52,081 research outputs found
Computation of Mixed Type Functional Differential Boundary Value Problems
This is the published version, also available here: http://dx.doi.org/10.1137/040603425.We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model, and Frenkel--Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals
Analytical and numerical investigation of mixed-type functional differential equations
NOTICE: this is the authorās version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234 (2010), doi: 10.1016/j.cam.2010.01.028This journal article is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments
Mixed-type functional differential equations: A numerical approach
This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses mixed-type functional equations
The numerical solution of forwardābackward differential equations: Decomposition and related issues
NOTICE: this is the authorās version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234,(2010), doi: 10.1016/j.cam.2010.01.039This journal article discusses the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of āforwardā solutions and ābackwardā solutions
On the Volterra property of a boundary problem with integral gluing condition for mixed parabolic-hyperbolic equation
In the present work we consider a boundary value problem with gluing
conditions of integral form for parabolic-hyperbolic type equation. We prove
that the considered problem has the Volterra property. The main tools used in
the work are related to the method of the integral equations and functional
analysis.Comment: 18 page
DOLFIN: Automated Finite Element Computing
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of
computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code
An Integral Spectral Representation of the Massive Dirac Propagator in the Kerr Geometry in Eddington-Finkelstein-type Coordinates
We consider the massive Dirac equation in the non-extreme Kerr geometry in
horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive
a functional analytic integral representation of the associated propagator
using the spectral theorem for unbounded self-adjoint operators, Stone's
formula, and quantities arising in the analysis of Chandrasekhar's separation
of variables. This integral representation describes the dynamics of Dirac
particles outside and across the event horizon, up to the Cauchy horizon. In
the derivation, we first write the Dirac equation in Hamiltonian form and show
the essential self-adjointness of the Hamiltonian. For the latter purpose, as
the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon,
we cannot use standard elliptic methods of proof. Instead, we employ a new,
general method for mixed initial-boundary value problems that combines results
from the theory of symmetric hyperbolic systems with near-boundary elliptic
methods. In this regard and since the time evolution may not be unitary because
of Dirac particles impinging on the ring singularity, we also impose a suitable
Dirichlet-type boundary condition on a time-like inner hypersurface placed
inside the Cauchy horizon, which has no effect on the dynamics outside the
Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the
projector onto a finite-dimensional, invariant spectral eigenspace of the
angular operator and the radial Green's matrix stemming from Chandrasekhar's
separation of variables. Applying Stone's formula to the spectral measure of
the Hamiltonian in the spectral decomposition of the Dirac propagator, that is,
by expressing the spectral measure in terms of this resolvent, we obtain an
explicit integral representation of the propagator.Comment: 31 pages, 1 figure, details added, references added, minor
correction
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