Fractional Calculus and Applied Analysis最新文献

There is a well-established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case of these chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times.

In this paper, which can be considered as an extension of our previous publication (Liu and Yu in Fract Calc Appl Anal 25:2040-2061, 2022) in same journal, we analyze the stability and synchronization for the discrete fractional-order neural network systems with mixed time delays. By new techniques, we give the proof of the discrete fractional-order Halanay inequality with mixed time delays, which contains both discrete and distributed time delays. Then, using this fractional-order Halanay inequality and constructing an appropriate Lyapunov function, we give the sufficient criteria of Mittag-Leffler stability and synchronization for the discrete fractional-order neural network systems with mixed time delays. Finally, an example is provided to illustrated one of the results.

We are concerned with a space-time fractional parabolic initial-boundary value problem of Sturm-Liouville type in a general star graph with mixed Dirichlet and Neumann boundary controls. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. Using the notion of no-regret control introduced by Lions, we prove the existence, uniqueness, and characterize the low regret control of a quadratic boundary optimal control problem, then we prove that this low regret control converges to the no-regret control and we provide the associated optimality systems and conditions that characterize that no-regret control.

We consider a p-fractional Choquard-type equation

where (0<s<1<p<p_gle p_s^*), (Nge max {2ps+alpha , p^2 s}), (a,b,varepsilon _gin (0,infty )), (K(x)= |x|^{-(N-alpha )}), (alpha in (0,N)) and F(u) is a doubly critical nonlinearity in the sense of the Hardy-Littlewood-Sobolev inequality. It is noteworthy that the local nonlinearity may also have critical growth. Combining Brezis-Nirenberg’s method with some new ideas, we obtain ground state solutions via the mountain pass lemma and a new generalized Lions-type theorem.

We introduce a novel class of stochastic partial differential equations (SPDEs) driven by space-only fractional Lévy noise. In contrast to the prevalent focus on space-time noise in the existing literature, our work explores the unique challenges and opportunities presented by purely spatial perturbations. We establish the existence and uniqueness of the solution to the stochastic heat equation by rigorously establishing the well-definedness and equivalence of mild and weak solution concepts, utilizing a blend of stochastic, deterministic, and fractional calculus techniques. Specifically, we derive explicit expressions for the covariance and variance functions, and characterize the solution’s law. These results constitute a first step towards a comprehensive understanding of SPDEs with space-only fractional Lévy noise.

In this paper, we investigate various classes of the abstract multi-term fractional difference equations and the abstract higher-order difference equations with integer order derivatives. The abstract difference equations under our consideration can be unsolvable with respect to the highest derivative. We use the Riemann-Liouville and Caputo fractional derivatives, provide some new applications of Poisson like transforms and clarify certain results about the existence and uniqueness of almost periodic type solutions to the abstract difference equations.

In this article, we consider a multi-term (varphi )-Caputo fractional dynamical system with non-instantaneous impulses. Firstly, we derive the solution for the linear (varphi )-Caputo fractional differential equation by using the generalized Laplace transform. Then, some necessary and sufficient conditions have been examined for the controllability of the linear multi-term (varphi )-Caputo fractional dynamical system with non-instantaneous impulses. Further, we establish some sufficient conditions for the controllability of the nonlinear system by utilizing the Schauder’s fixed point theorem and Gramian matrix. Finally, a simulated example is used to validate the obtained results of this article.

We introduce an efficient discretisation of a novel fractional-order adaptive exponential (FrAdEx) integrate-and-fire model, which is used to study the fractional-order dynamics of neuronal activities. The discretisation is based on an extension of L1-type methods that can accurately handle exponential growth and the spiking mechanism of the model. This new method is implicit and uses adaptive time stepping to robustly handle the stiff system that arises due to the exponential term. The implicit nonlinear system can be solved exactly, without iterative methods, making the scheme efficient while maintaining accuracy. We present a complete error model for the numerical scheme that can be extended to other integrate-and-fire models with minor changes. To show the feasibility of our approach, the numerical method has been rigorously validated and used to investigate the diverse spiking oscillations of the model. We observed that the fractional-order model is capable of predicting biophysical activities, which are interpreted through phase diagrams describing the transition from one firing type to another. This simple model shows significant promise, as it has sufficient expressive dynamics to reproduce several features qualitatively from a biophysical dynamical perspective.

The fractional Laplacian and fractional gradient are operators which play fundamental role in modeling of anomalous diffusion in d-dimensional space with the fractional exponent (alpha in (1,2)). The principal-value integrals are split into singular and regular parts where we avoid using any weight function for the approximation in the singularity neighborhood. The resulting approximation coefficients are calculated from optimal value of the singular domain radius which is only a function of the exponent (alpha ) and a given grid topology. Various difference schemes are presented for the regular rectangular grids with mesh size (h>0), and also for the hexagonal and the dodecahedral ones. This technique enables to evaluate the fractional operators with the approximation error (textrm{O}(h^{4-alpha })) which is verified using testing functions with known analytical expression of their fractional Laplacian and fractional gradient. Resulting formulas can be also used for the numeric solution of the fractional partial differential equations.

Let ({e^{-t{mathcal {L}}^{alpha }}}_{t>0}) be the heat semigroup related to the fractional Schrödinger operator (mathcal {L}^{alpha }:=(-varDelta +V)^{alpha }) with (alpha in (0,1)), where V is a non-negative potential belonging to the reverse Hölder class. In this paper, we analyze the convergence of the following type of the series

for (beta >0) and for any (N=(N_{1},N_{2})in mathbb {Z}^{2}) with (N_{1}<N_{2}), where ({t_{j}}_{jin mathbb {Z}}) is an increasing sequence in ((0,infty )) and ({v_{j}}_{jin mathbb {Z}}) is a bounded sequence of real numbers. The symbol (partial _{t}^{beta }) denotes the Caputo time-fractional derivative. We prove that the maximal operator (T_{*,t}^{alpha ,beta }(f)=sup _{begin{array}{c} Nin mathbb {Z}^{2} N_{1}<N_{2} end{array}}|T_{N,t}^{alpha ,beta }(f)|) is bounded on weighted Lebesgue spaces (L^{p}_{w}({mathbb {R}}^{n})), and is a bounded operator from (BMO_{{mathcal {L}},w}^{gamma }({mathbb {R}}^{n})) into (BLO_{{mathcal {L}},w}^{gamma }({mathbb {R}}^{n})), where (gamma in [0,1)) and w belongs to the class of weights associated with the auxiliary function (rho (x,V)).