In this study,the wave motion in elastodynamics for unbounded media is modeled using an unsplit-field perfectly matched layer(PML)formulation that is solved by employing an isogeometric analysis(IGA).In the adopted co...In this study,the wave motion in elastodynamics for unbounded media is modeled using an unsplit-field perfectly matched layer(PML)formulation that is solved by employing an isogeometric analysis(IGA).In the adopted combination,the non-uniform rational B-spline(NURBS)functions are employed as basis functions.Moreover,the unbounded and artificial domains,defined in the PML method,are contained in a single patch domain.Based on the proposed scheme,the approximation of the geometry problem is set in a new scheme in which the PML’s absorbing and attenuation properties and the description of traveling waves can be represented.This includes a higher continuity and smoother approximation of the computed domain.As high-order NURBS basis functions are non-interpolatory,a penalty method is present to apply a time-dependent displacement load.The performance of the NURBS-based PML is analyzed through numerical examples for 1D and 2D domains,considering homogeneous and heterogeneous media.Further,we verify the long-time numerical stability of the present method.The developed method can be used to simulate hypothetical stratified domains commonly encountered in soil-structure interaction analyses.展开更多
In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not b...In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.展开更多
In this paper we investigate the existence and stability of periodic solutions(on a half-line R_(+))and almost periodic solutions on the whole line time-axis R to the Boussinesq system on several classes of unbounded ...In this paper we investigate the existence and stability of periodic solutions(on a half-line R_(+))and almost periodic solutions on the whole line time-axis R to the Boussinesq system on several classes of unbounded domains of R^(n) in the framework of interpolation spaces.For the linear Boussinesq system we combine the L^(p)—L^(q)-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions.Then,we prove the existence of periodic solutions by invoking Massera’s principle.We also prove the existence of almost periodic solutions.Then we use the results of the linear Boussinesq system to establish the existence,uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces.Our results cover and extend the previous ones obtained in[13,34,38].展开更多
This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum ...This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum principles,the global boundary Holder estimates and the boundary Harnack inequalities of the equations,we show that all solutions bounded from below are linear combinations of two special solutions(exponential growth at one end and exponential decay at the other)with a bounded solution to the degenerate equations.展开更多
On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three c...On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.展开更多
Since [1] established the Pohozaev identity in bounded domains, this identity has been the principal tool to deal with the non-existence of the equation
Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bound...Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers,which typically require significant effort,and the presence of nonlinearity in the equation makes such designs even more challenging.Emerging deep learning-based methods for solving PDEs,with the physics-informed neural networks(PINNs)method as a representative,still face significant challenges when directly used to solve PDEs on unbounded domains.Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs.In light of this,this paper proposes a novel and effective data generationbased operator learning method for solving PDEs on unbounded domains.The key idea behind this method is to generate high-quality training data.Specifically,we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term.Then,using these constructed data comprising exact solutions,initial conditions,and source terms,we train an operator learning model called MIONet,which is capable of handling multiple inputs,to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest.Finally,we utilize the generalization ability of this model to predict the solution of the target PDE.The effectiveness of this method is exemplified by solving the wave equation and the Schr¨odinger equation defined on unbounded domains.More importantly,the proposed method can deal with nonlinear problems,which has been demonstrated by solving Burgers’equation and Korteweg-de Vries(KdV)equation.The code is available at https://github.com/ZJLAB-AMMI/DGOL.展开更多
Examines the development of the composite legendre approximation in unbounded domains. Proof of the stability and convergence of a proposed scheme; Discussion of two-dimensional exterior problems; Error estimations.
With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We prove...With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.展开更多
Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Bouss...Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.展开更多
The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains,and shows exponential rate of convergence of the approximate solution toward the solution.
