Département de mathématiques
MASTER
Domain : Mathematics and informatics
Faculty : Mathematics
Speciality: Functional Analysis
Fractional Power of Linear Operator
The
study of fractional powers of operators has a rich history.However, it is only
currently that the general theory was developed. The fractional powers of
closed linear operators were first constructed by Bochner and afterwards Feller, for the Laplacian operator.These
constructions relyon the fact that theLaplacian generates a semigroup.When A is the negative
of the infinitesimal generator of a bounded semi-group of operators, Hille and Phillips revealed that fractional powers could be considered in the framework of an operational
calculus which they originated. This program was carried out thoroughly by
Balakrishnan, He gave later a new definition and enhanced his theory to a larger class of
operators.The goal of fractional calculus, which is around 300 years old, is to
understand the problem of non-whole orders of traditional derivatives.
As is well known, the extension of the notion of derivative to non-whole orders is not done in a unique way.
Fractional
calculus had played a very important role in various fields such as physics, chemistry,
mechanics, electricity,economics, control theory, signal and image processing, biophysics,
blood flow phenomena, aerodynamics, fitting of experimental data, etc.
ut =
−a(−∆)αu + g(u, v) inΩ × R+
vt =
−a(−∆)βv + f(u, v) in Ω × R+
supplemented
with the boundary and initial conditions
∂u/∂η (x, t) = ∂v/∂η(x, t) = 0 on ∂Ω × R+ (2)
u(x, 0) =
u0(x), v(x, 0) = v0(x) in Ω (3)
Since we
are in the period of the epidemic, we focus on its role in biomedicine with
regard to the spread of epidemics .
Take an
example :
g(u, v) =
−λuv
f(u, v) =
−λuv − µv
this system
(1) (2) (3) describes the spread of epidemics with in a confined population.
The functions u(x, t), v(x, t) represent densities
of susceptible and infected individuals. The positive constants λ and µ represent
the infection rate and the removal rate respectively (see [18]). The Neumann
boundary conditions implies that there is no infection across the
boundary.
In this
work, We will study the following questions:
Does the
fractional Cauchy problem accept a local solution?
Does the
problem accept a local solution also if the operator raises the fractional
power?
- This work is divided into four chapter:
- In the
first chapter, we presented some definitions and theorems that we will use in this
note.
- In the
secend chapter, mainly introduces definitions and basic properties of
fractional powers of closed operators.
- In the
third chapter, the main purpose is to study the existence and uniqueness of mild
solutions and classical solutions of Cauchy problems (LCP) and (SLCP).
- In fourth
chapter, the main purpose is to study the existence of local in time positive solution
of the time fractional reaction–diffusion system with a balance law.
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