Fractional Calculus and Applied Analysis最新文献

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.

This paper investigates stochastic averaging principle for a class of mixed slow-fast stochastic differential equations driven simultaneously by a multidimensional standard Brownian motion and a multidimensional fractional Brownian motion with Hurst parameter (1/2<H<1). The stochastic averaging principle shows that the slow component strongly converges to the solution of the corresponding averaged equations under a weaker condition than the Lipschitz one.

In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source, we establish the uniqueness of the inverse problem, and also a Hölder-type conditional stability of the inverse problem is firstly given. Meanwhile, we prove a stability estimate of optimal order for the inverse problem. Then some convergence estimates for the regularized solution are proved under an a-priori and an a-posteriori regularization parameter choice rule. Finally, several numerical experiments illustrate the effectiveness of the proposed method in one-dimensional case.

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the (1+alpha ) order convergence rate for fractional order (alpha in (0,1)), of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments.