Fractional Calculus and Applied Analysis最新文献

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations

has not been known and remained as an open problem for a few decades, where (Nge 2), (Delta _{alpha }=-left( -Delta right) ^{alpha /2}) denotes the fractional Laplace operator with (0<alpha le 2), (psi ) is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows:

We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index (alpha in [1,2)) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of (L^p([0,T];{{mathcal {C}}}_b^{alpha +beta }({{mathbb {R}}}^d))cap W^{1,p}([0,T];{{mathcal {C}}}_b^beta ({{mathbb {R}}}^d))) solution under the assumptions that the drift coefficient and nonhomogeneous term are in (L^p([0,T];{{mathcal {C}}}_b^{beta }({{mathbb {R}}}^d))) with (pin [alpha /(alpha -1),+infty ]) and (beta in (0,1)). As applications, we prove the unique strong solvability as well as Davie’s type uniqueness of time inhomogeneous stochastic differential equation with the drift in (L^p([0,T];{{mathcal {C}}}_b^{beta }({mathbb R}^d;{{mathbb {R}}}^d))) and driven by the (alpha )-stable process for (beta > 1-alpha /2) and (p>2alpha /(alpha +2beta -2)).

The maximal (B_{p,q}^{s})-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in ( B_{p,q}^{s}) and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the (B_{p,q}^{s})-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called (alpha )-fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the (alpha )-fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on ({mathcal {C}}(Delta , {mathbb {R}})), the space of all real-valued continuous functions defined on a triangular domain (Delta ), is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of (alpha )-fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for ({mathcal {C}}(Delta , {mathbb {R}})).

In this paper, we consider the inverse problem for identifying the source term and the initial value of time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator. Firstly, we prove that the problem is ill-posed and give the conditional stability result. Then, we choose the Tikhonov regularization method to solve this ill-posed problem, and give the error estimates under a priori and a posteriori regularization parameter selection rules. Finally, we give numerical examples to illustrate the effectiveness of this method.

This article proposes a predictor-corrector scheme for solving the fractional differential equations ({}_0^C D_t^alpha y(t) = f(t,y(t)), alpha >0) with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though ({}_0^C D_t^alpha y(t)) is not smooth at (t=0). Numerical examples are carried out to verify the theoretical analysis.

This study delves into the origin, evolution, and practical applications of fractional difference inequalities based on recent literature. The review provides an overview of existing inequalities proposed under various definitions. Furthermore, to enhance this potent mathematical tool, a series of new inequalities have been introduced. Additionally, leveraging renowned Lyapunov functions in continuous-time domain, their discrete-time counterparts have been formulated. Moreover, several new potential Lyapunov functions have been identified. This review aims to aid readers in selecting suitable inequalities and Lyapunov functions to analyze the stability of nabla fractional order systems.

In this paper, the problem of identifying Multiple-Input-Single-Output (MISO) systems with fractional models from noisy input-output available data is studied. The proposed idea is to use Higher-Order-Statistics (HOS), like fourth-order cumulants (foc), instead of noisy measurements. Thus, a fractional fourth-order cumulants based-simplified and refined instrumental variable algorithm (frac-foc-sriv) is first developed. Assuming that all differentiation orders are known a priori, it consists in estimating the linear coefficients of all Single-Input-Single-Output (SISO) sub-models composing the MISO model. Then, the frac-foc-sriv algorithm is combined with a nonlinear optimization technique to estimate all the parameters: coefficients and orders. The performances of the developed algorithms are analyzed using numerical examples. Thanks to fourth-order cumulants, which are insensitive to Gaussian noise, and the iterative strategy of the instrumental variable algorithm, the parameters estimation is consistent.

Unstable first order time delay systems are frequently encountered in industrial applications, such as chemical plants, hydraulic processes, in satellite communications or economic systems, to name just a few. The control of such processes is a challenging issue. In this paper a filtered Smith Predictor control structure is used to compensate for the process time delays and to ensure the stability of the overall closed loop system. A simplified type of a fractional order PI controller is then designed to meet zero steady state error and an overshoot requirement. The tuning is based on the root locus analysis of the closed loop fractional order system. Simulation examples are provided to validate the proposed method and to demonstrate the efficiency of the proposed control method. Comparisons with two existing methods are included to highlight the possibility of using the proposed method as an alternative solution for controlling these types of processes.

Let (({{mathcal {X}}},d,mu )) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral (I_beta ) associated with admissible functions and its commutators. Similarly to (I_beta ), corresponding results for Calderón–Zygmund operators T associated with admissible functions are also included in this article.