Related papers: An iterative method for the solution of Laplace-li…

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be…

In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in…

We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g} u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$ equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a…

We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -\Delta u + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ \Omega \subset…

We approximate the solution of the equation $$ -\Delta_S u+u = f $$ on a two-dimensional, embedded, orientable, closed surface $S$ where $-\Delta_S$ denotes the Laplace Beltrami operator on $S$ by using continuous, piecewise linear finite…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems,…

In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…

For each $\lambda \in \mathbb N^*$, we consider the integral equation: \[ \int_{\lambda y} ^{\lambda x} f(t)\, d t = f(x) - f(y) \mbox{ for every $(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two continuous functions…

In this paper, we present a fast multipole method (FMM) for solving the two-dimensional Laplace equation in a half-plane with Robin boundary conditions. The method is based on a novel expansion theory for the reaction component of the…

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} \Delta u=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{\eta-1}u+f\,\,\mbox{ on…

In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-\Delta)^{p+{\frac{\alpha}{2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{R}^n,\\…

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple "islands" are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a…

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed…

In this paper, we focus on the partial differential equation \begin{equation*} (-\Delta)^\frac{\alpha}{2} u(x)=f(x,u(x))\;\;\;\;\text{ in }\mathbb{R}^n, \end{equation*} where $0<\alpha\leq 2$. By the direct method of scaling spheres…

Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle…

In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -\Delta_1 u = F(x,u) & \text{in}\;\Omega,\\ \newline…

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle…

We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our…

In this paper, we deal with the following elliptic type problem $$ \begin{cases} (-\Delta)_{q(.)}^{s(.)}u + \lambda Vu = \alpha \left\vert u\right\vert^{p(.)-2}u+\beta \left\vert u\right\vert^{k(.)-2}u & \text{ in }\Omega, \\[7pt] u =0 &…