mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The...

A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation...

occurs on a different side of the equation. A differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form...

pair of integrals Separable partial differential equation, a class of equations that can be broken down into differential equations in fewer independent...

those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent...

the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: ∇ 2 f = − k 2...

methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs)...

equations Sine-Gordon equation Sturm–Liouville theory of orthogonal polynomials and separable partial differential equations Universal differential equation...

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium...

Hamilton–Jacobi–Bellman equation from dynamic programming. The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation − ∂ S ∂ t = H...

optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation. In discrete time any multi-stage...

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its...

with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a...

logistic differential equation. When n = 0 {\displaystyle n=0} , the differential equation is linear. When n = 1 {\displaystyle n=1} , it is separable. In...

The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system.: 1–2  Its...

ordinary differential equations List of nonlinear partial differential equations List of named differential equations List of stochastic differential equations...

{e^{2}}{4\pi \varepsilon _{0}r}}\psi =E\psi } This is a separable, partial differential equation which can be solved in terms of special functions. When...

An inexact differential equation is a differential equation of the form: M ( x , y ) d x + N ( x , y ) d y = 0 {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}...

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written...

in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry....

basis with which one is working; this corresponds to a separable partial differential equation. Therefore, a key technique to understanding operators...

probabilist's Hermite polynomials are solutions of the Sturm–Liouville differential equation ( e − 1 2 x 2 u ′ ) ′ + λ e − 1 2 x 2 u = 0 , {\displaystyle \left(e^{-{\frac...

analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class...

m}(\theta ,\phi ).} The differential equation which characterises the function R ( r ) {\displaystyle R(r)} is called the radial equation. The kinetic energy...

Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications...

the kinetic theory of gases. Einstein derived a differential equation, known as a diffusion equation, for describing the probability of finding a particle...

functions) which are determined using the separable coordinate systems for the linear partial differential equation. There are many expansions in terms of...

limits is called the correspondence principle. The solution of this differential equation is given by ψ ( t ) = e − i H t / ℏ ψ ( 0 ) . {\displaystyle \psi...

orbital. The Schrödinger equation for the wavefunction of an atom with one electron is a separable partial differential equation. (This is not the case...

number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied...