A novel delay-dependent asymptotic stability conditions for differential and Riemann-Liouville fractional differential neutral systems with constant delays and nonlinear perturbation

The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.This work is supported by Science Achievement Scholarship of Thailand (SAST), Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2020 and National Research Council of Thailand and Khon Kaen University, Thailand (6200069)

Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique

A class of semilinear fractional difference equations is introduced in this paper. The fixed point theorem is adopted to find stability conditions for fractional difference equations. The complete solution space is constructed and the contraction mapping is established by use of new equivalent sum equations in form of a discrete Mittag-Leffler function of two parameters. As one of the application, finite-time stability is discussed and compared. Attractivity of fractional difference equations is proved, and Mittag-Leffler stability conditions are provided. Finally, the stability results are applied to fractional discrete-time neural networks with and without delay, which show the fixed point technique’s efficiency and convenience

Some fundamental properties on the sampling free nabla Laplace transform

Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the N-transform more effective and efficient. On the other hand, they enrich the discrete fractional order system theor

On well-posedness of vector-valued fractional differential-difference equations

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form (*) {Delta(alpha) u(n) = Au(n + 2) + f(n, u(n)), n is an element of N-0, 1 < alpha <= 2; u(0) = u(0); u(1) = u(1); where A is a closed linear operator defined on a Banach space X. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by A, and natural restrictions on the nonlinearity f. Finally we present some original examples to illustrate our results

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Mathematical Economics

This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus

Application of Fractional Calculus to Rainfall-Streamflow Modelling

There is evidence that hydrologic systems exhibit memory processes that may be represented by fractional order systems. A new theory is developed in this work that generalises the classical unit hydrograph technique for the rainfall-runoff transformation. The theory is based upon a fractional order linear deterministic systems approach subject to an initial condition and is taken to apply to the entire rainfallstreamflow transformation (i.e. including baseflow). The general equation for a cascade of time-lagged linear reservoirs of fractional order subject to a constant initialisation function is derived, and is shown to be a form of fractional relaxation model. Dooge's (1959) general theory of the instantaneous unit hydrograph is shown to fit within the new theoretical framework. Similarly the relationship to the general storage equation of Chow and Kulandaiswamy (1971) is demonstrated. It is shown that the correct initialisation of cascade models requires a substantial number of initial conditions which may limit the viability of applying them in practice. Consequently, the differential formulation of the classical Nash cascade has been corrected and reinterpreted. The unbounded nature of the solution to the convolution integral form of the single fractional relaxation model is overcome by application of the Laplace transform of the pulse rainfall hyetograph following Wang and Wu (1983). The model parameters are fitted using the genetic algorithm. The fractional order cascade equations are tested for classical rainfall-runoff modelling using a set of 22 events for the River Nenagh. The cascade of 2 unequal fractionalorder reservoirs is shown to converge to that of the integer order case, whilst the cascade of equal reservoirs shows some differences. For the modelling of the total rainfall-streamflow process the single fractional order reservoir model with a constant initialisation function is tested on a selection of events for a range of UK catchment scales (22km^ to 510km ). A rainfall loss model is incorporated to account for infiltration and evapotranspiration. The results show that the new approach is viable for modelling the rainfall-streamflow transformation at the lumped catchment scale, although the parameter values are not constant for a given catchment. Further work is recommended on determining the nature of the initialisation function using field studies to improve the identification of the model parameters on an event-by-event basis