The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficie...The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation,and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method,and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.展开更多
In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possib...In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛV<min ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε0>0 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.展开更多
By constructing a special cone and using cone compression and expansion fixed point theorem, this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for...By constructing a special cone and using cone compression and expansion fixed point theorem, this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for a class of first order differential equation. As applications of the main results, two examples are given at the end of this paper.展开更多
We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi,Laguerre and Hermite functions.A detailed comparison of the convergence rates of these spectral ...We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi,Laguerre and Hermite functions.A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out,both theoretically and computationally.A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.展开更多
The author studies the infinite element method for the boundary value problems of second order elliptic equations on unbounded and multiply connected domains. The author makes a partition of the domain into infinite n...The author studies the infinite element method for the boundary value problems of second order elliptic equations on unbounded and multiply connected domains. The author makes a partition of the domain into infinite number of elements. Without dividing the domain, as usual, into a bounded one and an exterior one, he derives an initial value problem of an ordinary differential equation for the combined stiffness matrix, then obtains the approximate solution with a small amount of computer work. Numerical examples are given.展开更多
The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the...The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.展开更多
In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the...In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.展开更多
We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate tran...We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate transformation,which maps an unbounded domain into a unit square.Arbitrary geometries are defined by suitable level-set functions.The equations are discretized by classical nine-point stencil on interior points,while boundary conditions and high order reconstructions are used to define the field variables at ghost-points,which are grid nodes external to the domain with a neighbor inside the domain.The linear system arising from such discretization is solved by a multigrid strategy.The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources.The method is suitable to treat problems in which the geometry of the source often changes(explore the effects of different scenarios,or solve inverse problems in which the geometry itself is part of the unknown),since it does not require complex re-meshing when the geometry is modified.Several numerical tests are successfully performed,which asses the effectiveness of the present approach.展开更多
In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poinca...In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaréinequality in unbounded cylindrical domains,we first study the asymptotic property of the first eigenvalueλp,s(ωδ)with respect to the domainωδ.Then,by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains,we prove the existence results.The present work complements the results of Mosconi–Perera–Squassina–Yang[The Brezis–Nirenberg problem for the fractional p-Laplacian.C alc.Var.Partial Differential Equations,55(4),25 pp.2016]to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem[Positive solutions for elliptic equations with critical growth in unbounded domains.In:Chapman Hall/CRC Press,Boca Raton,2000,192–199]to the fractional p-Laplacian setting.展开更多
文摘In this study,the wave motion in elastodynamics for unbounded media is modeled using an unsplit-field perfectly matched layer(PML)formulation that is solved by employing an isogeometric analysis(IGA).In the adopted combination,the non-uniform rational B-spline(NURBS)functions are employed as basis functions.Moreover,the unbounded and artificial domains,defined in the PML method,are contained in a single patch domain.Based on the proposed scheme,the approximation of the geometry problem is set in a new scheme in which the PML’s absorbing and attenuation properties and the description of traveling waves can be represented.This includes a higher continuity and smoother approximation of the computed domain.As high-order NURBS basis functions are non-interpolatory,a penalty method is present to apply a time-dependent displacement load.The performance of the NURBS-based PML is analyzed through numerical examples for 1D and 2D domains,considering homogeneous and heterogeneous media.Further,we verify the long-time numerical stability of the present method.The developed method can be used to simulate hypothetical stratified domains commonly encountered in soil-structure interaction analyses.
文摘In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.
基金financially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2021.04financially supported by Vietnam Ministry of Education and Training under Project B2022-BKA-06.
文摘In this paper we investigate the existence and stability of periodic solutions(on a half-line R_(+))and almost periodic solutions on the whole line time-axis R to the Boussinesq system on several classes of unbounded domains of R^(n) in the framework of interpolation spaces.For the linear Boussinesq system we combine the L^(p)—L^(q)-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions.Then,we prove the existence of periodic solutions by invoking Massera’s principle.We also prove the existence of almost periodic solutions.Then we use the results of the linear Boussinesq system to establish the existence,uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces.Our results cover and extend the previous ones obtained in[13,34,38].
文摘This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum principles,the global boundary Holder estimates and the boundary Harnack inequalities of the equations,we show that all solutions bounded from below are linear combinations of two special solutions(exponential growth at one end and exponential decay at the other)with a bounded solution to the degenerate equations.
基金the National Natural Science Foundation of China(Grant Nos.1140116211301136)the ScienceFoundation of Hebei Province(Grant No.A2014208158)
文摘On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.
基金This work is supported in port by the Foundation of Zhongshan University Advanced Research Center.
文摘Since [1] established the Pohozaev identity in bounded domains, this identity has been the principal tool to deal with the non-existence of the equation
基金supported by the China Postdoctoral Science Foundation(No.2023M743266)Zhejiang Provincial Postdoctoral Research Project Merit-based Funding(No.ZJ2023067)+1 种基金Exploratory Research Project(No.2022RC0AN02)Research Initiation Project(No.K2022RC0PI01)of Zhejiang Lab.
