Foundations

Local Lipschitz continuity ensures one local trajectory.

The matrix exponential solves a finite linear system.

A PDE becomes a solution problem only with a domain and data.

For y′=f(t)g(y) on a region where g(y)≠0, integrating dy/g(y)=f(t)dt gives an implicit solution.

For y′+p(t)y=q(t) with continuous p,q, μ(t)=exp(∫p(t)dt) transforms the equation into (μy)′=μq.

For ay″+by′+cy=0, a≠0, roots of ar²+br+c=0 give the solution modes: distinct real roots give eʳ¹ᵗ,eʳ²ᵗ; a repeated root gives eʳᵗ,teʳᵗ; conjugate α±iβ give eᵅᵗcos βt,eᵅᵗsin βt.

For a linear equation Ly=f, any one particular solution yₚ plus the general homogeneous solution y_c gives every solution y=y_c+yₚ.

In a C¹ planar autonomous system, equilibria are zeros of the vector field.

For a 2L-periodic piecewise smooth real f, aₙ=L⁻¹∫₋ᴸᴸf(t)cos(nπt/L)dt and bₙ=L⁻¹∫₋ᴸᴸf(t)sin(nπt/L)dt.

The one-sided Laplace transform is F(s)=∫₀∞e⁻ˢᵗf(t)dt.

For u_t=κu_xx on 0<x<L with zero endpoint temperatures, separation gives modes sin(nπx/L)e⁻κ⁽ⁿπ/L⁾²ᵗ.

For u_tt=c²u_xx with zero ends on [0,L], extend initial displacement f and velocity g oddly with period 2L to F,G.

For a smooth rapidly decreasing f whose derivatives also decrease faster than any power, define f̂(ξ)=∫Rⁿe⁻ⁱˣ·ξf(x)dx.

For y″+p(x)y′+q(x)y=f(x), use independent homogeneous solutions y₁,y₂ and set yₚ=u₁y₁+u₂y₂.

For −(py′)′+qy=λwy on [a,b] with homogeneous Dirichlet endpoints, real p,p′,q,w continuous, p,w>0 and q≥0, eigenvalues increase to infinity and eigenfunctions are unique up to scale.

For the stated regular Sturm–Liouville problem and continuous piecewise smooth f, coefficients cₙ=∫f yₙw dx/∫yₙ²w dx give f=Σcₙyₙ at interior points.

For continuous real 2π-periodic boundary data g, u(r,θ)=(2π)⁻¹∫₋π^π[(1−r²)/(1−2r cos(θ−α)+r²)]g(α)dα is harmonic on 0≤r<1 and extends continuously to u(1,θ)=g(θ).

For a complex 2π-periodic Riemann-integrable f, cₙ=(2π)⁻¹∫₋π^πf(x)e⁻ⁱⁿˣdx and s_N=Σ|n|≤N cₙeⁱⁿˣ satisfy ∫|f−s_N|²→0.

If Nʳ=0 and A=λI+N, then eᵗᴬ=e^(λt)Σₖ₌₀ʳ⁻¹tᵏNᵏ/k!.

For i u-t=−Delta-u−lambda|u|^alpha u on R-n, real lambda and 0<alpha<4/n, every L2 initial datum has a unique local mild solution in C([0,T];L2) intersected with L-q(0,T;L^(alpha+2)), where q=4(alpha+2)/(n alpha).

On a compact smooth Riemannian manifold without boundary, every de Rham class has a unique harmonic representative.

For the heat equation u-t−kappa Delta-u=f with zero Dirichlet data, testing by u yields a balance between the time derivative of half the squared L2 norm, gradient dissipation and forcing.

An unbounded operator is specified by a linear domain D(T) in a Hilbert space and a map T:D(T)→H.

For a closed densely defined symmetric operator T, deficiency indices n-plus and n-minus are the dimensions of ker(T*−iI) and ker(T*+iI).

A self-adjoint operator has a projection-valued spectral measure E on R and can be represented as an integral of lambda dE(lambda).

If P is a constant-coefficient elliptic differential operator of order m on R-n and u is a distribution on an open U, P-u in local H-s implies u in local H-(s+m).

A smooth elliptic operator on a compact manifold without boundary maps H^(s+m) to H-s with finite-dimensional kernel and cokernel and closed range.

For incompressible viscous flow, divergence-free initial data and pressure enforce the constraint in u-t+(u·grad)u=nu Delta-u−grad p+f.