Showing papers on "Laplace's equation published in 2021"
TL;DR: In this paper, a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks is introduced, which is used to improve the training in deep learning for partial differential equations.
23 citations
TL;DR: In this paper, the phase behavior of non-associating and associating fluids in nanopores is described using the conventional cubic plus association equation of state (CPA EoS).
Abstract: The conventional cubic‒plus‒association equation of state (CPA EoS) is extended to describe the phase behavior of non‒associating and associating fluids in nanopores. To consider the effects of molecule‒wall interaction, a new pressure term was introduced into the CPA EoS. The modified Laplace equation was used to describe the capillary condensation in confined space. A thermodynamic model, i.e., mCPA/Laplace, based on modified CPA EoS and Laplace equation was presented. Furthermore, while correlations among the tensile strength, fluid critical point, and liquid spinodal point were discussed, the negative pressure for the mCPA/Laplace and mSRK models was studied. The calculations of the mCPA/Laplace model for pure O2, Ar, N2, Kr, CO2, C2H6, C3H8, n‒C4H10, n‒C5H12, n‒C6H14, n‒C7H16 and certain binary mixtures agreed with experimental data. Using the parameter database from the experimental data, this promising study can be extended to both shale oil and shale gas.
21 citations
TL;DR: In this paper, a radial integration boundary element method (RI-IGABEM) based on precise integration method (PIM) is proposed to solve transient heat conduction problems of functionally gradient materials (FGMs) with heat source.
19 citations
TL;DR: In this article, another fundamental change was applied to address straight normal deferential conditions with consistent coefficients and SEE change of incomplete derivative is inferred and its appropriateness showed utilizing three equations: wave equation, heat equation and Laplace equation.
Abstract: In this paper another fundamental change in particular SEE change was applied to address straight normal deferential conditions with consistent coefficients and SEE change of incomplete derivative is inferred and its appropriateness showed utilizing three is inferred and its appropriateness showed utilizing: wave equation, heat equation and Laplace equation, we find the particular solutions of these equations.
16 citations
TL;DR: In this article, the Stokes equations and the energy transport equation do abide by the coordinate-transformation invariant theory if the former is replaced with the pressure Laplace equation, which enables us to take advantage of the merit of this theory and to analytically design metamaterial thermo-hydrodynamic cloaks.
15 citations
TL;DR: The most important properties of the conformable derivative and integral of Laplace's integral have been recently introduced in this paper, where the solution of this mathematical problem with Dirichlet-type and Neumann-type conditions are discussed.
Abstract: The most important properties of the conformable derivative and integral have been recently introduced. In this paper, we propose and prove some new results on conformable Laplace’s equation. We discuss the solution of this mathematical problem with Dirichlet-type and Neumann-type conditions. All our obtained results will be applied to some interesting examples.
14 citations
TL;DR: In this article, a fast method for computing the electrostatic energy and forces for a collection of charges in doubly-periodic slabs with jumps in the dielectric permittivity at the slab boundaries was developed.
Abstract: We develop a fast method for computing the electrostatic energy and forces for a collection of charges in doubly-periodic slabs with jumps in the dielectric permittivity at the slab boundaries. Our method achieves spectral accuracy by using Ewald splitting to replace the original Poisson equation for nearly-singular sources with a smooth far-field Poisson equation, combined with a localized near-field correction. Unlike existing spectral Ewald methods, which make use of the Fourier transform in the aperiodic direction, we recast the problem as a two-point boundary value problem in the aperiodic direction for each transverse Fourier mode, for which exact analytic boundary conditions are available. We solve each of these boundary value problems using a fast, well-conditioned Chebyshev method. In the presence of dielectric jumps, combining Ewald splitting with the classical method of images results in smoothed charge distributions which overlap the dielectric boundaries themselves. We show how to preserve spectral accuracy in this case through the use of a harmonic correction which involves solving a simple Laplace equation with smooth boundary data. We implement our method on Graphical Processing Units, and combine our doubly-periodic Poisson solver with Brownian Dynamics to study the equilibrium structure of double layers in binary electrolytes confined by dielectric boundaries. Consistent with prior studies, we find strong charge depletion near the interfaces due to repulsive interactions with image charges, which points to the need for incorporating polarization effects in understanding confined electrolytes, both theoretically and computationally.
12 citations
TL;DR: In this article, a fast method for computing the electrostatic energy and forces for a collection of charges in doubly periodic slabs with jumps in the dielectric permittivity at the slab boundaries was developed.
Abstract: We develop a fast method for computing the electrostatic energy and forces for a collection of charges in doubly periodic slabs with jumps in the dielectric permittivity at the slab boundaries. Our method achieves spectral accuracy by using Ewald splitting to replace the original Poisson equation for nearly singular sources with a smooth far-field Poisson equation, combined with a localized near-field correction. Unlike existing spectral Ewald methods, which make use of the Fourier transform in the aperiodic direction, we recast the problem as a two-point boundary value problem in the aperiodic direction for each transverse Fourier mode for which exact analytic boundary conditions are available. We solve each of these boundary value problems using a fast, well-conditioned Chebyshev method. In the presence of dielectric jumps, combining Ewald splitting with the classical method of images results in smoothed charge distributions, which overlap the dielectric boundaries themselves. We show how to preserve the spectral accuracy in this case through the use of a harmonic correction, which involves solving a simple Laplace equation with smooth boundary data. We implement our method on graphical processing units and combine our doubly periodic Poisson solver with Brownian dynamics to study the equilibrium structure of double layers in binary electrolytes confined by dielectric boundaries. Consistent with prior studies, we find strong charge depletion near the interfaces due to repulsive interactions with image charges, which points to the need for incorporating polarization effects in understanding confined electrolytes, both theoretically and computationally.
