Related papers: Complex and real valued solutions for fractoinal H…

We show that the bounded solutions to the fractional Helmholtz equation, $(-\Delta)^s u= u$ for $0<s<1$ in $\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-\Delta)u= u$ in $\mathbb{R}^n$ for $n \ge…

A fractional diffusion equation based on Riemann-Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of $H$-functions. It differs from the known…

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form \[ \partial_t^\alpha w(t,x) = \phi(\Delta) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \]…

Cet article est consacre a l'etude des mesures limites associees a la solution de l'equation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est suppose regulier et l'operateur non-captif. La solution de…

We study the electric Helmholtz equation $\Delta u + Vu + \lambda u =f$ and show that, for certain potentials, the solution $u$ given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical…

We prove the existence of $L^2$ solutions to the Helmholtz equation $(-\Delta - 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$…

We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schroedinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We…

This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and…

The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a…

A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized…

A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM).…

This paper is concerned with the Fourier-Bessel method for the boundary value problems of the Helmholtz equation in a smooth simply connected domain. Based on the denseness of Fourier-Bessel functions, the problem can be approximated by…

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of Helmholtz equation by the WKBJ or EWT methods. We obtain the complex-valued solutions (called…

This paper gives the existence and uniqueness results for solution of fractional differential equations with Hilfer derivative. Using some new techniques and generalizing the restrictive conditions imposed on considered function, the…

In this paper we present in one-dimensional space a numerical solution of a partial differential equation of fractional order. This equation describes a process of anomalous diffusion. The process arises from the interactions within the…

We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the…

A major challenge of many diffraction calculations, using some form of the Rayleigh-Sommerfeld formulas, is the integration of a highly oscillatory integrand. Here we derive a potentially useful alternative form of solution to the Helmholtz…

Fundamental rules and definitions of Fractional Differintegrals are outlined. Factorizing 1-D and 2-D Helmholtz equations four fractional eigenfunctions are determined. The functions exhibit incident and reflected plane waves as well as…

This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view,…

The study of the limiting absorption principle for elliptic equations with periodic structures is very challenging when the dimension is greater than 1. The fundamental reason for the dimensional barrier is the mismatch between directional…