Dynamical generalized functions and the multiplication problem

This book is designed for senior undergraduate and graduate students pursuing courses in mathematics, physics, engineering and biology. The text begins with a study of ordinary differential equations. The concepts of first- and second-order equations are covered initially. It moves further to linear systems, series solutions, regular Sturm–Liouville theory, boundary value problems and qualitative theory. Thereafter, partial differential equations are explored. Topics such as first-order partial differential equations, classification of partial differential equations and Laplace and Poisson equations are also discussed in detail. The book concludes with heat equation, one-dimensional wave equation and wave equation in higher dimensions. It highlights the importance of analysis, linear algebra and geometry in the study of differential equations. It provides sufficient theoretical material at the beginning of each chapter, which will enable students to better understand the concepts and begin solving problems straightaway.

Existence and stability analysis of [formula omitted]th order multi term fractional delay differential equation

The paper aims to to study the concept of asymptotic almost automorphy in the context of generalized functions. We introduce an algebra of asymptotically almost automorphic generalized functions which contains the space of smooth asymptotically almost automorphic functions as a subalgebra. The fundamental importance of this algebra, is related to the impossibility of multiplication of distributions; it also contains the asymptotically almost automorphic Sobolev--Schwartz distributions as a subspace. Moreover, it is shown that the introduced algebra is stable under some nonlinear operations. As a by pass result, the paper gives a Seeley type result on extension of functions in the context of the algebra of bounded generalized functions and the algebra of bounded generalized functions vanishing at infinity, these results are used to prove the fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. As applications, neutral difference-differential systems are considered in the framework of the algebra of generalized functions.

We study a nonautonomous equation with generalized coefficients in an algebra of generalized functions. The solutions of the equation can be rather different depending on the interpretation of the equation. We show that all these solutions can be obtained from the solution of this equation in the algebra of generalized functions.

We study the problem of division in Colombeau's space of new generalized functions in the associated sense: more precisely we solve the equation of the formF·G≈pH with adequate assumptions onF. By using the Fourier transformation we construct, by simple methods, the solution in the p-associated sense of a partial differential equation with constant coefficients.

ADMISSIBILITY THEORY AND THE GLOBAL BEHAVIOR OF SOLUTIONS OF FUNCTIONAL EQUATIONS

Introduction The Common Lisp Object System is an object-oriented extension to Common Lisp as defined in Common Lisp: The Language, by Guy L. Steele Jr. It is based on generic functions, multiple inheritance, declarative method combination, and a meta-object protocol. The first two chapters of this specification present a description of the standard Programmer Interface for the Common Lisp Object System. The first chapter contains a description of the concepts of the Common Lisp Object System, and the second contains a description of the functions and macros in the Common Lisp Object System Programmer Interface. The chapter "The Common Lisp Object System Meta-Object Protocol" describes how the Common Lisp Object System can be customized. The fundamental objects of the Common Lisp Object System are classes, instances, generic functions, and methods. A class object determines the structure and behavior of a set of other objects, which are called its instances. Every Common Lisp object is an instance of a class. The class of an object determines the set of operations that can be performed on the object. A generic function is a function whose behavior depends on the classes or identities of the arguments supplied to it. A generic function object contains a set of methods, a lambda-list, a method combination type, and other information. The methods define the class-specific behavior and operations of the generic function; a method is said to specialize a generic function. When invoked, a generic function executes a subset of its methods based on the classes of its arguments. A generic function can be used in the same ways that an ordinary function can be used in Common Lisp; in particular, a generic function can be used as an argument to funcall and apply and can be given a global or a local name. A method is an object that contains a method function, a sequence of parameter speclalizers that specify when the given method is applicable, and a sequence of qualifiers that is used by the method combination facility to distinguish among methods. Each required formal parameter of each method has an associated parameter specializer, and the method will be invoked only on arguments that satisfy its parameter specializers. The method combination facility controls the selection of methods, the order in which they are run, and the values that are returned by the generic function. The Common Lisp Object System offers a default method combination type and provides a facility for declaring new types of method combination.

26 Difference Equations with Public Health Applications

Among recurrent functions the most sophisticated are Poisson stable functions. For discontinuous functions, there are very few results, for the stability. Discontinuous compartmental Poisson stable functions are in the focus of this research. As the discontinuity points of the functions, a special time sequences, Poisson sequences, are considered. It the first time, the discontinuous functions of two compartments, periodic and Poisson stable, are investigated. To combine periodicity and Poisson stability, in the case of continuous functions, a convergence sequence with a special kappa property was used [1,2]. For discontinuous functions, this property is not enough, because we also should consider the discontinuity points of the function. For this reason, we need a new concept known as Poisson couple, that is, a couple of a sequence of discontinuity points and convergence sequence that has the kappa property. Moreover, we meet the challenges for the stability by considering functions on diagonals in the space of arguments. Examples of Poisson stable functions are given to illustrate the theoretical results. The method and results can be effectively used in the study of different types of functional differential equations, impulsive differential equations and differential equations generalized piecewise constant argument, as well as their application.

Course 13 Elements of nonequilibrium statistical mechanics

Integral Equations for Nonlinear Problems in Partial Differential Equations

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.

On Favard's theorems

Nonlinear dynamics undergoes a period of explosive growth today. However some federative concepts have emerged : solitons for nonlinear waves, strange attractors for dissipative phenomena, resonant tori in bamiltonien theory, scaling laws and fractals in many physical systems from low dimensional chaos to fully developed turbulence. In particular, nonlinear oscillator differential equations provided a source of inspiration for perturbation methods in the early work of Diiffing, Van der Pol and Rayleigh, then for bifurcation and modern qualitative analysis in the work of Poincar~, Birkhoff and others russian mathematiciens including Liapourov, Andronov and more recently Kolmogorov and Arnold. 1,2 Most of the modern approach turns around the idea of periodic orbits and resonances in hamiltonien systems. From Kolmogorov-Arnold-Moser (K_AM) theory we expect to preserve the structural stability of nonresonant tori against small hamiltonien conservative perturbations. 3 Perturbed resonant trajectories were first studied by Poincard who predicted the phenomenon of splitting of separatrices and local stochasticity for two degrees of freedom (2-tori). One popular example of Poincar~-Birkhoff theory of resonance is given by the so-called Henon mapping as a model of the motion of stars in the gravitational field of the galaxy. 4,5 For n-tori (n > 2) a random wandering in the resonant regions between invariant tori (Arnold diffusion and stochastic webs) was predicted. 3 More recently the minimal dimensionality for stochastic webs to form in hamiltonien systems reached its lower bound (n = 1.5), the half degree of freedom refering to a time periodic perturbation. 6

Generic non-extendability and total unboundedness in function spaces