scholarly journals On the Green function of the killed fractional Laplacian on the periodic domain

2021 ◽  
Vol 24 (5) ◽  
pp. 1629-1635
Author(s):  
Thomas Simon
Keyword(s):  
Green Function ◽  
Simple Proof ◽  
Fractional Laplacian ◽  
Contour Integration ◽  
Triple Product ◽  
Periodic Domain ◽  
Probabilistic Representation ◽  
Positive Part ◽  
Completely Monotone ◽  
The Green Function

Abstract We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

scholarly journals Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function

2021 ◽  
Vol 24 (5) ◽  
pp. 1507-1534
Author(s):  
Uyen Le ◽  
Dmitry E. Pelinovsky
Keyword(s):  
Green Function ◽  
Linear Operator ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Maximum Point ◽  
Periodic Domain ◽  
Caputo Fractional Derivatives ◽  
The Green Function

Abstract The linear operator c + (−Δ) α/2, where c > 0 and (−Δ) α/2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

Sign in / Sign up

Export Citation Format

Share Document