In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older...

that every solution is a Pell multiple of a solution from that set. In particular, if ( u , v ) {\displaystyle (u,v)} is the fundamental solution to u 2 −...

(\mathbf {x} )\end{cases}}} The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e., Φ ( x , t ) = Φ ( x...

navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation...

particle), which is the solution of the Euler equations in two-dimensional incompressible flow. A Green's function is a fundamental solution that also satisfies...

the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function...

the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below. A fundamental solution usually has eight...

is a differential operator on Rn, is to seek first a fundamental solution, which is a solution of the equation L [ u ] = δ . {\displaystyle L[u]=\delta...

differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential...

Source: The fundamental solution due to a singular point force embedded in an Oseen flow is the Oseenlet. The closed-form fundamental solutions for the generalized...

Green's functions are studied largely from the point of view of fundamental solutions instead. Under many-body theory, the term is also used in physics...

value. The second conclusion asserts that the Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations. Note that for smooth complex-valued...

two variables that defines an integral transform Heat kernel, the fundamental solution to the heat equation on a specified domain Convolution kernel Stochastic...

electrolyte solutions. Using a Green's function, the potential at distance r from a central point charge Q (i.e., the fundamental solution) is φ ( r )...

mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary...

mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. It is one of the...

}}=A(t)\mathbf {x} } is a linear non-autonomous dynamical system in Rn with fundamental solution matrix Φ(t), Φ(0) = I, then the equilibrium point 0 is said to have...

see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related. Denote the...

origin, the Newtonian kernel Γ {\displaystyle \Gamma } which is the fundamental solution of the Laplace equation. It is named for Isaac Newton, who first...

physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector...

\cdot )\to f,&{\text{as }}t\to 0;\end{cases}}} to have a smooth fundamental solution, i.e. a real-valued function p (0, +∞) × R2d → R such that p(t, ·...

Dirac's delta function under the transformation, analogous to the fundamental solution of a partial differential operator. It is usually easier to analyze...

integral transforms and infinite series, or by employing appropriate fundamental solutions. For example, the Dirichlet problem of the heat equation on the...

In physics, the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response...

matrix. The Green's functions, or fundamental solutions, are often problematic to integrate as they are based on a solution of the system equations subject...

point force embedded in a Stokes flow. From its derivatives, other fundamental solutions can be obtained. The Stokeslet was first derived by Oseen in 1927...

variation of parameters usually involves the fundamental solution of the homogeneous problem, the infinitesimal solutions x s {\displaystyle x_{s}} then being...

law has the same mathematical form as the Heat equation and its fundamental solution is the same as the Heat kernel, except switching thermal conductivity...

w^{2}=u+v{\sqrt {(609)(7766)}},} where ( u , v ) {\displaystyle (u,v)} is the fundamental solution of the Pell equation u 2 − ( 609 ) ( 7766 ) v 2 = 1. {\displaystyle...

physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having...