Fractional Calculus and Applied Analysis最新文献

We establish the existence and uniqueness of a renormalized solution to an evolution equation featuring the non-local fractional p(x, y)-Laplacian and nonnegative (L^1)-data. The definition of renormalized solutions is adapted to the non-local nature to bypass the use of chain rules which is unavailable. The fractional p(x, y)-Laplacian well encapsulates the fractional p-Laplacian with a constant exponent p. Hence our result extends [25] to the setting of variable exponents.

The current study discusses the approximate solutions for a class of fractional stochastic integro-differential equation with short memory driven by non-instantaneous impulses (NIIs) defined on a separable Hilbert space. The approximation to the nonlinear functions is obtained using orthogonal projection operator. The existence and convergence of the sequence of approximate solutions is proved using a fixed point theorem and analytic semigroup theory. Moreover, we show that finite-dimensional approximations converge, guaranteeing both computational feasibility and theoretical soundness. This study emphasises on short-memory systems, which are very significant for modelling fading memory effects. We demonstrate the practical importance and versatility of our method by applying it on fractional stochastic Burgers’ and subdiffusion equations.

In this paper, we consider the long time behavior of solutions of the three dimensional (3D) generalized Navier-Stokes equations with damping. This family of 3D generalized Navier-Stokes equations with damping can be viewed as an interpolation model between subcritical (if (alpha >frac{5}{4})), critical (if (alpha =frac{5}{4})), and supercritical dissipations (if (alpha <frac{5}{4})) and it may reduce to many models by varying the parameters. First, in a periodic bounded domain, we study the existence and uniqueness of weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since our system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu called the evolutionary system to obtain various attractors and their properties. Moreover, the determining wavenumbers are also investigated here and this is the first result for a fractional equation.

This study introduces a novel family of exponential sampling type neural network Kantorovich operators, leveraging Hadamard fractional integrals to significantly enhance function approximation capabilities. By incorporating a flexible parameter (alpha ), derived from fractional Hadamard integrals, and utilizing exponential sampling, introduced to tackle exponentially sampled data, our operators address critical limitations of existing methods, providing substantial improvements in approximation accuracy. We establish fundamental convergence theorems for continuous functions and demonstrate effectiveness in pth Lebesgue integrable spaces. Approximation degrees are quantified using logarithmic moduli of continuity, asymptotic expansions, and Peetre’s K-functional for r-times continuously differentiable functions. A Voronovskaja-type theorem confirms higher-order convergence via linear combinations. Extensions to multivariate cases are proven for convergence in ({L}_{{p}})-spaces ((1le {p}<infty ).) MATLAB algorithms and illustrative examples validate theoretical findings, confirming convergence, computational efficiency, and operator consistency. We analyze the impact of various sigmoidal activation functions on approximation errors, presented via tables and graphs for one and two-dimensional cases. To demonstrate practical utility, we apply these operators to image scaling, focusing on the “Butterfly” dataset. With fractional parameter (alpha =2), our operators, activated by a parametric sigmoid function, consistently outperform standard interpolation methods. Significant improvements in Structural Similarity Index Measure (SSIM) and Peak Signal-to-Noise Ratio (PSNR) are observed at ({m}=128), highlighting the operators’ efficacy in preserving image quality during upscaling. These results, combining theoretical rigor, computational validation, and practical application to image scaling, showcase the performance advantage of our proposed operators. By integrating fractional calculus and neural network theory, this work advances constructive approximation and image processing.

This paper is devoted to exploring a new class of Hilfer fractional stochastic switched dynamical systems with the Rosenblatt process and abrupt changes, where the abrupt changes occur suddenly at specific points and extend over finite time intervals. Initially, we established solvability outcomes for the proposed switched dynamical systems by employing the fractional calculus, fixed point method, and Mittag-Leffler function. Moreover, we derived the Ulam-Hyers stability criteria for considered switched dynamical systems. Finally, we provide an example to illustrate the obtained results.

In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator etc. Several distributional properties that include the probability generating function, probability mass function and their governing differential equations are obtained for these variants. It is shown that some of these time-changed processes are Lévy and for such processes we derived the associated Lévy measure. The explicit expressions for the covariance and codifference of the component processes for some of these time-changed variants are obtained. An application of the multivariate generalized space fractional counting process to a shock model is discussed.

In this paper, we focus on a type of fractional nonlinear Schrödinger equation with odd periodic boundary conditions, where the nonlinearity periodically depending on spatial variable x. By an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems with unbounded perturbation, we obtain that there exists a lot of smooth quasi-periodic solutions with small amplitude for fractional nonlinear Schrödinger equations.

We deal with the boundary value problem

where (Omega ) is an open and bounded subset of (mathbb {R}^N) with smooth boundary, ((-Delta )^s), (sin (0,1)) denotes the fractional Laplacian, (lambda ge 0) and f is locally Lipschitz and continuous. We provide necessary and sufficient conditions on f to ensure existence and multiplicity of bounded solutions between two zeros of f.

This paper aims to study the eigenvalue problems of a mixed local and nonlocal operator in the exterior of a nonempty, bounded, simply connected domain (varOmega subset {mathbb {R}}^N) with Lipschitz boundary (partial varOmega ne emptyset ). By employing the variational methods combined with the Ljusternik-Schnirelmann principle, we prove the existence of a non-decreasing sequence of eigenvalues. In particular, we prove the principal eigenvalue is simple and isolated. We establish the positivity of the first eigenfunction by obtaining a strong maximum principle. The results obtained here are new even for the case (p=2).

In the present paper, we study inverse problems related to determining the time-dependent coefficient and unknown source function of fractional heat equations. Our approach shows that having just one set of data at an observation point ensures the existence of a weak solution for the inverse problem. Furthermore, if there is an additional datum at the observation point, it leads to a specific formula for the time-dependent source coefficient. Moreover, we investigate inverse problems involving non-local data of the fractional heat equation.