Mathematical Sciences
http://hdl.handle.net/11141/40
2019-01-15T04:38:13ZBoundary Value Problems in a Multidimensional Box for Higher Order Linear and Quasi-Linear Hyperbolic Equations
http://hdl.handle.net/11141/2619
Boundary Value Problems in a Multidimensional Box for Higher Order Linear and Quasi-Linear Hyperbolic Equations
Aljaber, Noha
Boundary value problems in a multidimensional box for higher order linear hyperbolic
equations are considered. The concept of associated problems are introduced.
For general boundary value problems there are established:
(i) Necessary and sufficient conditions for a linear problem to have the Fredholm property
in two–dimensional case;
(ii) Necessary and sufficient conditions of well–posedness in two–dimensional case;
(iii) Unimprovable sufficient conditions for a linear problem to have the Fredholm property;
(iv) Unimprovable sufficient conditions of well–posedness and α–well–posedness;
(v) Effective sufficient conditions of unqie solvability of two–point, periodic and Dirichlet
type problems.
(iv) Unimprovable conditions of unique solvability of two dimensional ill–posed periodic
problems.
For the Dirichlet type problem in a two–dimensional smooth convex domain:
(i) Sufficient conditions for a linear problem to have the Fredholm property;
(ii) sufficient conditions of unique solvability.
For quasi–linear boundary value problems there are established:
(i) Optimal sufficient conditions of solvability and unique solvability;
(ii) Effective sufficient conditions of solvability of periodic and Dirichlet type problems
in case, where the righthand side of the equation has arbitrary growth order with
respect to some phase variables.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-05-01T00:00:00ZLower Order Perturbations of Critical Fractional Laplacian Equations
http://hdl.handle.net/11141/2616
Lower Order Perturbations of Critical Fractional Laplacian Equations
Al Oweidi, Khalid Fanoukh
We give suffcient conditions for the existence of nontrivial solutions to a
class of critical nonlocal problems of the Brezis-Nirenberg type. Our result
extends some results in the literature for the local case to the nonlocal setting.
It also complements the known results for the nonlocal case.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-07-01T00:00:00ZLyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays
http://hdl.handle.net/11141/2590
Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays
Agarwal, Ravi P.; Hristova, Snezhana G.; O'Regan, Donal
One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks.
2018-05-09T00:00:00ZOn the Qualitative Theory of the Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes
http://hdl.handle.net/11141/2482
On the Qualitative Theory of the Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes
Aal-Rkhais, Habeeb Abed
We consider nonlinear second order degenerate or singular parabolic equation
ut − a(um)xx + buβ + c(up)x = 0, a, m, β, p > 0, b, c ∈ R
describing reaction-diffusion-convection processes arising in many areas of science
and engineering, such as filtration of oil or gas in porous media, transport of
thermal energy in plasma physics, flow of chemically reacting fluid, evolution of
populations in mathematical biology etc. We apply the methods developed in
U.G. Abdulla, Journal of Differential Equations, 164, 2(2000), 321-354 for the
reaction-diffusion equation (c = 0) and prove the existence, uniqueness, boundary
regularity and comparison theorems for the initial-boundary value problems in non-cylindrical
domains with non-smooth boundary curves under the minimal restriction on the boundary. Constructed weak solutions are continuous up to the non-smooth
boundary if at each interior point the left modulus of the lower (respectively upper)
semicontinuity of the left (respectively right) boundary curve satisfies an upper
(respectively lower) H¨older condition near zero with H¨older exponent ν > 1
2
. The
value 1
2
is critical as in the classical theory of heat equation, and is independent
of nonlinearity parameters m, β, p, and from the degeneration or singularity of the
PDE. General theory is applied to the problem on the initial development and
asymptotics of the interfaces and local solutions near the interfaces for the reaction-diffusion-convection
equation with compactly supported initial function. Depending
on the relative strength of three competing forces such as diffusion, convection, and
reaction, the interface may expand, shrink or remain stationary. The methods used
are rescaling and blow-up techniques for the identification of the asymptotics of
the solution along the class of interface type curves, construction of the barriers and
application of the comparison theorem in non-cylindrical domains with characteristic
boundary curves, as they are developed in papers U.G. Abdulla & J.King, SIAM J.
Math. Anal., 32, 2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002),
541-560.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-05-01T00:00:00Z