136 research outputs found
Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
In this chapter, we mainly review theoretical results on inverse source
problems for diffusion equations with the Caputo time-fractional derivatives of
order . Our survey covers the following types of inverse
problems: 1. determination of time-dependent functions in interior source terms
2. determination of space-dependent functions in interior source terms 3.
determination of time-dependent functions appearing in boundary condition
Unraveling Forward and Backward Source Problems for a Nonlocal Integrodifferential Equation: A Journey through Operational Calculus for Dzherbashian-Nersesian Operator
This article primarily aims at introducing a novel operational calculus of
Mikusi\'nski's type for the Dzherbashian-Nersesian operator. Using this
calculus, we are able to derive exact solutions for the forward and backward
source problems (BSPs) of a differential equation that features
Dzherbashian-Nersesian operator in time and intertwined with nonlocal boundary
conditions. The initial condition is expressed in terms of Riemann-Liouville
integral (RLI). Solution is presented using Mittag-Leffler type functions
(MLTFs). The outcomes related to the existence and uniqueness subject to
certain conditions of regularity on the input data are established.Comment: 13 page
Some inverse source problems of determining a space dependent source in fractional-dual-phase-lag type equations
The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the formh(t)f(x)with a known functionh(t). The unknown space dependent sourcef(x)is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions onh(t)(monotonically increasing character ofh(t)was removed) in a case ofdominant parabolicbehavior. The proof technique was based on spectral analysis. Section Modified Model for tau q>tau Tshows that an analogy of Theorem 2 fordominant hyperbolicbehavior (fractional Cattaneo-Vernotte equation) is not possible
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Singular Kernel Problems in Materials with Memory
In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integro-differential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integro-differential equations. An initial boundary value problem with homogeneneous Neumann boundary conditions is studied.In recent years the interest on devising and study
new materials is growing since they are widely used in different applications
which go from rheology to bio-materials or aerospace applications.
In this framework,
there is also a growing interest in understanding the behaviour of materials with memory, here
considered. The name
of the model aims to emphasize that the behaviour of
such materials crucially depends on time not only
through the present time but also through the past history. Under the
analytical point of view, this corresponds to model problems represented by
integro-differential
equations which exhibit a kernel non local in time. This is the case of rigid
thermodynamics with memory as well as of isothermal viscoelasticity; in the two different
models the kernel represents, in turn, the heat flux relaxation function and
the relaxation modulus. In dealing with
classical materials with memory these kernels are regular function of both the present
time as wel
Energy dissipation for hereditary and energy conservation for non-local fractional wave equations
Using the method of a priori energy estimates, energy dissipation is proved
for the class of hereditary fractional wave equations, obtained through the
system of equations consisting of equation of motion, strain, and fractional
order constitutive models, that include the distributed-order constitutive law
in which the integration is performed from zero to one generalizing all linear
constitutive models of fractional and integer orders, as well as for the
thermodynamically consistent fractional Burgers models, where the orders of
fractional differentiation are up to the second order. In the case of non-local
fractional wave equations, obtained using non-local constitutive models of
Hooke- and Eringen-type in addition to the equation of motion and strain, a
priori energy estimates yield the energy conservation, with the reinterpreted
notion of the potential energy
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