Fractional Calculus and Applied Analysis最新文献

This paper aims to establish the existence of a weak solution for the following problem:

in ({mathbb R}^{N}) where (Nge 1), (sin (0,1), lambda in (0,N), mathcal {H}(x,y,t)=int _{0}^{|t|} h(x,y,r)r dr,) ( h:{mathbb R}^{N}times {mathbb R}^{N}times [0,infty )rightarrow [0,infty )) is a generalized N-function and ((-Delta )^{s}_{mathcal {H}}) is a generalized fractional Laplace operator. The functions (V,K:{mathbb R}^{N}rightarrow (0,infty )), non-linear function (f:{mathbb R}rightarrow {mathbb R}) are continuous and ( F(t)=int _{0}^{t}f(r)dr.) First, we introduce the homogeneous fractional Musielak–Sobolev space and investigate their properties. After that, we pose the given problem in that space. To establish our existence results, we use variational technique based on the mountain pass theorem. We also prove the existence of a ground state solution by the method of Nehari manifold.

In this work, the exact solutions of time fractional Fokker-Planck equation are investigated using the symmetry approach. Also, the convergence of the reported solutions is proved along with the graphical interpretation of the obtained solutions.

In this paper, we investigate the existence of mild solutions of the nonlinear fractional Rayleigh-Stokes problem for a generalized second grade fluid on an infinite interval. We firstly show the boundedness and continuity of solution operator. And then, by using a generalized Arzelà-Ascoli theorem and some new techniques, we get the compactness on the infinite interval. Moreover, we prove the existence of global mild solutions of nonlinear fractional Rayleigh-Stokes problem.

In this paper, we are concerned with a two-term fractional differential equation with functional boundary conditions. We discuss the existence of two kinds of solutions with respect to this type of equation. In this sense, for a class of two-term problems with specific boundary conditions, we use Matlab software to calculate the eigenvalues of the boundary value problems with Riemann-Liouville fractional derivative as an application of the two-term fractional differential equation, and further, we derive the dependence of eigenvalues on some parameters. Finally, we indicate that this method of estimating the eigenvalues by means of such two-term fractional order equations will also help in solving other eigenvalue problems.

This paper delves into the existence of three weak solutions for a fractional ((varphi , psi ))-like system involving the fractional (varphi ) Laplacian and the fractional (psi ) Laplacian respectively within the fractional Orlicz-Sobolev space. The proof is achieved by the well-known Bonanno-Marano techniques.

Being based on coupling by change of measure and an approximation technique, the Harnack inequalities for a class of stochastic functional differential equations driven by fractional Ornstein-Uhlenbeck process with Hurst parameter (0<H<1/2) are established. By using a transformation formulas for fractional Brownian motion, the Harnack inequalities for stochastic functional differential equations driven by fractional Ornstein-Uhlenbeck process with Hurst parameter (1/2<H<1) are established.

In this investigation, we conduct a rigorous analysis of a class of non-homogeneous generalized double phase problems, characterized by the inclusion of the fractional (phi _{x ,y}^i(cdot ))-Laplacian operator (where (i=1,2)) and a Choquard-logarithmic nonlinearity, along with a real parameter. Our methodology involves establishing a set of precise conditions related to the Choquard nonlinearities and the continuous function (phi _{x ,y}^i), under which we are able to confirm the existence of multiple distinct solutions to the problem. The analysis is situated within the realm of fractional modular spaces. Key to our approach is the application of the mountain pass theorem, which allows us to circumvent the necessity of the Palais-Smale condition, beside this we lay in the strategic use of the Hardy-Littlewood-Sobolev inequality to underpin the theoretical framework of our study.

We consider fractional partial differential equations posed on the full space (mathbb {R}^d). Using the well-known Caffarelli-Silvestre extension to (mathbb {R}^d times mathbb {R}^+) as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on (mathbb {R}^d times (0,mathcal {Y})) converge to the solution of the original problem as (mathcal {Y}rightarrow infty ). Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem.

We consider multi-term fractional differential equations with continuous variable coefficients and differential operators of Erdélyi–Kober type and multiple independent fractional orders. We solve such equations in a general framework, obtaining explicit solutions in the form of uniformly convergent series. By considering several particular cases, we verify the consistency of our results with others previously obtained in the literature.

This paper investigates the existence of infinitely many pairs of nontrivial solutions for a class of nonsmooth fractional Hamiltonian systems, where the energy functional associated with the system is not continuously differentiable and does not satisfy the Palais-Smale condition. By considering a potential function of the form (V(t,x)=-K(t,x)+W(t,x)), where K and W are continuously differentiable functions with specific growth conditions, we extend existing results to cover cases involving nonsmoothness and certain types of nonlocal interactions. The study is based on variational methods and critical point theory, and we establish several theorems that guarantee the existence of multiple solutions under appropriate hypotheses on the nonlinearities of the system. These results contribute to the understanding of nonsmooth fractional Hamiltonian systems, particularly when traditional compactness conditions fail.