Dissipativity of Volterra functional differential equations

This paper is concerned with the dissipativity of Volterra functional differential equations in a Hilbert space. A sufficient condition for dissipativity of one class of such equations is obtained. This result is applied to delay differential equations and integro-differential equations to obtain dissipativity results that are more general and deeper than related results in the previous literature.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Generalized Halanay inequalities for dissipativity of Volterra functional differential equations

This paper is concerned with the dissipativity of theoretical solutions to nonlinear Volterra functional differential equations (VFDEs). At first, we give some generalizations of Halanay's inequality which play an important role in study of dissipativity and stability of differential equations. Then, by applying the generalization of Halanay's inequality, the dissipativity results of VFDEs are obtained, which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay-integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice.     

Frobenius integrable decompositions for ninth-order partial differential equations of specific polynomial type

Frobenius integrable decompositions are presented for a kind of ninth-order partial differential equations of specific polynomial type. Two classes of such partial differential equations possessing Frobenius integrable decompositions are connected with two rational Bäcklund transformations of dependent variables. The presented partial differential equations are of constant coefficients, and the corresponding Frobenius integrable ordinary differential equations possess higher-order nonlinearity. The proposed method can be also easily extended to the study of partial differential equations with variable coefficients.     

Polynomial differential equations compute all real computable functions on computable compact intervals

In the last decade, there have been several attempts to understand the relations between the many models of analog computation. Unfortunately, most models are not equivalent. Euler's Gamma function, which is computable according to computable analysis, but that cannot be generated by Shannon's General Purpose Analog Computer (GPAC), has often been used to argue that the GPAC is less powerful than digital computation. However, when computability with GPACs is not restricted to real-time generation of functions, it has been shown recently that Gamma becomes computable by a GPAC. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models.     

A note on fuzzy differential equations

We study the effect of the forcing term to the solution of a fuzzy differential equation.     

What is a universal computing machine?

A computer is classically formalised as a universal Turing machine or a similar device. However over the years a lot of research has focused on the computational properties of dynamical systems other than Turing machines, such cellular automata, artificial neural networks, mirrors systems, etc.In this paper we propose a unifying formalism derived from a generalisation of Turing’s arguments. Then we review some of universal systems proposed in the literature and show that are particular case of this formalism. Finally, we review some of the attempts to understand the relation between dynamical and computational properties of a system.     

The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II

We provide a qualitative analysis of the -dimensional dynamical system:

where is an arbitrary positive integer. Under mild algebraic conditions on the constant matrix , we show that every solution , , extends to a solution on , such that , for . Moreover, the difference between any two solutions approaches as . We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.

     

On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.