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The heat equation and its homogeneous Dirichlet boundary conditions are linear, so superposition of the separated modes givesThe Fourier sine basis represents the initial profile on the interior of the interval. At $t=0$ its Fourier sine series must equal $x(1-x)$. Multiplying by $\sin (m\pi x/L)$, integrating from $0$ to $L$, and using orthogonality givesTherefore the required Fourier sine coefficients areFor every $t>0$, the exponential factors ensure convergence of the differentiated series, and every term satisfies the partial differential equation and the boundary conditions. The Fourier sine series recovers the smooth initial profile at every $0<x<L$ as $t\downarrow 0$.

The scan explicitly prints $x(1-x)$ on $(0,L)$; it must not silently become $x(L-x)$. For $L\ne 1$, the interior profile tends to $L(1-L)\ne 0$ at the right endpoint, whereas the prescribed boundary condition is zero for $t>0$. The solution is therefore not continuous at the corner $(L,0)$, although it satisfies every condition on the printed open ranges. A solution continuous on the closed space-time domain with these data would require $L=1$.^[ref]

For a nonzero separable solution $u=XT$, separation of variables givesThese Dirichlet boundary conditions follow because $T$ is not identically zero. If $\kappa =0$, the affine function $X=A+Bx$ obeys both conditions only when $A=B=0$. If $\kappa =-{\rho }^2<0$, $X=A\cosh (\rho x)+B\sinh (\rho x)$ also has only the zero solution under both conditions.

For $\kappa ={\omega }^2>0$, $X=A\cos (\omega x)+B\sin (\omega x)$. The first boundary condition forces $A=0$, and the second forces $\omega L=n\pi$ for a nonzero solution. Hence all nonzero spatial modes areThe identically zero function is also a solution. Allowing $B_n=0$ includes it in the displayed family. Here $L>0$ is the interval length and $n$ labels the eigenvalues of the operator $-d^2/d{x}^2$ with these boundary conditions.

The absorbing states give $e_0=e_n=0$. Multiplying the difference equation in (i) by $3$ givesThe second finite difference of $k^2$ is $2$, so a particular solution is $-3k^2/2$; the homogeneous solutions are $A+Bk$. Imposing the two boundary conditions gives $A=0,B=3n/2$. ThereforeUniqueness follows because the difference of any two solutions has zero second finite difference, hence is affine function, and vanishes at both endpoints. This is the mean absorption time of a lazy symmetric random walk with holding probability $1/3$.

The endpoint boundary conditions areThe finite difference equation can be written $y_{n+1}-2y_n+y_{n-1}=2h^2$. Set $d_n=y_n-y_{n-1}$. Then $d_{n+1}-d_n=2h^2$, so $d_n=d_1+2h^2(n-1)$. Summing from $1$ to $n$ givesUsing $y_N=5$ and $h=2/N$ yields $N{d}_1=4-h^{2}N(N-1)=4/N$, hence $d_1=4/N^2=h^2$. ThereforeThis also covers $N=1$, where there are no interior equations. Uniqueness follows because the difference of two solutions has zero second finite difference, hence is linear in $n$, and vanishes at both endpoints. This is an exact finite difference solution for constant second derivative: the method is exact at the nodes here because the centered second finite difference differentiates this quadratic polynomial exactly.

For $y=1+x^2$, the derivatives are $y^{\prime }=2x$ and $y^{\prime \prime }=2$. Its endpoint values are $y(0)=1$ and $y(2)=5$, so it satisfies both the ordinary differential equation and the boundary conditions. Moreover any solution of $y^{\prime \prime }=2$ has the form $x^2+Ax+B$; the boundary conditions force $B=1$ and $A=0$. Thus the unique solution of the boundary value problem is

Put $a=\pi /2$. Substituting $x=-a$ and $x=a$ into part (ii), using $\cos (\pi /4)=\sin (\pi /4)=1/\sqrt{2}$, givesThus $A-B=\sqrt{2}(1+e^{-a})$ and $A+B=\sqrt{2}(1-e^a)$. Solving these two linear equations givesConsequently the solution of the boundary value problem isThe two independent boundary conditions determine $A,B$ uniquely.

