Fractional Calculus and Applied Analysis最新文献

In this paper we study existence and uniqueness of solution stochastic differential equations involving fractional integrals driven by Riemann-Liouville multifractional Brownian motion and a standard Brownian. Then, we obtain approximate numerical solution of our problem and colon cancer chemotherapy effect model are presented to confirm our results. We show that considering time dependent Hurst parameters play an important role to get more realistic results.

We investigate the fractional stochastic heat equation, driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. This class of process includes White-colored noise, fractional colored noise and other related processes. We give a sufficient condition for the existence of the mild solution and we establish some properties of its. Then, we study the self similarity and the path regularity of this solution with respect to time variable on the particular case when the noise behaves as a fractional Brownian motion in time.

The causal shift-invariant convolution is studied from the point of view of inversion. Abel’s algorithm, used in the tautochrone problem, is considered and Sonin’s existence condition is deduced. To generate pairs of functions verifying Sonin’s condition, the class of Mittag-Leffler type functions is used. In particular, functions that are impulse responses of ARMA(N,N) systems serve as a basis. The possible use of Abel’s procedure as a support for introducing generalized fractional derivatives is evaluated.

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order (alpha in (0,1)) in ({mathbb {R}}^d) was shown by the authors [4] to generate a semi-group on the space ({mathfrak {C}}) of continuous functions (f:{mathbb {R}}^+rightarrow {mathbb {R}}^d) with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) (equiv ) (id_{x_0}) for (x_0) (in ) ({mathbb {R}}^d). Here it is shown that this semi-dynamical system has a global Caputo attractor in ({mathfrak {C}}), which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition.

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown.

We consider the ill-posed inverse problem of determining an unknown source term appearing in abstract fractional diffusion-wave equations from a general noise assumption. Based on a Hölder-type source condition, we give the theoretical order optimality as well as the conditional stability result. To solve the problem, we propose fractional filter regularization methods, which can be regarded as an extension of the classical Tikhonov and Landweber methods. The idea is first to transform the problem into an ill-posed operator equation, then construct the regularization methods for the operator equation by introducing a suitable fractional filter function. As a natural further step, we study the convergence of the regularization methods, for which we derive order optimal rates of convergence under both a priori and a posteriori parameter choice rules. Applications of our fractional filter functions to both the fractional Tikhonov and the fractional Landweber filters are also investigated. Finally, three numerical examples in one-dimensional and two-dimensional cases are tested to validate our theoretical results.

Consider the following fractional Hamiltonian system:

Here, (_{t}D_{infty }^{alpha }) and (_{-infty }D_{t}^{alpha }) represent the Liouville-Weyl fractional derivatives of order (frac{1}{2}< alpha < 1), (L in C(mathbb {R}, mathbb {R}^{N^2})) is a symmetric matrix, and (W in C^{1}(mathbb {R} times mathbb {R}^N, mathbb {R})). By applying the Fountain Theorem and the Dual Fountain Theorem, we demonstrate that this system admits two distinct sequences of solutions under the condition that L meets a new non-coercive criterion, and the potential W(t, x) exhibits combined nonlinearities.

In this article, we present least fractional nonlinearity for exhibiting chaos in a memristor-based hyper-chaotic multi-stable hidden system. When implementing memristor-based systems, distinct dimensions/order define the memristor nonlinearity. In this work, the memristor dimension has been changed fractionally to identify the lowest order of nonlinearity required to induce chaos in a proposed system. The two-parameter frequency scanning helps in understanding both oscillation and non-oscillation regimes. The system fractional nonlinearity strength will help in deeper understanding of mathematical modelling and control. In addition, multistability and hidden oscillations were thoroughly investigated in the proposed system. The current work combines analytical, numerical, and experimental methods to demonstrate the system dynamics.

Let (p,qin [1,infty )), s be a nonnegative integer, (alpha in mathbb {R}), and (mathcal {X}) be (mathbb {R}^n) or a cube (Q_0subsetneqq mathbb {R}^n). In this article, the authors introduce the localized special John–Nirenberg–Campanato spaces via congruent cubes, (jn_{(p,q,s)_{alpha }}^{textrm{con}}(mathcal {X})), and show that, when (pin (1,infty )), the predual of (jn_{(p,q,s)_{alpha }}^{textrm{con}}(mathcal {X})) is a Hardy-kind space (hk_{(p',q',s)_{alpha }}^{textrm{con}}(mathcal {X})), where (frac{1}{p}+frac{1}{p'}=1=frac{1}{q}+frac{1}{q'}). As applications, in the case (mathcal {X}=mathbb {R}^n), the authors obtain the boundedness of local Calderón–Zygmund singular integrals and local fractional integrals on both (jn_{(p,q,s)_{alpha }}^{textrm{con}}(mathbb {R}^n)) and (hk_{(p,q,s)_{alpha }}^{textrm{con}}(mathbb {R}^n)). One novelty of this article is to find the appropriate expression of local Calderón–Zygmund singular integrals on (jn_{(p,q,s)_{alpha }}^{textrm{con}}(mathbb {R}^n)) and the other novelty is that, for the boundedness on (hk_{(p,q,s)_{alpha }}^{textrm{con}}(mathbb {R}^n)), the authors use the duality theorem to overcome the essential difficulties caused by the deficiency of both the molecular and the maximal function characterizations of (hk_{(p,q,s)_{alpha }}^{textrm{con}}(mathbb {R}^n)).

This paper discusses some aspects of Taylor’s formulas in fractional calculus, focusing on use of Caputo’s definition. Such formulas consist of a polynomial expansion whose coefficients are values of the fractional derivative evaluated at its starting point multiplied by some coefficients determined through the Gamma function. The properties of fractional derivatives heavily affect the expansion’s coefficients. In the first part of the paper, we review the currently available formulas in fractional calculus with a particular focus on the Caputo derivative. In the second part, we prove why the notion of sequential fractional derivative (i.e., n-fold fractional derivative) is required to build Taylor expansions in terms of fractional derivatives. Such properties do not seem to appear in the literature. Furthermore, some new properties of the expansion coefficients are also shown together with some computational examples in Wolfram Mathematica.