Fractional Calculus and Applied Analysis最新文献

This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional differentiable functions, the comparison principle, counterfactual reasoning, and the spectral analysis method (concerning the integral presentations of basic solutions). Some numerical examples are also provided to demonstrate the validity of the proposed results.

In this paper, we deal with a time dependent inverse source problem for a nonlinear p-Laplacian pseudoparabolic equation containing a fractional derivative in time of order (alpha in (0,1)). Moreover, the equation is perturbed by a power-law damping (reaction) term, which, depending on whether its sign is positive or negative, may account for the presence of a source or an absorption within the system. The equation is supplemented with a measurement in a form of an integral over space domain along with the initial and Dirichlet boundary conditions, to determine both the solution of the equation and the unknown source term. For the associated inverse source problem, under suitable assumptions on the data, we establish global and local in time existence and uniqueness of weak solutions for different values of exponents and coefficients.

This paper investigates the Cauchy problem of time-space fractional Keller-Segel-Navier-Stokes system in ({mathbb {R}}^d~(dge 2)), which describes both memory effect and Lévy process of the system. The local and global existence of mild solutions are obtained by the (L^p-L^q) estimates of Mittag-Leffler operators combined with Banach fixed point theorem and Banach implicit function theorem, respectively. Furthermore, some properties are established, such as mass conservation, decay estimates, stability and self-similarity of mild solutions.

In this note, we consider a pseudo-differential operator (T_a) defined as

It is well-known that (T_a) is not bounded on (L^2) in general when a belongs to the forbidden Hörmander class (S^{n(rho -1)/2}_{rho ,1},0le rho le 1). In this note, when (s>0,0le rho le 1,1le rle 2) and (ain S^{n(rho -1)/r}_{rho ,1}), we prove that (T_a) is bounded on the Triebel-Lizorkin space (F^s_{p,q}) if (r<p,q<infty ) or (r<ple infty ,q=infty ). As the most important special example, when (ain S^{n(rho -1)/2}_{rho ,1}) and (s>0), if (2<p,q<infty ) or (2<ple infty ,q=infty ), then (T_a) is bounded on (F^s_{p,q}). When (rho <1), this result is entirely new.

In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrix K(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case of K(x) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.

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.