0 & \kappa & 0 & 0 \\ 0 & 0 & \kappa & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad b = \begin{pmatrix} 0 \\ 0 \\ 0 \\ -1 \end{pmatrix}, \qquad a = f = 0.

For the heat equation all of these coefficients are constant: they do not vary with $\mathbf{x}$ , which is part of what makes it such a tractable example.

Three types of PDE

The matrix $A$ of second-derivative coefficients is what classifies a second-order PDE. The type is read off from the eigenvalues of $A$ , and there are three cases.

A second-order PDE is elliptic when the coefficient matrix $A$ is positive or negative definite — that is, all its eigenvalues share one sign (all positive, or all negative).

A second-order PDE is hyperbolic when $A$ has exactly one positive and $d - 1$ negative eigenvalues, or exactly one negative and $d - 1$ positive — one eigenvalue stands against all the others.

A second-order PDE is parabolic when one eigenvalue of $A$ is zero and all the others share the same sign, and the rank of $A$ taken together with the first-derivative vector $b = (a_i)$ is maximal (full, i.e. equal to $d$ ). Equivalently: where a second derivative is missing in some variable, a first derivative in that variable must be present to restore full rank.

To check the elliptic case we do not test the definiteness of $A$ directly; we determine it with the Eigenvalue Criterion from optima, which is simpler — a symmetric matrix is positive definite exactly when all its eigenvalues are positive, and negative definite when all are negative. So the signs of the eigenvalues settle the type at a glance.

The three classic equations are the cleanest examples of each type:

Elliptic — the Laplace (potential) equation $\Delta u = 0$ . The coefficient matrix is the identity (all eigenvalues $+1$ ), which is positive definite.
Parabolic — the heat equation $\Delta u = u_t$ . As computed above, $A = \mathrm{diag}(\kappa, \kappa, \kappa, 0)$ has eigenvalues $\kappa, \kappa, \kappa, 0$ : one zero, the rest of the same sign. On its own $A$ has rank $3$ ; the missing time direction is supplied by the first-derivative term $-u_t$ (the vector $b$ ), bringing the combined rank to the full $d = 4$ . This is precisely why a parabolic equation must carry a first derivative in the variable whose second derivative is absent — here, time.
Hyperbolic — the wave equation $\Delta u = u_{tt}$ . The second time derivative $u_{tt}$ enters with the opposite sign to the spatial ones, giving one eigenvalue against the rest. This is the equation behind ripples spreading from a stone dropped in a lake, and it reappears later in the modeling of traffic.

Where the names come from — conic sections

The labels elliptic, parabolic, and hyperbolic are borrowed from the classification of conic sections. If you forget for a moment that the $T_{x_i x_j}$ are derivatives and instead read the second-order coefficients as a quadratic form in ordinary variables, the equation $\sum a_{i,j}\, x_i x_j = \text{const}$ describes a conic. A positive-definite form traces an ellipse; a form with mixed signs traces a hyperbola; a degenerate form with a vanishing direction traces a parabola. The PDE inherits the name of the conic its leading-coefficient matrix would generate — which is why the type depends only on $A$ and its eigenvalue signs.