Related papers: Fractional extensions of some boundary value probl…
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for…
The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$, appears in a wide class of physical systems, including L\'evy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which…
Our work concerns the study of inverse problems of heat and wave equations involving the fractional Laplacian operator with zeroth order nonlinear perturbations. We recover nonlinear terms in the semilinear equations from the knowledge of…
This article is concerned with the existence and uniqueness of solutions to some fractional order boundary value problems. Our results are based on some fixed point theorems. For the applicability of our results, we provide an example.
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
In this paper, a boundary-value problem for the 3-D wave equation with Caputo and Bessel operators is investigated. Sufficient conditions on the initial data are established for the existence of a unique solution to the considered problem.
There has been considerable recent interest in solving non-local equations of motion which contain an infinite number of derivatives. Here, focusing on inflation, we review how the problem can be reformulated as the question of finding…
This paper presents a new technique to investigate the existence of solutions to fractional three-point boundary value problems at resonance in a Hilbert space. Based on the proposed method, the restricted conditions…
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the…
We present a numerical method for the solution of diffusion problems in unbounded planar regions with complex geometries of absorbing and reflecting bodies. Our numerical method applies the Laplace transform to the parabolic problem,…
The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal's necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using…
In order to solve fractional variational problems, there exist two theorems of necessary conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that…
We consider an inverse boundary value problem for diffusion equations with multiple fractional time derivatives. We prove the uniqueness in determining a number of fractional time-derivative terms, the orders of the derivatives and…
Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The…
In this work we study the following fractional critical problem $$ (P_{\lambda})=\left\{\begin{array}{ll} (-\Delta)^s u=\lambda u^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} \Omega\\ u=0 & \mbox{in} \RR^n\setminus \Omega\,,…
In this paper we consider second-order field theories in a variational setting. From the variational principle the Euler-Lagrange equations follow in an unambiguous way, but it is well known that this is not true for the Cartan form. This…
Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of…
Fractional differential equations model processes with memory effects, providing a realistic perspective on complex systems. We examine time-delayed differential equations, discussing first-order and fractional Caputo time-delayed…
We describe a simple way of constructing exponentially growing solutions of the second order systems with the Laplacian as the principal term.
We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the…