Fractional Calculus and Applied Analysis最新文献

A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability.

Various constitutive models have been proposed to quantify a wide range of non-Newtonian fluids, but there is lack of a systematic classification and evaluation of these competing models, such as the quantitative comparison between the classical integer-order constitutive models and the newly proposed fractional derivative equations for non-Newtonian fluids. This study reviews constitutive equation models for non-Newtonian fluids, including time-independent fluids, viscoelastic fluids, and time-dependent fluids. A comparison between fractional derivative non-Newtonian fluid constitutive equations and traditional constitutive equations is also provided. Results show that the space fractional derivative model is equivalent to some classical constitutive models under reasonable assumptions. Further discussions are made from the perspective of the industrial and biomedical applications of non-Newtonian fluids. Advantages and limitations of the constitutive models are also explored to help users to select proper models for real-world applications.

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain (varOmega ) such that V has no critical point in (partial varOmega ). To overcome the “lack of compactness" of the problem, we combine Del Pino-Felmer’s penalization technique with Moser’s iteration method and some ideas from Alves [1].

In order to accurately capture non-local properties and long-term memory effects, this study combines the tempered fractional-order operator with delayed neural networks to investigate its stability, leveraging the introduced decay term of the tempered fractional-order operator. Firstly, the discrete-time tempered fractional-order neural networks model (DTFNNs) is presented. Furthermore, in an effort to better understand the dynamic behavior of complex systems, solutions to discrete-time tempered fractional non-homogeneous equations are obtained. The stability conditions for systems are subsequently established, contributing novel insights to the field. To validate the robustness of these conditions, numerical experiments are conducted, underscoring the practical relevance of the proposed model.

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker (L^{p})-Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities.

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain (varOmega subset {mathbb {R}}^N) are also proved. As application, we measure explicitly how small has to be (text {diam}(varOmega )) so that weak and strong maximum principles associated to this problem hold in (varOmega ).

The pricing problem of European option is investigated under the generalized fractional jump-diffusion model. First of all, the generalized fractional jump-diffusion model is proposed, with the assumption that the underlying asset price follows this model, and the explicit solution is derived using the Itô formula. Then, the partial differential equation (PDE) of the European option price is obtained by using the delta-hedging strategy, and the analytical solutions of the European call and put option prices are obtained through the risk-neutral pricing principle. Moreover, the accuracy of closed-form formula for European option pricing is verified by the Monte Carlo simulation. Furthermore, the properties of the pricing formulas are discussed and the impact of main parameters on the option pricing model are analyzed via calculations of Greeks. Finally, the rationality and validity of the established option pricing model are verified by numerical analysis.

A generalization of fractional linear viscoelasticity based on Scarpi’s approach to variable-order fractional calculus is presented. After reviewing the general mathematical framework, a variable-order fractional Maxwell model is analysed as a prototypical example for the theory. Some physical considerations are then provided concerning the fractionalisation procedure and the choice of the transition functions. Lastly, the material functions for the considered model are derived and numerically evaluated for exponential-type and Mittag-Leffler-type order functions.

In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied.

We consider a problem which describes the heat diffusion in a complex media with fading memory. The model involves a fractional time derivative of order between zero and one instead of the classical first order derivative. The model takes into account also the effect of a neutral delay. We discuss the existence and uniqueness of a mild solution as well as a classical solution. Then, we prove a Mittag-Leffler stability result. Unlike the integer-order case, we run into considerable difficulties when estimating some problematic terms. It is found that even without the memory term in the heat flux expression, the stability is still of Mittag-Leffler type.