31,945 research outputs found
Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives
On a series of examples from the field of viscoelasticity we demonstrate that
it is possible to attribute physical meaning to initial conditions expressed in
terms of Riemann-Liouville fractional derivatives, and that it is possible to
obtain initial values for such initial conditions by appropriate measurements
or observations.Comment: LaTeX2e, 14 page
Modified Optimal Energy and Initial Memory of Fractional Continuous-Time Linear Systems
Fractional systems with Riemann-Liouville derivatives are considered. The
initial memory value problem is posed and studied. We obtain explicit steering
laws with respect to the values of the fractional integrals of the state
variables. The Gramian is generalized and steering functions between memory
values are characterized.Comment: Submitted 30/Nov/2009; Revised (major revision) 24/April/2010;
Accepted (after minor revision) 22/July/2010; for publication in Signal
Processin
Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives
In this work we study a generalized nonlocal thermistor problem with
fractional-order Riemann-Liouville derivative. Making use of fixed-point
theory, we obtain existence and uniqueness of a positive solution.Comment: Submitted 17-Jul-2011; revised 09-Oct-2011; accepted 21-Oct-2011; for
publication in the journal 'Differential Equations & Applications'
(http://dea.ele-math.com
Coupled systems of fractional equations related to sound propagation: analysis and discussion
In this note we analyse the propagation of a small density perturbation in a
one-dimensional compressible fluid by means of fractional calculus modelling,
replacing thus the ordinary time derivative with the Caputo fractional
derivative in the constitutive equations. By doing so, we embrace a vast
phenomenology, including subdiffusive, superdiffusive and also memoryless
processes like classical diffusions. From a mathematical point of view, we
study systems of coupled fractional equations, leading to fractional diffusion
equations or to equations with sequential fractional derivatives. In this
framework we also propose a method to solve partial differential equations with
sequential fractional derivatives by analysing the corresponding coupled system
of equations
Boundary Conditions for Fractional Diffusion
This paper derives physically meaningful boundary conditions for fractional
diffusion equations, using a mass balance approach. Numerical solutions are
presented, and theoretical properties are reviewed, including well-posedness
and steady state solutions. Absorbing and reflecting boundary conditions are
considered, and illustrated through several examples. Reflecting boundary
conditions involve fractional derivatives. The Caputo fractional derivative is
shown to be unsuitable for modeling fractional diffusion, since the resulting
boundary value problem is not positivity preserving
Fractional differential equations solved by using Mellin transform
In this paper, the solution of the multi-order differential equations, by
using Mellin Transform, is proposed. It is shown that the problem related to
the shift of the real part of the argument of the transformed function, arising
when the Mellin integral operates on the fractional derivatives, may be
overcame. Then, the solution may be found for any fractional differential
equation involving multi-order fractional derivatives (or integrals). The
solution is found in the Mellin domain, by solving a linear set of algebraic
equations, whose inverse transform gives the solution of the fractional
differential equation at hands.Comment: 19 pages, 2 figure
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