Fractional Calculus and Applied Analysis最新文献

We propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle (P=[0,a]times [0,b]) ((a>0, b>0)). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, On the single partial Caputo derivatives for functions of two variables, Periodica Mathematica Hungarica 87(2), (2023), 324–339].

In this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order s with (sin (0,1)). The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains non-smooth terms. As the mappings involved may not be Gâteaux differentiable, we use a regularization technique to regularize these nonlinear terms, aiming to obtain Gâteaux differentiable mappings. By employing this regularization technique, we are able to derive the first-order optimality condition for the regularized control problem by using the associated adjoint system. Furthermore, we conduct a limit analysis on the regularized term resulting in an optimality system for the non-smooth problem of C-stationary type. Subsequently, we establish a primal optimality condition, specifically B-stationarity. Under the assumption of “constraint qualification”, we derive the strong stationarity conditions for the non-smooth optimization problem with control constraints and establish the equivalence between B-stationarity and strong stationarity conditions.

This paper addresses the questions of well-posedness to fractional m-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity:

where (R(u)=left| uright| ^{r}ln (|u|)). Guided by the made assumptions, we arrive at the conclusions of the local and global solvability of solutions within the framework of Galerkin approximation. In addition, this study considers weak solutions’ asymptotic stability and explosion in finite time. Significantly, we not only figure out the relationship between the non-local fractional operator and singular potential term, but generalize and improve earlier results in the literature.

In this paper, our goal is to study the following class of Hardy–Hénon type problems

when (mu ge alpha {>-1}), and the nonlinearity f has exponential critical growth in the sense of the Trudinger-Moser inequality. This way, with an appropriate change of variable, due to the behavior of the weight (|x|^{alpha }), one can obtain a version of this inequality in the radial context that allows us to treat the problem with a nonlocal framework and a critical exponent depending on (alpha ). When (alpha >0), we have a Hénon problem and this exponent becomes larger than usual. This fact is a counterpart for Ni [37] result for the local case and (mathbb {R}^N) ((Nge 3)). If (-1<alpha <0), we have a Hardy equation. In this case, the exponent is smaller than usual, but we treat a problem with a singularity at 0. The main difficulty is to overcome the lack of compactness inherent to problems involving nonlinearities with critical growth. For this, we apply the variational methods to control the minimax level using Moser functions (see [48]). Then, we guarantee the existence of at least one radial solution under suitable hypotheses for the constants (lambda , mu ,alpha ), as well as, on f, combined with the interaction of the spectrum of the fractional Laplacian operator via the Mountain Pass Theorem or the Linking Theorem (see [41, 42]). Thus, we study a version of problems treated in [3] for nonlinearities with exponential critical growth and in [25] in a nonlocal context.

Here we introduce a notion of fractional k-dimensional measure, (0le k<n), that depends on a parameter (sigma ) that lies between 0 and 1. When (k=n-1) this coincides with the notions of fractional area and perimeter, and when (k=1) this coincides with the notion of fractional length. It is shown that, when multiplied by the factor (1-sigma ), this (sigma )-measure converges to the k-dimensional Hausdorff measure up to a multiplicative constant that is computed exactly. We also mention several future directions of research that could be pursued using the fractional measure introduced.

In this paper, an important property of fractional order operators involving discontinuous functions is discussed, First, a pioneering work of impulsive fractional differential equations is recalled to illuminate the incorrectness of notation ({^C_{t_k}D}^{q}_t). Second, a class of piecewise-defined equations with Caputo fractional derivative is contrastively investigated, and it is revealed that the additivity of integration on intervals for integer-order integral does not hold for fractional integrals, not to mention fractional derivatives. Third, by utilizing the Heaviside step function, an interesting property of fractional integral involving piecewise-defined functions is correspondingly presented. Finally, illustrative examples are given for validation of the derived results, which may lead to a new way to reconsider the dynamic behavior of fractional hybrid systems, as well as discontinuous control design for fractional systems.

We investigate a semilinear problem for a fractional diffusion equation with variable order Caputo fractional derivative (left( partial _t^{beta (t)} uright) (t)) subject to homogeneous Dirichlet boundary conditions. The right-hand side of the governing PDE is nonlinear (Lipschitz continuous) and it contains a weakly singular Volterra operator. The whole process takes place in a bounded Lipschitz domain in ({{mathbb {R}}}^d). We establish the existence of a unique solution in (Cleft( [0,T],L^{2} (varOmega )right) ) if (u_0in L^{2} (varOmega )). Moreover, if (mathcal {L}^{gamma }u_0in L^{2} (varOmega )) for some (0<gamma <1-frac{delta }{beta (0)}) ((delta ) depends on the right-hand-side of the PDE) then (mathcal {L}^{gamma }uin Cleft( {[}0,T{]},L^{2} (varOmega )right) ).

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in ({mathbb {R}}^{n}), (nge 2). The global existence and (L^{infty })-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) (b>1-frac{alpha }{n}), for any initial value and birth rate; (ii) (0<ble 1-frac{alpha }{n}), for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the (L^{infty })-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong.

This research aims to investigate the stabilization problem of the Riemann-Liouville spatial (beta )-fractional output with order (beta in (0, 1)) for a class of bilinear dynamical systems with a time Caputo (alpha )-fractional derivative. Initially, we provide definitions and establish the well-posedness of the problem addressed. Furthermore, we introduce a feedback control strategy that ensures both weak and strong stabilization of the (beta )-fractional output, under a broad set of sufficient conditions. Additionally, we present numerical computations to elucidate the effectiveness of the obtained results.

In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition, the improved regularity of solution is derived. Furthermore, to overcome the weak singularity and enhance convergence precision, a second-order fitted scheme based on L2-(1_sigma ) approximation and order reduction method is applied to such problems, which is an improvement for the work [6]. Ultimately, examples are presented to verify the effectiveness of our theoretical results.