Fractional Calculus and Applied Analysis最新文献

This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation

subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in (B_1(0)) for any (tin mathbb {R}). Another one is to establish the Liouville theorem for homogeneous master equation

which states that all bounded solutions must be constant. We propose a new methodology for a direct method of moving planes applicable to the fully fractional heat operator ((partial _t-Delta )^s), and the proof of our main results based on this direct method involves the perturbation technique, limit argument as well as Fourier transform. This study opens up a way to investigate the geometric behavior of master equations, and provides valuable insights for establishing qualitative properties of solutions and even for deriving important Liouville theorems for other types of fractional order parabolic equations.

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay (I_nu ) Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay (I_nu ) random variable and the so–called Neumann series of modified Bessel functions of the first kind are established, providing, consequently, a new integral representation for such cdf in terms of a definite integral. Another fashion expression for the given cdf is derived in terms of the Grünwald-Letnikov fractional derivative of the widely applicable Marcum Q–function, which represents a certain simplification of the already existing relationship between McKay (I_nu ) random variable and a Marcum Q–functions. The exposition ends with some open questions, drawing the interested reader’s attention, among others, to the summation of some Neumann series.

We extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain (varOmega ) with non-local dynamic conditions on the boundary (partial varOmega ). Due to the pioneering nature of the present research, we propose here the apparently simple case of (varOmega =(0, infty )) with boundary ({0}) of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

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.