Laplace Operator


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Related to Laplace Operator: divergence, Laplace transform, Nabla operator

Laplace operator

[lə′pläs ‚äp·ə‚rād·ər]
(mathematics)
The linear operator defined on differentiable functions which gives for each function the sum of all its nonmixed second partial derivatives. Also known as Laplacian.

Laplace Operator

 

(also Laplacian), a linear differential operator, which associates to the function Φ(x1, x2, . . ., xn) of η variables x1, x2, . . ., xn the function

In particular, if Φ = Φ (x, y) is a function of two variables x, y, then the Laplace operator has the form

and if Φ= Φ (x) is a function of one variable, then the Laplacian of Φ coincides with the second derivative, that is,

The Laplace operator is encountered in those problems of mathematical physics where the properties of an isotropic homogeneous medium (for example, the propagation of light, heat flow, the motion of an ideal incompressible fluid) are studied.

The equation ΔΦ = 0 is usually called the Laplace equation and hence the name Laplace operator.

References in periodicals archive ?
Using property (8), we can define the fractional Laplace operator by the equation
where [DELTA] is the Laplace operator, [r.sub.s] is a fixed point in the open domain [OMEGA], [delta](r) is the Dirac delta function, [partial derivative]/[partial derivative]n is the outward normal derivative on the surface S, and [g.sub.S](r) is a given function specified on the surface S satisfying the constraint
Holder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to derive the stochastically exponential stability criterion of the complex system.
We shall suppose for simplicity that [DELTA] is the self--adjoint realization of the Laplace operator on [L.sup.2](0, 2[pi]) with periodicity condition
By using the standard relation between the Dirichlet Laplacian on [R.sub.+] and the restriction of the Laplace operator -[DELTA] on [R.sup.3] to spherically symmetric functions (see for example [2, Sec.
where [Y.sub.U](s) is dynamic mobility, [V.sub.U](s) is admittance, s is variable determined as the Laplace operator, described by the formula s = i[omega], [Z.sub.U](s) is mechanical impedance, and [U.sub.U](s) is dynamic stiffness.
In this work, we will focus our attention on some special aspects of controllability problems for parabolic system involving Laplace operator with differend type of observation.
When we consider using all 27 sampling fields in the basic cube, the discrete Laplace operator can be derived in a similar way as the second-order accurate seven-point formula [18].
Keywords: Self adjoiness, Hamiltonian operator, potential, Hilbert space, Norm, Laplace Operator.
where R(s) and P(s) are relatively right prime (rrp) polynomial matrices in the Laplace operator s with P(s) column proper.
Drawn from their lectures given at the May 2005 summer school, these three expository articles connect the basic knowledge of a graduate student in mathematics with isoperimetric inequalities for eigenvalues of the Laplace operator, random Schrodinger operators, and the stability of matter in terms of quantum mechanics and quantum electrodynamics.
We shall use a number of properties of spherical harmonics and of eigenfunctions of the Laplace operator on the unit ball.