A Fourier sine series automatically meets the zero-temperature boundary conditions because $\sin 0=\sin (n\pi )=0$. Equivalently, extend ${\Theta }_0$ oddly from $(0,L)$ to $(-L,L)$ and then periodically with period $2L$. Its Fourier series has only sine terms, withThe coefficient formula follows from the orthogonality identityFor piecewise smooth initial data the series equals ${\Theta }_0$ at interior continuity points and the average of its one-sided values at interior jumps. More general square-integrable data have convergence in L2 norm convergence. A classical solution continuous at the initial corners additionally requires ${\Theta }_0(0)={\Theta }_0(L)=0$, in agreement with the prescribed boundary conditions.

The wave equation is second order in time. Even one normal mode obeys a second-order ordinary differential equation, so its initial displacement and initial velocity are independent data. Physically they specify stored elastic deformation and the initial motion.

The heat equation is first order in time; its initial temperature already fixes the instantaneous time derivative through the spatial Laplacian. It needs one initial field. In either case spatial boundary conditions remain necessary and are not substitutes for initial data. Classical fixed-end string data must satisfy the appropriate endpoint compatibility, including $f(0)=f(L)=0$ and, when endpoint velocities exist, $g(0)=g(L)=0$.

For constant stratification and $U=U_0+\Lambda z$, a decaying zero-interior-PV mode has vertical structure $e^{-Kz}$. Its topographic boundary condition gives $A^{\prime }+ik(U_0+\Lambda /K)A=ikϵN^2{U}_0/(Kf)$. Resonance requires $K{U}_0+\Lambda =0$ and nonzero forcing $kϵU_0$. With zero initial disturbance, $A$ grows linearly and is imaginary relative to cosine topography. Its meridional velocity is therefore opposite in phase to the relief for positive $f,U_0,ϵ$. A zero zonal wavenumber or zero boundary current removes the forcing and invalidates an unconditional claim of secular growth.

A designed spatial arrangement of dielectric or other electromagnetic material controls optical propagation, confinement or emission. Examples include waveguides, distributed Bragg reflectors, photonic crystals and defect cavities. Spatial interference and mode boundary conditions act on electromagnetic states; they are distinct from the electronic band structure of the material itself.

A surface displacement constrained to vanish on all four sides of an ideal square has sine factors $\sin (n\pi x/L)\sin (m\pi y/L)$, with $n,m\ge 1$. The fundamental therefore has wavenumber $\sqrt{2}\pi /L$. This differs from the impermeable-wall condition in a closed tank, which permits cosine modes with a zero index. For an elastic plate a simply supported edge also supplies its bending boundary condition; clamping imposes a different problem.

A plasma oscillation is a collective oscillation of charge density and its restoring electric field. In a three-dimensional bulk charged superfluid, long-range Coulomb forces give the longitudinal collective mode a nonzero long-wave plasma frequency. Different geometry and electromagnetic boundary conditions can change that limit.

Invariance under spacetime translations and Lorentz transformations. A scalar action with constant coefficients and contracted Lorentz indices has this symmetry when its domain and boundary conditions also permit the transformations.

Let $Y_{\ell }$ be a spherical harmonic of degree $\ell$ on the unit sphere. The Laplacian in spherical coordinates separates its angular part from a radial power, giving the displayed identity for $r>0$. Thus a Poisson equation $\Delta \Phi =r^q{Y}_{\ell }$ has the particular solution $\Phi =r^{q+2}{Y}_{\ell }/[(q+2)(q+3)-\ell (\ell +1)]$ when the denominator is nonzero. Add $A{r}^{\ell }{Y}_{\ell }+B{r}^{-\ell -1}{Y}_{\ell }$ to impose the radial boundary conditions. If the denominator vanishes at $p=q+2$, differentiating the displayed identity with respect to $p$ gives instead $\Delta (r^p\log r{Y}_{\ell })=(2p+1)r^{p-2}{Y}_{\ell }$. Regularity at the origin and decay at infinity must be checked separately; the radial identity alone is local away from zero.

For field functionals, a Poisson operator $J$ is an operator whose displayed pairing of variational derivatives defines a Poisson bracket. It must be skew under the integral pairing and satisfy the functional Jacobi identity. A constant skew differential operator, such as $J={\partial }_x$ with periodic or vanishing boundary conditions, gives the simplest examples. The Hamiltonian evolution is $u_t=J(\delta H/\delta u)$.

Potential-vorticity inversion recovers a balanced velocity field from potential vorticity and specified boundary and global data. In two-dimensional vortex dynamics this is a Poisson equation; shallow-water quasi-geostrophic inversion and stratified quasi-geostrophic inversion use different elliptic boundary value problems. Advance the materially conserved scalar, then invert again to recover the evolving flow. Boundary buoyancy can induce flow with zero interior anomaly. The balance relation and boundary conditions are essential: unrestricted gravity waves are additional degrees of freedom. McIntyre's potential-vorticity account explains the general invertibility principle.

For real wave impedances at normal incidence, the electric-field transmission coefficient $t$ gives transmitted power fraction $T=(Z_1/Z_2)|t{|}^2$. Reflection gives $R=|r{|}^2$. Enforcing both tangential electric and magnetic boundary conditions gives $r=(Z_2-Z_1)/(Z_2+Z_1)$ and $t=2Z_2/(Z_1+Z_2)$, which satisfy $R+T=1$.

Consider assemblies of congruent polygonal tiles with same-labelled faces paired by reflection and pure Neumann boundary conditions on unpaired faces. An invertible intertwiner for all face permutation matrices preserves the Euler characteristic. The number of tiles is the dimension of the representation. For each face label the number of edge classes is $(n+\mathrm{tr}{M}_s)/2$. For each vertex label the number of vertex classes is the number of orbits of the subgroup generated by incident-face involutions, hence the dimension of its invariant vectors. Intertwining preserves all these numbers. Summing the vertex and edge counts proves equal $V-E+F$, including boundary corners. This argument uses genuine permutation matrices; signed matrices encoding mixed boundary conditions do not give the same topological count.

Quantum mechanics describes a quantum system using a quantum state, with observables represented by self-adjoint operators. For a single particle in one dimension, a pure state can be described by a wavefunction $\psi (x,t)$: its squared modulus gives the probability density for position. The Schrödinger equation predicts how this probability amplitude evolves between interventions; measurement in quantum mechanics connects it to experimental outcomes through the Born rule. A wavefunction therefore predicts a distribution of possible positions rather than a single trajectory as in classical mechanics.

The infinite square well makes three central ideas visible with elementary sine functions. Confinement selects discrete energy eigenvalues; wavefunction normalization converts a shape into quantitative probabilities; and a quantum superposition can have a changing probability density even when each constituent energy eigenstate has a stationary one. This combination explains why quantum mechanics uses both linear algebra and differential equations: eigenfunctions describe the building blocks, while quantum phases determine how they interfere. The same tools help explain the discrete energies of chemical atoms and molecules, although their Quantum Hamiltonians require different potentials and often many particles.

In his first 1926 paper, Erwin Schrödinger formulated quantization as an eigenvalue problem for a hydrogen atom in nonrelativistic quantum mechanics. On its opening page he compared the emergence of integers with the number of nodes of a standing wave on a vibrating string. The infinite square well below realizes that appealing idea especially transparently: the allowed integers follow from the boundary conditions. This historical analogy concerns the origin of discrete energies; the Born rule supplies the statistical interpretation of the modern wavefunction. ^[ref].

When the free angular-momentum phases all recur at the driving period, successive identical kicks add coherently. For the full revival $\hslash T/(2I)=2\pi$ and an initial zero-angular-momentum state, the kinetic energy after $N$ kicks is ${\hslash }^2{\eta }^2{N}^2/(4I)$. This ballistic law is specific to the stated resonance and boundary conditions.