Glossary

A rule that turns one function into another, usually by differentiating and combining it -- the function-space analogue of a matrix.

The physics of sound -- pressure waves traveling through a medium such as air or a pipe of gas.

The input a function is applied to -- the x in f(x) -- not a logical argument or a proof step.

The bounded solution of Bessel's equation -- the radial ODE that falls out of separating a polar eigenvalue problem.

A constraint the solution (or its derivative) must satisfy at the edge of the spatial region.

A simplified standard form for a second-order PDE, reached by changing to coordinates built from its characteristic curves.

The familiar (x, y, z) grid of coordinates, measured along fixed perpendicular axes.

The problem of finding a solution to a first-order PDE that matches prescribed data along a given initial curve.

A curve along which a first-order PDE reduces to an ordinary differential equation for u.

A fixed ratio between a wave's two paired quantities -- voltage and current on a transmission line, or pressure and velocity in acoustics -- that depends only on the medium, not on the particular wave passing through.

A number or known function multiplying a variable or an unknown term, rather than the variable itself.

The full Fourier series rewritten with complex exponentials in place of separate sine and cosine terms.

A statement that a quantity's rate of change inside a region equals the net rate it flows across that region's boundary, plus any source.

A way of combining two functions by sliding one across the other and integrating their product at every offset -- the operation that turns a heat kernel into a solution formula.

One of the numbers that pin down a point's location, together with the labeled axes or directions those numbers are measured along.

How much of a quantity -- mass, charge, or a heat source -- is packed into each unit of length, area, or volume.

The instantaneous rate of change of a function -- how fast its output changes per unit change in an input.

The number of independent directions or coordinates a space needs -- a line is one-dimensional, a plane two-dimensional, ordinary space three-dimensional.

A boundary condition that prescribes the value of u itself on the boundary.

The quantity D = B^2 - 4AC, built from a second-order linear PDE's principal-part coefficients, whose sign determines the equation's type.

The interval of initial data that determines a wave solution's value at a single space-time point.

A method for solving a forced (source-term) evolution equation by treating the source as a continuous stream of fresh initial conditions.

Writing a function as an infinite series in the eigenfunctions of a Sturm-Liouville problem, with each coefficient found the same way a Fourier coefficient is.

A scalar (the eigenvalue) and a nonzero vector (the eigenvector) such that a matrix or operator acting on the vector just rescales it by that scalar, without changing its direction.

The classification for a PDE whose discriminant D = B^2 - 4AC is negative, e.g. Laplace's equation.

The quantity E(t), combining kinetic and potential energy, tracked as the string vibrates.

A proof technique showing the energy of the difference of two candidate solutions is constant and starts at zero, forcing the two solutions to coincide.

An extra rule, borrowed from thermodynamics, that picks out the one physically correct weak solution when a conservation law's shock has more than one mathematically valid jump.

Extending a function given on [0, L] to [-L, L] by mirroring it across x = 0.

An explicit formula computes the next unknown directly from already-known values; an implicit one defines the next unknown only as the solution of an equation (or system of equations) that must still be solved.

A function of the form e to the power of something -- growth or decay (or, with an imaginary exponent, rotation) at a rate proportional to the function's own current value.

A series representation of a function on [0, L] using a constant term plus cosines of integer multiples of pi*x/L.

A series representation of a function on [0, L] using only sines of integer multiples of pi*x/L.

A transform turning a function of x into a function of a frequency variable ξ, replacing x-differentiation with multiplication by iξ.

Heat flux is proportional to the negative gradient of temperature -- heat flows from hot toward cold.

How many complete oscillations a periodic wave or signal makes per unit time (or, for angular frequency, per unit time in radians).

A series representation of a function on [-L, L] using sines and cosines together.

A fixed-size overshoot that a Fourier partial sum makes right next to a jump discontinuity, no matter how many terms are added.

The PDE u_t = ku_{xx} (or its higher-dimensional analog) modeling how a quantity spreads out by diffusion over time.

The whole-line heat equation's solution for initial data concentrated at a single point -- a spreading Gaussian.

A linear PDE whose every term contains the unknown function or one of its derivatives -- no term depends only on the independent variables.

The sharp difference in how a wave's influence fades, depending only on the parity of the spatial dimension.

The classification for a PDE whose discriminant D = B^2 - 4AC is positive, e.g. the wave equation.

An equation that is true for every allowed value of its variables, not just for some special values -- a restatement, not a fact to solve for.

The variable you are free to choose (independent, like x or t) versus the variable whose value that choice determines (dependent, like u).

A linear PDE with a nonzero term depending only on the independent variables -- a forcing or source term.

A constraint that specifies the solution (and, for a second-order-in-time PDE, its time derivative) at the starting time.

A way of pairing two functions to produce a single number, built from an integral instead of a finite sum.

A running total built by adding up a quantity's contribution over every point of an interval, area, or volume.

The points of a region that are not on its edge or boundary -- every point close enough to still have room on all sides.

An unbroken stretch of the number line between two endpoints, such as all the numbers from a to b.

The 3D wave equation's closed-form solution, built from the initial data's average over a shrinking sphere rather than a single point.

The PDE \nabla^2u=0, satisfied by a quantity in steady state with no sources.

The polynomial solution of Legendre's equation -- the polar-angle ODE that falls out of separating a spherical eigenvalue problem, in the axisymmetric case.

The solid cone in space-time swept out by every point a single source can ever affect, once the wave equation's speed limit is extended to three spatial dimensions.

A sum of building-block pieces, each multiplied by its own constant -- c1 * v1 + c2 * v2 + ... -- with no other operation involved.

A PDE where the unknown function and its partial derivatives combine only by addition and multiplication by known functions of the independent variables.

A sequence of partial sums converging to a function in the sense that the average squared difference between them shrinks to zero, even where they still disagree pointwise.

Recovering a lower-dimensional wave-equation solution formula by treating its data as a special, direction-independent case of the higher-dimensional formula.

A technique for building a Green's function on a half-space or other simple region by adding a fictitious mirror-image source outside the region, chosen to force the boundary condition automatically.

Mass times velocity -- the physical quantity a fluid's own equation of motion balances when a force acts on it.

A boundary condition that prescribes the value of a directional derivative of u on the boundary, rather than u itself.

A curve inside a vibrating membrane that stays motionless -- displacement zero for all time -- for one particular mode.

A PDE where the unknown function or one of its derivatives is multiplied by another copy of itself, raised to a power, or fed through a nonlinear function.

One term in a separated series solution -- a single fixed spatial shape oscillating in time at a single natural frequency.

Rescaling a function or vector by a constant factor so a chosen measure of its size -- often its integral or norm -- comes out to exactly 1.

A way of computing an approximate answer with arithmetic on a grid of numbers, used when no exact formula for the solution exists.

Extending a function given on [0, L] to [-L, L] by mirroring it across x = 0 and flipping its sign.

The highest order of any partial derivative appearing in the equation.

Two functions whose inner product is zero.

The classification for a PDE whose discriminant D = B^2 - 4AC is exactly zero, e.g. the heat equation.

An identity equating a function's mean-square average to the sum of the squares of its Fourier coefficients.

An equation relating an unknown function of two or more variables to some of its partial derivatives.

Extending a function given on one interval to the entire real line by repeating it over and over.

A small change or disturbance added to a known solution or a piece of data, small enough that only its first-order effect matters.

What a Fourier series adds up to at one specific x -- the function's own value where f is continuous, and the average of the two one-sided limits at a jump.

A function P_r(\psi) that reproduces the disk-series solution of Laplace's equation as a single integral against the boundary data, instead of an infinite sum.

A scalar function whose rate of change in each direction gives a physical field's strength there -- for electrostatics, the function whose gradient is the electric field.

The collection of a PDE's highest-order derivative terms -- the only terms the discriminant test looks at.

An expression or equation where the unknown appears to the second power (squared) and no higher -- ax^2 + bx + c.

A first-order PDE whose coefficients on the derivatives may depend on u itself, while the derivatives still appear only to the first power.

Pointing straight out from a fixed center or axis -- the direction the coordinate r measures in polar or spherical coordinates.

The set of future space-time points a single piece of initial data at one location can possibly affect.

The equation forcing a shock's speed to equal the jump in flux divided by the jump in the conserved quantity.

A spreading fan of characteristics, rather than a sharp jump, that forms when a conservation law's initial data calls for characteristics to spread apart instead of collide.

A ratio of integrals, built from a trial function, whose value equals a Sturm-Liouville problem's eigenvalue exactly when the trial function is an eigenfunction.

A function chosen to satisfy given nonhomogeneous boundary data on its own, subtracted off to reduce the remaining problem to homogeneous boundary conditions.

How a traveling disturbance changes sign, or doesn't, when it reaches a fixed or free end and heads back the other way.

A forced vibration's amplitude growing without bound because the forcing frequency matches one of the system's own natural frequencies.

A combination of a system's unknowns that stays constant along a specific family of curves, even though each unknown alone changes.

A boundary condition that prescribes a fixed linear combination of u and its normal derivative on the boundary.

A trial solution u(x,t) = X(x)T(t) -- a product of a function of x alone and a function of t alone -- substituted into a PDE to split it into two ordinary differential equations.

The single constant that a separated PDE's x-dependent side and t-dependent side must both equal, since one depends only on x and the other only on t.

A discontinuity that forms in a conservation law's solution where characteristic curves carrying different values cross.

A Sturm-Liouville problem where p or sigma vanishes at an endpoint, or the interval is unbounded -- the regular problem's hypotheses fail, even though a version of orthogonality often still holds.

A necessary condition boundary data must satisfy before a Neumann problem for Laplace's equation has any solution at all.

An eigenfunction on the sphere, built by combining an azimuthal exponential with an associated Legendre factor.

A solution that no longer changes with time -- every time derivative is zero.

The fact that eigenfunctions of a regular Sturm-Liouville problem belonging to distinct eigenvalues are orthogonal, once the weight function is included.

A boundary value problem of the form d/dx[p(x)y'] + q(x)y + lambda*sigma(x)y = 0 together with boundary conditions, whose nontrivial solutions exist only for special values of lambda.

For a homogeneous linear PDE, any linear combination of solutions is itself a solution.

A property or expression stays exactly the same after a specific change -- swapping two variables, reflecting a domain, or rotating it -- and that unchanged-ness is often enough to prove a result with no extra computation.

Differentiating (or integrating) a Fourier series one term at a time, which is not always valid the way it is for a finite sum.

A physical line (a wire pair, or a pipe carrying gas) along which voltage-and-current (or pressure-and-velocity) waves travel, governed by the same hyperbolic system as other paired-wave problems.

The requirement that the initial curve nowhere runs tangent to a characteristic curve -- needed for the Cauchy problem to have a unique solution near that curve.

A solution whose entire shape moves at a constant speed without changing form -- a function of x minus (speed times t) alone.

The sine, cosine, and related functions built from a right triangle's side ratios, or equivalently from motion around a circle.

The rate of change of position with respect to time, including direction -- not just how fast something moves, but which way.

The PDE u_{tt} = c^2u_{xx} (or its higher-dimensional analog) modeling a disturbance that propagates at a fixed speed c.

The function sigma(x) multiplying lambda*y in a Sturm-Liouville equation, which the eigenfunctions turn out to be orthogonal with respect to.

A problem with a solution, only one solution, and a solution that changes only a little when the given data changes only a little.