347 research outputs found
Semilinear ordinary differential equation coupled with distributed order fractional differential equation
System of semilinear ordinary differential equation and fractional
differential equation of distributed order is investigated and solved in a mild
and classical sense. Such a system arises as a distributed derivative model of
viscoelasticity and in the system identfica- tion theory. Also, the existence
and uniqueness of a solution to a general linear fractional differential
equation in the space of tempered distributions is given
Forced oscillations of a body attached to a viscoelastic rod of fractional derivative type
We study forced oscillations of a rod with a body attached to its free end so
that the motion of a system is described by two sets of equations, one of
integer and the other of the fractional order. To the constitutive equation we
associate a single function of complex variable that plays a key role in
finding the solution of the system and in determining its properties. This
function could be defined for a linear viscoelastic bodies of
integer/fractional derivative type
On a system of equations arising in viscoelasticity theory of fractional type
We study a system of partial differential equations with integer and
fractional derivatives arising in the study of forced oscillatory motion of a
viscoelastic rod. We propose a new approach considering a quotient of relations
appearing in the constitutive equation instead the constitutive equation
itself. Both, a rod and a body are assumed to have finite mass. The motion of a
body is assumed to be translatory. Existence and uniqueness for the
corresponding initial-boundary value problem is proved within the spaces of
functions and distributions
Nano and viscoelastic Beck's column on elastic foundation
Beck's type column on Winkler type foundation is the subject of the present
analysis. Instead of the Bernoulli-Euler model describing the rod, two
generalized models will be adopted: Eringen non-local model corresponding to
nano-rods and viscoelastic model of fractional Kelvin-Voigt type. The analysis
shows that for nano-rod, the Herrmann-Smith paradox holds while for
viscoelastic rod it does not
On a initial value problem arising in mechanics
We study initial value problem for a system consisting of an integer order
and distributed-order fractional differential equation describing forced
oscillations of a body attached to a free end of a light viscoelastic rod.
Explicit form of a solution for a class of linear viscoelastic solids is given
in terms of a convolution integral. Restrictions on storage and loss moduli
following from the Second Law of Thermodynamics play the crucial role in
establishing the form of the solution. Some previous results are shown to be
special cases of the present analysis
Space-time fractional Zener wave equation
Space-time fractional Zener wave equation, describing viscoelastic materials
obeying the time-fractional Zener model and the space-fractional strain
measure, is derived and analyzed. This model includes waves with finite speed,
as well as non-propagating disturbances. The existence and the uniqueness of
the solution to the generalized Cauchy problem are proved. Special cases are
investigated and numerical examples are presented
Distributional framework for solving fractional differential equations
We analyze solvability of a special form of distributed order fractional
differential equations within the space of tempered distributions supported by
the positive half-line
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