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Let (Ω,F,P)$(\Omega ,\mathcal {F},\mathbb {P})$ be a complete probability space and let p(·),q(·):[0,1]→(0,∞)$p(\cdot),q(\cdot):[0,1]\rightarrow (0,\infty)$ be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces Lp(·),q(·)(Ω)$\mathcal {L}_{p(\cdot),q(\cdot)}(\Omega)$, which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces Hp(·),q(·)(Ω)$\mathcal {H}_{p(\cdot),q(\cdot)}(\Omega)$. [ABSTRACT FROM AUTHOR]

In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests. [ABSTRACT FROM AUTHOR]

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative CDaγx(t)+q(t)x(t)=0,a

We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces. To do this, we prove the weak–weak type modular inequality of the Hardy–Littlewood maximal operator with respect to the Young function. Orlicz–Morrey spaces contain Lp$L^p$ spaces (1≤p≤∞$1\le p\le \infty$), Orlicz spaces, and generalized Morrey spaces as special cases. Hence, we get necessary and sufficient conditions on these function spaces as corollaries. [ABSTRACT FROM AUTHOR]

A new alternative numerical method for solving variable‐order fractional optimal control problems (VO‐FOCPs) is introduced. We use fractional‐order Chebyshev wavelets for solving these VO‐FOCPs. By applying the regularized beta functions, an exact value for the fractional integration of the given wavelets is provided. By applying this formula and the given wavelets, we reduce the given VO‐FOCP to a system of algebraic equations which can be solved by using known methods. For the function approximation of our method, we also give the convergence analysis. By using several numerical examples, we show that this method is more accurate than the other existing methods in the literature. [ABSTRACT FROM AUTHOR]

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether α(x)p(x)−n+λ(x)p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0⩽α(x)p(x)−n+λ(x)p(x)⩽1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. [ABSTRACT FROM AUTHOR]

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length. [ABSTRACT FROM AUTHOR]

A new homogenization scheme is proposed to model the effects of surface phenomena occurring at the pore fluid/solid interface on the effective transport properties of multiscale and reactive geosystems, e.g., clayey geomaterials. This new scheme is a generalization of the so‐called differential scheme (DS) used to infer the effective properties of a composite made up of several materials. It is named the fractional differential scheme and is based on two key elements: (i) the concept of realizability of the DS itself and (ii) a fractional integral formulation of the DS for a porous medium considered as a two‐component composite. The formulation of the fractional DS introduces two parameters: a cementation exponent m and a fractional order α. The cementation exponent m is related to the microstructure of the material. The fractional order α accounts for the amplitude of the "surface" transport of ions resulting from the physico‐chemical interactions between hydrated cations and swelling clay minerals. Both parameters m and α are inverted from two sets of data: electrical conductivity measurements obtained on clayey rocks and effective diffusion coefficient measurements acquired from a Na‐montmorillonite geomaterial. The inversion results demonstrate that the fractional DS model is able to capture the dependence of the cation concentration on the effective transport properties of the clayey materials under study. Our results also show that the fractional order α can be considered an indirect indicator of the amplitude of physico‐chemical interactions between hydrated cations and swelling clay minerals occurring at the pore fluid/solid interface. [ABSTRACT FROM AUTHOR]