In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations...

method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these...

)},} for each respective substitution. Both may be solved via Separation of Variables. A linear differential equation is homogeneous if it is a homogeneous...

be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon...

only on space. Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts Ψ ( r , t...

time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example...

problem. When the problem allows additive separation of variables, the HJE leads directly to constants of motion. For example, the time t can be separated...

mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian...

single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions....

coordinates. The 3-variable Laplace equation ∇ 2 Φ = 0 {\displaystyle \nabla ^{2}\Phi =0} admits solution via separation of variables in toroidal coordinates...

general) Separation of variables, in mathematics to solve certain (separable) differential equations Separation principle, in control theory Separation process...

^{2}}}=0.} Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result...

functions u = u (x, y, z, t) of a time variable t (a variable representing time) and one or more spatial variables x, y, z (variables representing a position...

"Separation of church and state" is a metaphor paraphrased from Thomas Jefferson and used by others in discussions of the Establishment Clause and Free...

equations of lower dimension, preferably ordinary differential equations, which can often be solved exactly. This can sometimes be done using separation of variables...

used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions...

mass of the string. This problem is amenable to the method of separation of variables. If we assume that h(x, t) can be written as the product of the form...

ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear...

(differential geometry), a type of set allowing an intrinsic Riemannian-geometry characterisation of the additive separation of variables in the Hamilton–Jacobi...

three-variable Laplace operator, one can for instance expand it in the rotationally invariant coordinate systems which allow separation of variables. For...

form may be more useful, depending on application. Performing a separation of variables will give ∫ y ( 0 ) y ( t ) d y 6 A 5 y 5 / 3 + C 0 = t {\displaystyle...

Zbl 0671.15005 Wolsson, Kenneth (1989b), "Linear dependence of a function set of m variables with vanishing generalized Wronskians", Linear Algebra and...

a first order ordinary differential equation and solves it by separation of variables. Both steps may be difficult or even impossible. In such cases...

with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either...

equation may be solved exactly using separation of variables or a similarity solution. However, the separation of variables solution is known to blow up to...

homogeneous by subtracting the steady solution. Then, applying separation of variables leads to the solution: u ( y , t ) = U y h − 2 U π ∑ n = 1 ∞ 1...

= − x 2 {\displaystyle y'y^{-2}-{\frac {2}{x}}y^{-1}=-x^{2}} Changing variables gives the equations u = 1 y ,   u ′ = − y ′ y 2 − u ′ − 2 x u = − x 2...

associative algebras of the notion of a separable field extension Separable differential equation, in which separation of variables is achieved by various...

solution of Laplace's equation of the static potential, ∇2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where...

Neumann boundary condition Stefan problem Wiener–Hopf problem Separation of variables Green's function Elliptic partial differential equation Singular...