Fractional Calculus and Applied Analysis最新文献

This article studies topological properties of the solution set for a class of Caputo fractional delayed evolution inclusions. Firstly, in the scenario when the cosine family is noncompact, the compactness and (R_{delta })-property are obtained for the mild solution set. Then, as an application of the above obtained results, the approximative controllability is demonstrated. Finally, an example is given as an illustration.

The paper addresses structural properties of distributed order controllers. A Distributed Order PID (DOPID) controller is a control structure in which a continuum of “differintegral” actions of orders between -1 and 1 are integrated together, and where relative contributions of different orders is determined by a weighting function. This stands in sharp contrast to conventional proportional-integral-derivative controllers, or even fractional order PID (FPID) controller and multi-term FPID, in which discrete actions appear only and a finite set of real parameters, controller gains, are sufficient to specify contributions of each action. The paper presents an in-depth analysis of the DOPID controller, emphasizing its theoretical properties and distinctions with integer and fractional order PID. It is shown that DOPID can be considered a generalization of these controllers only if the weighting function is a sequence of Dirac pulses. Some structural deficiencies of DOPID in case of a wide class of weighting functions have been emphasized. A modified DOPID structure — which we refer to as the DOPID of the Second Kind — is proposed and analyzed as well. Among other things, it has been shown that such modified DOPID controller provides better generalization to discrete order controllers (PID and FPID).

This work is an extended and supplemented version of the paper presented at ICFDA 2024 at Bordeaux University, July 2024 (see [27]).

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity

where (mathfrak {M}) is the Kirchhoff function, (Q=mathbb {R}^{2N}setminus ((mathbb {R}^Nsetminus Omega )times (mathbb {R}^Nsetminus Omega ))), (Omega subset mathbb {R}^N), is a bounded domain with Lipschitz boundary, (lambda >0), (N>ps), (0<s,gamma <1), ((-Delta )_{p}^{s}) is the fractional p-Laplacian for (1<p<infty ) and (p_s^*=frac{Np}{N-ps}) is the critical Sobolev exponent. We employ a cut-off argument to obtain the existence of k (being arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove an uniform (L^{infty }({Omega })) bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.

This paper investigates the existence and multiplicity of positive solutions to the following semilinear problem:

where (fin C([0,infty ),{mathbb {R}})) represents an oscillating nonlinearity that satisfies a type of area condition. Our main analytical tools include variational methods and the sub-supersolution method.

In this study, we establish an explicit representation of solutions to (psi )-Hilfer type linear fractional differential equations with variable coefficients in weighted spaces. Furthermore, we prove the existence and uniqueness of solutions for these equations. As a special case, we derive corresponding results for (psi )-fractional differential equations with variable coefficients. To demonstrate the practical applications of our theoretical results, we derive explicit solutions for several representative cases, including the voltmeter equation in electrochemistry, the equation around an (alpha )-ordinary point, and the fractional Ayre equation. Furthermore, we provide numerical simulations.

In this paper, we study the well-posedness and discretization of the space-time fractional parabolic equations (STFPEs) of the Sturm-Liouville type on a metric star graph. The considered problem involves the fractional time derivative in the Caputo sense, and the spatial fractional derivative is of the Sturm-Liouville type consisting of the composition of the right-sided Caputo derivative and left-sided Riemann-Liouville fractional derivative. By introducing the appropriate function spaces for the involved fractional operators in both the time and spatial variable, we prove the well-posedness of the weak solution of the considered STFPEs by using the Galerkin approximation method. Moreover, we propose a difference scheme to find the numerical solution of the STFPEs on a metric star graph by approximating the Caputo time derivative using the L1 method and spatial fractional derivative with the Grünwald-Letnikov formula. Finally, we illustrate the performance and the accuracy of the proposed difference scheme via examples.

In this paper we study an evolution problem (FDIVHVI) which constitutes of the nonlinear fractional differential inclusion with damping driven by a variational-hemivariational inequality (VHVI) in Banach spaces. More precisely, first, it is shown that the solution set for VHVI is nonempty, bounded, convex and closed under the surjectivity theorem and the Minty formula. Then, we introduce an associated multivalued map with the solution set of the VHVI, and prove that it is upper semicontinuous and measurable. Finally, by utilizing the fixed point theorem of condensing multivalued operators, properties of ((alpha , mu ))-regularized families of operators and properties of measure of noncompactness, we show the existence of mild solutions for FDIVHVI.

In this note, we present a new uniqueness criterion for nonlinear fractional differential equations, which can be seen as an improvement of the result given by Ferreira [Bull. London Math. Soc. 45, 930–934 (2013)].

In this paper, Cauchy problem for incompressible Navier-Stokes equations with time fractional differential operator and fractional Laplacian in (mathbb {R}^n) ((nge 2)) is investigated. The global and local existence and uniqueness of mild solutions are obtained with the help of Banach fixed point theorem when the initial data belongs to (L^{p_{c}}(mathbb {R}^n)) ((p_c=frac{n}{alpha -1})). In addition, the decay properties of mild solutions to the considered time-space fractional equations are constructed. Moreover, it is shown that when the initial data belongs to (L^{p_{c}}(mathbb {R}^n)cap L^{p}(mathbb {R}^n)) with (1<p<p_c), the existence and uniqueness of global and local mild solutions can also be established. At the end of this paper, the integrability of mild solutions is discussed.

Our main concern is the existence of a new (PC_{2-v})-mild solution for Hilfer fractional impulsive evolution equations of order (alpha in (1,2)) and (beta in [0,1]) as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of (PC_{2-v})-mild solution for the objective equations by the Laplace transform and probability density function. Secondly, we rely on the Banach contraction mapping principle to discuss a new existence and uniqueness result of (PC_{2-v})-mild solution when the sine family is compact. Thirdly, a sufficient condition for the approximate controllability result of impulsive evolution equations is formulated and proved under the assumptions that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximately controllable. Finally, two examples are given to illustrate the validity of the obtained results in the application.