Fractional Calculus and Applied Analysis最新文献

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.

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.