Fractional Calculus and Applied Analysis最新文献

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)).

This study focuses on the existence, uniqueness, and quenching behavior of solution to the time-space fractional Kawarada problem, where the time derivative is the Caputo-Hadamard derivative and the spatial derivative is the fractional Laplacian. The mild solution represented by Fox H-function, based on the fundamental solution, is considered in space (Cleft( [a, T], L^r(mathbb {R}^d)right) ). We use the fractional maximum principles to prove (u(textrm{x},t)ge u_a(textrm{x})) for the positive initial value. Then the relationship between quenching phenomena and the size of domain is examined. Finally, the finite difference scheme is established for solving the quenching solution to the considered problem in one and two space dimensions. The numerical simulations show the effectiveness and feasibility of the theoretical analysis.

This study examines the controllability criteria for linear and semilinear fractional dynamical systems with delays in both state and control variables in the framework of the Caputo fractional derivative. To establish the controllability criteria for linear fractional dynamical systems, the study derives necessary and sufficient conditions by employing the positive definiteness of the Grammian matrix. Extending this analysis to semilinear fractional dynamical systems, Krasnoselskii’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. Furthermore, in addressing semilinear fractional dynamical systems with delays in both state and control, Banach’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. In order to enhance the comprehension of the theoretical results, the study presents three specific examples along with appropriate graphical representations.

In this paper, we introduce a concept of the nth-level general fractional derivatives that combine the Djrbashian–Nersessian fractional derivatives and the general fractional derivatives with the Sonin kernels in one definition. Then some basic properties of these fractional derivatives including the fundamental theorems of fractional calculus and a formula for their Laplace transform are presented. As an example, all results derived for the nth-level general fractional derivatives are demonstrated on the important particular case of the Djrbashian–Nersessian fractional derivative.

In this paper, we provide new multiplicity results for a class of impulsive fractional Schrödinger-Kirchhoff-type equations involving p-Laplacian and Riemann-Liouville derivatives. By using the variational method and critical point theory, we obtain that the impulsive fractional problem has infinitely many solutions under appropriate hypotheses when the parameter (lambda ) lies in different intervals.

In this paper, we are concerned with 2D non-autonomous Reissner-Mindlin-Timoshenko plate systems with Laplacian damping terms and nonlinear sources terms. The global well-posedness is proved by using the theory of maximal monotone operators. And then we get the Lipschtiz stability of the solution. By establishing the existence of pullback absorbing sets and pullback asymptotic compactness of the process generated by the system, we obtain the existence of pullback attractors. The upper-semicontinuity of pullback attractors regarding the fractional exponent is also proved. It is the first time when the non-autonomous Reissner-Mindlin-Timoshenko plate systems are studied.