Fractional Calculus and Applied Analysis最新文献

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.

In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems.

In this paper, we study a class of backward problems for nonlinear fractional super-diffusion equations in Banach spaces. We consider the time fractional derivative in the sense of Caputo type. First, we establish some results for the existence of the mild solutions. Moreover, we obtain regularity results of the first order and fractional derivatives of mild solutions. These conclusions are mainly based on fixed point theorems and properties of (alpha )-resolvent family as well as Mittag-Leffler functions. Finally, two applications are provided to illustrate the efficiency of our results.

In this paper, we investigate a class of stochastic Riemann-Liouville type fractional differential equations driven by Lévy noise. By using Itô formula for the considered equation, we attempt to explore the non-confluence property of solution for the considered equation under some appropriate conditions. Our approach is to construct some suitable Lyapunov functions which is novel in exploring the non-confluence property of differential equations.

In this paper, we prove Hopf’s lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf’s lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf’s lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf’s lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method.

An initial-boundary value problem for the multi-term time-fractional wave equation on a bounded domain is considered. For the largest and smallest orders of the involved Caputo fractional time-derivatives, (alpha ) and (alpha _m), it is assumed (1<alpha <2) and (alpha -alpha _mle 1). Subordination principle with respect to the corresponding single-term time-fractional wave equation of order (alpha ) is deduced. Injectivity of the integral transform, defined by the subordination relation, is established. The subordination identity is used to prove uniqueness for a coefficient inverse problem for the multi-term equation, based on an analogous property for the related single-term one. In addition, the subordination relation is applied for deriving a regularity estimate.