scholarly journals VAGO METHOD FOR THE SOLUTION OF ELLIPTIC SECOND‐ORDER BOUNDARY VALUE PROBLEMS

2010 ◽  
Vol 15 (4) ◽  
pp. 533-545
Author(s):  
Nikolay Vabishchevich ◽  
Petr Vabishchevich
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Symmetric Tensor ◽  
Second Order ◽  
Diffusion Reaction ◽  
Vector Analysis ◽  
Difference Operators ◽  
Curl Operators ◽  
Grid Operators

Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection‐diffusion‐reaction equation the diffusion coefficient is a symmetric tensor of second order.

Non-well-ordered lower and upper solutions for semilinear systems of PDEs

2021 ◽  
pp. 2150080
Author(s):  
Alessandro Fonda ◽  
Giuliano Klun ◽  
Andrea Sfecci
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Parabolic Type ◽  
Existence Results ◽  
Second Order ◽  
Lower And Upper Solutions ◽  
Semilinear Systems ◽  
Systems Of Pdes ◽  
Abstract Framework

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

scholarly journals On sampling expansions of Kramer type

2004 ◽  
Vol 2004 (5) ◽  
pp. 371-385
Author(s):  
Anthippi Poulkou
Keyword(s):  
Boundary Value Problems ◽  
Discrete Spectrum ◽  
Differential Operators ◽  
Boundary Value ◽  
Sampling Theorem ◽  
Second Order ◽  
Joint Work ◽  
First Order ◽  
Survey Paper ◽  
Sampling Expansions

We treat some recent results concerning sampling expansions of Kramer type. The linkof the sampling theorem of Whittaker-Shannon-Kotelnikov with the Kramer sampling theorem is considered and the connection of these theorems with boundary value problems is specified. Essentially, this paper surveys certain results in the field of sampling theories and linear, ordinary, first-, and second-order boundary value problems that generate Kramer analytic kernels. The investigation of the first-order problems is tackled in a joint work with Everitt. For the second-order problems, we refer to the work of Everitt and Nasri-Roudsari in their survey paper in 1999. All these problems are represented by unbounded selfadjoint differential operators on Hilbert function spaces, with a discrete spectrum which allows the introduction of the associated Kramer analytic kernel. However, for the first-order problems, the analysis of this paper is restricted to the specification of conditions under which the associated operators have a discrete spectrum.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

scholarly journals Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

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