58,918 research outputs found
Fractional integral equations tell us how to impose initial values in fractional differential equations
The goal of this work is to discuss how should we impose initial values in
fractional problems to ensure that they have exactly one smooth unique
solution, where smooth simply means that the solution lies in a certain
suitable space of fractional differentiability. For the sake of simplicity and
to show the fundamental ideas behind our arguments, we will do this only for
the Riemann-Liouville case of linear equations with constant coefficients.
In a few words, we study the natural consequences in fractional differential
equations of the already existing results involving existence and uniqueness
for their integral analogues, in terms of the Riemann-Liouville fractional
integral. Under this scope, we derive naturally several interesting results.
One of the most astonishing ones is that a fractional differential equation of
order with Riemann-Liouville derivatives can demand, in principle,
less initial values than to have a uniquely determined
solution. In fact, if not all the involved derivatives have the same decimal
part, the amount of conditions is given by
where is the highest order in the differential equation such that
is not an integer
Distributed order equations as boundary value problems
This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Computers and mathematics with applications and is available at www.elsevier.comThis preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin
Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction
We propose a unified approach to nonlinear modal analysis in dissipative
oscillatory systems. This approach eliminates conflicting definitions, covers
both autonomous and time-dependent systems, and provides exact mathematical
existence, uniqueness and robustness results. In this setting, a nonlinear
normal mode (NNM) is a set filled with small-amplitude recurrent motions: a
fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In
contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a
NNM, serving as the smoothest nonlinear continuation of a spectral subspace of
the linearized system along the NNM. The existence and uniqueness of SSMs turns
out to depend on a spectral quotient computed from the real part of the
spectrum of the linearized system. This quotient may well be large even for
small dissipation, thus the inclusion of damping is essential for firm
conclusions about NNMs, SSMs and the reduced-order models they yield.Comment: To appear in Nonlinear Dynamic
On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations
This paper is mainly devoted to the study of the differentiation index and
the order for quasi-regular implicit ordinary differential algebraic equation
(DAE) systems. We give an algebraic definition of the differentiation index and
prove a Jacobi-type upper bound for the sum of the order and the
differentiation index. Our techniques also enable us to obtain an alternative
proof of a combinatorial bound proposed by Jacobi for the order.
As a consequence of our approach we deduce an upper bound for the
Hilbert-Kolchin regularity and an effective ideal membership test for
quasi-regular implicit systems. Finally, we prove a theorem of existence and
uniqueness of solutions for implicit differential systems
The Nahm Pole Boundary Condition
The Nahm pole boundary condition for certain gauge theory equations in four
and five dimensions is defined by requiring that a solution should have a
specified singularity along the boundary. In the present paper, we show that
this boundary condition is elliptic and has regularity properties analogous to
more standard elliptic boundary conditions. We also establish a uniqueness
theorem for the solution of the relevant equations on a half-space with Nahm
pole boundary conditions. These results are expected to have a generalization
involving knots, with applications to the Jones polynomial and Khovanov
homology.Comment: 60 pp, minor corrections in v.
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