12 citations
12 citations
11 citations
TL;DR: In this paper, the radial integration boundary element method (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source.
Abstract: The isogeometric analysis boundary element method (IGABEM) has great potential in the simulation of heat conduction problems due to its exact geometric representation and good approximation properties. In this paper, the radial integration IGABEM (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source. Similar to traditional BEM, the domain integrals cannot be avoided since the foundational solution for the Laplace equation is used to derive integral equation. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. In order to validate the accuracy and convergence of the RI-IGABEM in the analysis of the single or multiple boundary heat conduction problems, several 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number of applied points, the order of basis functions, and the position of internal applied points.
TL;DR: A recently proposed three-component Camassa-Holm equation is considered in this article, and it is shown that this system is a bi-Hamiltonian system, which is a more general formulation of the Camassah-Holme equation.
Abstract: A recently proposed three-component Camassa-Holm equation is considered It is shown that this system is a bi-Hamiltonian system
TL;DR: In this paper, the authors considered the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain, assuming that the object is occupied by an exotic material which obeys a nonlinear Ohms' law.
Abstract: In this work, we are concerned with the problem of estimating the size of an inclusion embedded in an object laying in the two dimensional domain. We assume that the object is occupied by an exotic material which obeys a nonlinear Ohms' law. In view of the assumption of the power law, we thus consider the weighted \begin{document}$ p $\end{document} -Laplace equation as a model problem in this case. Using only one voltage-current measurement, we give upper and lower bounds of the size of the inclusion.
01 Jan 2021
TL;DR: In this article, an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Laplace equation in the unit ball is obtained.
Abstract: In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.
TL;DR: In this paper, the authors introduced a general technique to transform a PDE problem on an unbounded domain to a poisson problem on a bounded domain using the Kelvin transform, which essentially inverts the distance from the origin.
Abstract: Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.
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TL;DR: In this paper, the geometric composition of a perfect fluid spacetime with torse-forming vector field is discussed in connection with conformal Ricci-Yamabe metric and conformal ∆-Ricci and Yamabe metric.
Abstract: The present paper is to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field {\xi} in connection with conformal Ricci-Yamabe metric and conformal {\eta}-Ricci-Yamabe metric. Here we have delineated the conditions for conformal Ricci-Yamabe soliton to be expanding, steady, or shrinking. Later, we have acquired Laplace equation from conformal {\eta}-Ricci-Yamabe soliton equation when the potential vector field {\xi} of the soliton is of gradient type. Lastly, we have designated perfect fluid with Robertson-Walker spacetime and some applications of physics and gravity.
TL;DR: In this paper, a solution of Laplace equation, after changing certain initial conditions in terms of wavelet transformation is obtained, which is further applied to denoise magnetic resonance (MR) images from brain web dataset at different noise levels.
TL;DR: An immersed boundary method is used to model non-stationary boundaries such as the free surface or the surface of a rigid body, and overlapping, body-fixed grids that are locally Cartesian to refine the solution near moving bodies to examine in depth the importance of nonlinear effects in the interaction between waves and rigid bodies.
TL;DR: In this paper, the authors established a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation, based on energy estimates and a parabolic nonlocal version of De Giorgi's method.
Abstract: We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi’s method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Holder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
TL;DR: In this article, the L q -mixed problem in domains in R n with C 1, 1 -boundary is considered and the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz.
Abstract: We consider the L q -mixed problem in domains in R n with C 1 , 1 -boundary. We assume that the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz. With these assump...
TL;DR: In this paper, the interior Holder regularity of spatial gradient of viscosity solution to the parabolic normalized p (x, t ) -Laplace equation u t = ( δ i j + ( p ( x, t ) − 2 ) u i u j | D u | 2 ).
TL;DR: In this article, the authors established local Lipschitz continuity of solutions for a class of sub-elliptic equations of divergence form, in the Heisenberg Group.
Abstract: The goal of this article is to establish local Lipschitz continuity of solutions for a class of sub-elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the sub-elliptic p -Laplace equation and more general quasilinear equations with polynomial or exponential type growth.
TL;DR: In this paper, the authors provide a rigorous justification of a new asymptotic model for low-cost numerical simulations of the acoustic wave scattering created by very small obstacles, which is based on near-field and far-field developments that are then matched by a key procedure that they describe and demonstrate.
Abstract: The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, includingthe transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev’s seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.
TL;DR: In this paper, a wavemaker theory for a moving boundary wavemaker of piston-type based on a nonlinear dispersive shallow water model, where the classical Boussinesq equations are employed as a starting point, is presented.
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TL;DR: In this paper, it was shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole in the Laplace equation with Neumann boundary condition.
Abstract: The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than $1+10^{-3}$. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
TL;DR: Analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces and use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes are investigated.
Abstract: The work is concerned with two problems: (a) analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces, (b) use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes. We use the ultraweak variational formulation for the model Laplace equation. The theoretical estimates are supported with 3D numerical experiments.
TL;DR: In this paper, the impact of the surrounding fluid of a dam-reservoir system on the stiffened lock gate structure subjected to an external sinusoidal acceleration is examined.
TL;DR: Numerical results validate the fast convergence of the MEs for the reaction components, and the O(N) complexity of the FMM with a given truncation number p for charge interactions in 3-D layered media.
TL;DR: In this article, a fully non-linear potential flow (FNPF) wave model is used to reproduce irregular sea states with different severity of wave breaking and a comprehensive procedure is introduced to ensure the quality of the reproduced fullscale sea states.
08 Jul 2021
TL;DR: In this paper, explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs, one of which hinges on the use of a radial integral operator introduced recently in the literature.
Abstract: We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lame–Navier equation in bounded and unbounded domains are discussed.