scholarly journals Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 464 ◽  
Author(s):  
Arsen Pskhu
Keyword(s):  
Wave Equation ◽  
Green Function ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Cylindrical Domain ◽  
Fractional Differentiation ◽  
Layer Potential ◽  
Diffusion Wave ◽  
Jump Relation ◽  
The Green Function

We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives.

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].

On the spectral properties and positivity of solutionsof a periodic boundary value problemfor a second-order functional differential equation

2020 ◽  
pp. 123-130
Author(s):  
Manuel J. Alves ◽  
Sergey M. Labovskiy
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Functional Differential Equation ◽  
Spectral Properties ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Functional Differential ◽  
The Green Function

For a functional-differential operator Lu = (1/ρ)(-(pu')' + ∫_0^l▒〖u(s)d_s r(x,s)〗) with symmetry, the completeness and orthogonality of the eigenfunctions is shown. Thepositivity conditions of the Green function of the periodic boundary value problem areobtained.

The green function of the boundary-value problem on a star-shaped graph

2013 ◽  
Vol 57 (2) ◽  
pp. 48-57 ◽  
Author(s):  
R. Ch. Kulaev
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Green Function ◽  
Shaped Graph

scholarly journals Oscillatory properties of the green function of discontinuous boundary value problem for equations of the fourth order

2015 ◽  
Author(s):  
Р.Ч. Кулаев
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Discontinuous Boundary ◽  
The Green Function ◽  
Oscillatory Properties

Работа посвящена изучению знаковых и осцилляционных свойств функции Грина разрывной краевой задачи для уравнения четвертого порядка, описывающей малые деформации системы, состоящей из двух жестко соединенных стержней, упруго подпертых в их общем конце. Получен критерий осцилляционности функции Грина. Показано, что если концы стержневой системы неподвижны, то осцилляционность функции Грина не зависит от способа закрепления концов.

scholarly journals Existence of Positive Solutions for m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Wafa M. Shammakh
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential ◽  
The Green Function

We investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

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