Fractional Calculus and Applied Analysis最新文献

In this paper, Cauchy problem for incompressible Navier-Stokes equations with time fractional differential operator and fractional Laplacian in (mathbb {R}^n) ((nge 2)) is investigated. The global and local existence and uniqueness of mild solutions are obtained with the help of Banach fixed point theorem when the initial data belongs to (L^{p_{c}}(mathbb {R}^n)) ((p_c=frac{n}{alpha -1})). In addition, the decay properties of mild solutions to the considered time-space fractional equations are constructed. Moreover, it is shown that when the initial data belongs to (L^{p_{c}}(mathbb {R}^n)cap L^{p}(mathbb {R}^n)) with (1<p<p_c), the existence and uniqueness of global and local mild solutions can also be established. At the end of this paper, the integrability of mild solutions is discussed.

Our main concern is the existence of a new (PC_{2-v})-mild solution for Hilfer fractional impulsive evolution equations of order (alpha in (1,2)) and (beta in [0,1]) as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of (PC_{2-v})-mild solution for the objective equations by the Laplace transform and probability density function. Secondly, we rely on the Banach contraction mapping principle to discuss a new existence and uniqueness result of (PC_{2-v})-mild solution when the sine family is compact. Thirdly, a sufficient condition for the approximate controllability result of impulsive evolution equations is formulated and proved under the assumptions that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximately controllable. Finally, two examples are given to illustrate the validity of the obtained results in the application.

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.