Fractional Calculus and Applied Analysis最新文献

We introduce and study the spaces of fractionally absolutely continuous functions of two variables of any order and the fractional Sobolev type spaces of functions of two variables. Our approach is based on the Riemann-Liouville fractional integrals and derivatives. We investigate relations between these spaces as well as between the Riemann-Liouville and weak derivatives.

This paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to the previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure (varPhi ). The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure (varPhi ). Extensions to general graph structures can be given by applying to our results a localisation technique.

In this paper, we present Lie symmetry analysis for time fractional Black-Scholes equation with time-dependent coefficients. The group classification is carried out by investigating the time-dependent coefficients (sigma (t)), r(t) and s(t). Then the obtained group generators are used to reduce the equation under study, some of the reduced equations are fractional ordinary equations with Erdélyi-Kober fractional derivative, and some exact solutions including power series solutions are constructed.

We are concerned with the inverse problem of reconstructing a fractional evolution equation with a source. To this end we use observations of the solution on the boundary to reconstruct the principal part of the operator and the fractional order of the time derivative, while an overdetermination at a time T is used to recover the source by a non iterative method. Numerical examples explain how to compute the fractional order and the source using finite data.

We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a ((p-1))-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to ’asymptotic’ weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for the existence of a solution. Our work extends classical results due to Brezis-Oswald [7] and Diaz-Saa [11] to the nonlinear nonlocal framework.

In this paper, we study a class of non-autonomous lower critical fractional Choquard equation with a pure-power nonlinear perturbation. Under some reasonable assumptions on the potential function h, we prove the existence and discuss asymptotic behavior of ground state solutions for our problem. Meanwhile, we also prove that the number of normalized solutions is at least the number of global maximum points of h when (varepsilon ) is small enough.

In this paper, we consider the following Schrödinger system involving the tempered fractional p-Laplacian

where (n ge 2), (a, b>0), (2<p<infty ), (0<s, t<1) and (lambda ) is a sufficiently small positive constant. To effectively utilize the direct method of moving planes, we first establish the narrow region principle and the decay at infinity. Then we prove the radial symmetry and monotonicity of positive solutions for the system in the unit ball and the whole space. Our results are an extension of some content in Ma and Zhang (Appl Math J Chin Univ 37: 52–72, 2022).

This survey aims to present various approaches to non-integer integrals and derivatives and their practical implementation within Wolfram Mathematica. It begins by short discussion of historical moments and applications related to fractional calculus. Different methods for handling non-integer powers of differentiation operators are presented, along with generalizations of fractional integrals and derivatives. The survey also delves into the diverse applications of fractional calculus in physics, engineering, medicine, and numerical calculations. Essential details of fractional integro-differentiation implemented in Wolfram Mathematica are highlighted. The Hadamard regularization of Riemann-Liouville operator is utilized as the foundation for creating the arbitrary order of integro-differential operator in Mathematica. The survey describes the application of fractional integro-differentiation to Taylor series expansions near zero using Hadamard regularization and the use of the Meijer G-function for evaluating derivatives of complex orders. We conclude with a discussion on applying fractional integro-differentiation to “differential constants” and provide generic formulas for fractional differentiation. The extensive list of references underscores the vast body of works on fractional calculus.

In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter (Hin (1/2,1)). By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present two important corollaries on the non-confluence property by the Poisson equation and M-matrix, respectively, which can verify the non-confluence property more effectively than the general condition. Finally, we provide an example to illustrate the practical usefulness of our theoretical results.

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration.