Fractional Calculus and Applied Analysis最新文献

This paper investigates stochastic averaging principle for a class of mixed slow-fast stochastic differential equations driven simultaneously by a multidimensional standard Brownian motion and a multidimensional fractional Brownian motion with Hurst parameter (1/2<H<1). The stochastic averaging principle shows that the slow component strongly converges to the solution of the corresponding averaged equations under a weaker condition than the Lipschitz one.

In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source, we establish the uniqueness of the inverse problem, and also a Hölder-type conditional stability of the inverse problem is firstly given. Meanwhile, we prove a stability estimate of optimal order for the inverse problem. Then some convergence estimates for the regularized solution are proved under an a-priori and an a-posteriori regularization parameter choice rule. Finally, several numerical experiments illustrate the effectiveness of the proposed method in one-dimensional case.

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the (1+alpha ) order convergence rate for fractional order (alpha in (0,1)), of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments.

In this paper, we establish sufficient conditions in order to guarantee the existence and uniqueness of discrete weighted pseudo S-asymptotically (omega )-periodic solution to the semilinear fractional difference equation

where (1<alpha <2,) A is a closed linear operator in a Banach space X which generates an ((alpha ,beta ))-resolvent sequence ({S^n_{alpha ,beta }}_{nin mathbb N_0}subset mathcal {B}(X)) and (g:mathbb N_0times Xrightarrow X) a discrete weighted pseudo S-asymptotically (omega )-periodic function satisfying suitable Lipschitz type conditions in the spatial variable (local and global), based in fixed point Theorems. In order to achieve this objective, we prove invariance by convolution and principle of superposition for a class of suitables function spaces.

In this work, we consider an inverse problem of determining the coefficient at the lower term of a fractional diffusion equation. The direct problem is the initial-boundary problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the unknown coefficient, an overdetermination condition of the integral form is specified with respect to the solution of the direct problem. Using Green’s function for an ordinary fractional differential equation with a non-local boundary condition and the Fourier method, the inverse problem is reduced to an equivalent problem. Further, by using the fixed-point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

This paper is devoted to the continuity of the weak solution for tempered fractional general diffusion equations driven by tempered fractional Brownian motion (TFBM). Based on the Feynman-Kac formula (1.2), by using the Itô isometry for the stochastic integral with respect to TFBM, Parseval’s identity and some ingenious calculations, we establish the continuities of the solution with respect to Hurst index H and tempering parameter (lambda ) of TFBM.

In this work, we employ a biorthogonal approach to construct the infinite-dimensional Fractional Pascal measure (mu ^{(alpha )}_{_{sigma }}, 0 < alpha le 1), defined on the tempered distributions space (mathcal {E}') over (mathbb {R} times mathbb {R}^{*}_{+}). The Hilbert space (L^{2}(mu ^{(alpha )}_{_{sigma }})) is characterized using a set of generalized Appell polynomials (mathbb {P}^{(alpha )}_{widehat{sigma }}={P^{(alpha )}_{n, widehat{sigma }}, nin mathbb {N}}) associated with the measure (mu ^{(alpha )}_{_{sigma }}). This paper presents novel properties of the kernels (P^{(alpha )}_{n, widehat{sigma }}) in infinite dimensions, offering valuable insights. Additionally, we delve into the discussion of the generalized dual Appell system, broadening the scope of our results.

This paper is to obtain sufficient conditions for the uniqueness and existence of solutions to a new nonlinear fractional partial integro-differential equation with boundary conditions. Our analysis relies on an equivalent implicit integral equation in series obtained from an inverse operator, the multivariate Mittag-Leffler function, Leray-Schauder’s fixed point theorem as well as Banach’s contractive principle. Several illustrative examples are also presented to show applications of the key results derived. Finally, we consider the generalized fractional wave equation in ({mathbb {R}}^n) and deduce the analytic solution for the first time based on the inverse operator method, which leads us a fresh approach to studying some well-known partial differential equations.

In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like memory. Generalized fractional maps, in which memory is not exactly the power-law memory but the asymptotically power-law-like memory, are used to model and investigate general properties of these systems. In this paper we extend the definition of the notion of generalized fractional maps of arbitrary positive orders that previously was defined only for maps which, in the case of integer orders, converge to area/volume-preserving maps. Fractional generalizations of Hénon and Lozi maps belong to the newly defined class of generalized fractional maps. We derive the equations which define periodic points in generalized fractional maps. We consider applications of our results to the fractional and fractional difference Hénon and Lozi maps.

We use a generalized Riemann-Liouville type derivative of an abstract function of one variable and existence of a weak solution to an abstract fractional parabolic problem on [0, T] containing Riemann-Liouville derivative of a function of one variable and spectral fractional powers of a weak Dirichlet-Laplace operator to study existence of a strong solution to this problem. Our goal in this regard is to provide conditions that allow the transition from a weak to a strong solution. Next, we passage from the abstract problem to a classical one on ([0,T]times varOmega ), containing partial (with respect to time (tin [0,T],)) Riemann-Liouville derivative of the unknown real-valued function of two variables and fractional powers of a weak Dirichlet-Laplacian of this function (with respect to spatial variable (xin varOmega )). The most important in this regard is a theorem on the relation of the fractional derivatives of an abstract function of one variable and real-valued one of two variables.