530 research outputs found
Dichotomous Hamiltonians with Unbounded Entries and Solutions of Riccati Equations
An operator Riccati equation from systems theory is considered in the case
that all entries of the associated Hamiltonian are unbounded. Using a certain
dichotomy property of the Hamiltonian and its symmetry with respect to two
different indefinite inner products, we prove the existence of nonnegative and
nonpositive solutions of the Riccati equation. Moreover, conditions for the
boundedness and uniqueness of these solutions are established.Comment: 31 pages, 3 figures; proof of uniqueness of solutions added; to
appear in Journal of Evolution Equation
Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions
A new generalised derivative operator μλ1,λ2n,m is introduced. This operator generalised many well-known operators studied earlier by many authors. Using the technique of differential subordination, we will study some of the properties of differential subordination. In addition we investigate several interesting properties of the new generalised derivative operator
Generalised Fractional Evolution Equations of Caputo Type
This paper is devoted to the study of generalised time-fractional evolution
equations involving Caputo type derivatives. Using analytical methods and
probabilistic arguments we obtain well-posedness results and stochastic
representations for the solutions. These results encompass known linear and
non-linear equations from classical fractional partial differential equations
such as the time-space-fractional diffusion equation, as well as their far
reaching extensions. \\ Meaning is given to a probabilistic generalisation of
Mittag-Leffler functions.Comment: To be published in 'Chaos, Solitons & Fractals
Univalence criteria of certain integral operator
In this paper, univalence criteria of certain integral operator defined by a generalised derivative
operator is obtained
Fractional Chemotaxis Diffusion Equations
We introduce mesoscopic and macroscopic model equations of chemotaxis with
anomalous subdiffusion for modelling chemically directed transport of
biological organisms in changing chemical environments with diffusion hindered
by traps or macro-molecular crowding. The mesoscopic models are formulated
using Continuous Time Random Walk master equations and the macroscopic models
are formulated with fractional order differential equations. Different models
are proposed depending on the timing of the chemotactic forcing.
Generalizations of the models to include linear reaction dynamics are also
derived. Finally a Monte Carlo method for simulating anomalous subdiffusion
with chemotaxis is introduced and simulation results are compared with
numerical solutions of the model equations. The model equations developed here
could be used to replace Keller-Segel type equations in biological systems with
transport hindered by traps, macro-molecular crowding or other obstacles.Comment: 25page
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