Related papers: Understanding and Resolving Singularities in 3D Di…
Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution…
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green…
The well-known expressions for the Green's functions for the Helmholtz equation in polar coordinates with Dirichlet and Neumann boundary conditions are transformed. The slowly converging double series describing these Green's functions are…
We review the integrable structure of the Dirichlet boundary problem in two dimensions. The solution to the Dirichlet boundary problem for simply-connected case is given through a quasiclassical tau-function, which satisfies the Hirota…
In this paper we analyse some possibilities of finding positive solutions for second order boundary value problems with Dirichlet and periodic boundary conditions, for which the correspondent Green's functions change sign. The obtained…
A method is given to obtain the Green's function for the Poisson equation in any arbitrary integer dimension under periodic boundary conditions. We obtain recursion relations which relate the solution in d-dimensional space to that in…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
A boundary integral equation method for the 3-D Helmholtz equation in multilayered media with many quasi-periodic layers is presented. Compared with conventional quasi-periodic Green's function method, the new method is robust at all…
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…
Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting…
Most Fredholm integral equations involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common…
We present a high order numerical method for the solution of the Neumann Green's function in two dimensions. For a general closed planar curve, our computational method resolves both the interior and exterior Green's functions with the…
In this paper, new representations of the Green's function for an acoustic d-dimensional half-space problem with impedance boundary conditions are presented. The main features of the new representation are: a) in addition to additive terms…
We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on…
As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness…
Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for…