文摘Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers,which typically require significant effort,and the presence of nonlinearity in the equation makes such designs even more challenging.Emerging deep learning-based methods for solving PDEs,with the physics-informed neural networks(PINNs)method as a representative,still face significant challenges when directly used to solve PDEs on unbounded domains.Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs.In light of this,this paper proposes a novel and effective data generationbased operator learning method for solving PDEs on unbounded domains.The key idea behind this method is to generate high-quality training data.Specifically,we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term.Then,using these constructed data comprising exact solutions,initial conditions,and source terms,we train an operator learning model called MIONet,which is capable of handling multiple inputs,to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest.Finally,we utilize the generalization ability of this model to predict the solution of the target PDE.The effectiveness of this method is exemplified by solving the wave equation and the Schr¨odinger equation defined on unbounded domains.More importantly,the proposed method can deal with nonlinear problems,which has been demonstrated by solving Burgers’equation and Korteweg-de Vries(KdV)equation.The code is available at https://github.com/ZJLAB-AMMI/DGOL.
文摘Examines the development of the composite legendre approximation in unbounded domains. Proof of the stability and convergence of a proposed scheme; Discussion of two-dimensional exterior problems; Error estimations.
文摘With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.
基金supported by M.E.C. (Spain), Project MTM 2006-07932supported by Junta de Andalucía, Project P06- FQM- 02373supported by Fondecyt-Chile (Grant No. 1080628)
文摘Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.
基金supported by the Ministry of Education and Science of the Russian Federation(No.02.a03.21.0008)
文摘The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains,and shows exponential rate of convergence of the approximate solution toward the solution.
基金supported by FRG of Hong Kong Baptist University,and RGC of Hong Kong.
文摘The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation,and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method,and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.
基金supported by National Natural Science Foundation of China(Grant Nos.11071095 and 11371159)Hubei Key Laboratory of Mathematical Sciences
文摘In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u>0 in Ω,N>p>2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛV<min ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε0>0 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.
基金Supported by the National Natural Science Foundation of China(No.10671167)the Natural Science Foundation of Liaocheng University(31805)
文摘By constructing a special cone and using cone compression and expansion fixed point theorem, this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for a class of first order differential equation. As applications of the main results, two examples are given at the end of this paper.
基金The work of J.S.is partially supported by the NFS grant DMS-0610646The work of L.W.is partially supported by a Start-Up grant from NTU and by Singapore MOE Grant T207B2202Singapore grant NRF 2007IDM-IDM002-010.
文摘We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi,Laguerre and Hermite functions.A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out,both theoretically and computationally.A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.
基金This work was supported by the China State Major Key Project for Basic Researches Science Fund of the Ministry of Education
文摘The author studies the infinite element method for the boundary value problems of second order elliptic equations on unbounded and multiply connected domains. The author makes a partition of the domain into infinite number of elements. Without dividing the domain, as usual, into a bounded one and an exterior one, he derives an initial value problem of an ordinary differential equation for the combined stiffness matrix, then obtains the approximate solution with a small amount of computer work. Numerical examples are given.
文摘The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.
基金supported by National Natural Science Foundation of China(Grant Nos.11771237 and 41390452)
文摘In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.
基金the OTRIONS project under the European Territorial Cooperation Programme Greece-Italy 2007-2013,and by PRIN 2009“Innovative numerical methods for hyperbolic problems with applications to fluid dynamics,kinetic theory and computational biology”.
文摘We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate transformation,which maps an unbounded domain into a unit square.Arbitrary geometries are defined by suitable level-set functions.The equations are discretized by classical nine-point stencil on interior points,while boundary conditions and high order reconstructions are used to define the field variables at ghost-points,which are grid nodes external to the domain with a neighbor inside the domain.The linear system arising from such discretization is solved by a multigrid strategy.The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources.The method is suitable to treat problems in which the geometry of the source often changes(explore the effects of different scenarios,or solve inverse problems in which the geometry itself is part of the unknown),since it does not require complex re-meshing when the geometry is modified.Several numerical tests are successfully performed,which asses the effectiveness of the present approach.
基金Natural Science Foundation of China(Grant No.12071185)。
文摘In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaréinequality in unbounded cylindrical domains,we first study the asymptotic property of the first eigenvalueλp,s(ωδ)with respect to the domainωδ.Then,by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains,we prove the existence results.The present work complements the results of Mosconi–Perera–Squassina–Yang[The Brezis–Nirenberg problem for the fractional p-Laplacian.C alc.Var.Partial Differential Equations,55(4),25 pp.2016]to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem[Positive solutions for elliptic equations with critical growth in unbounded domains.In:Chapman Hall/CRC Press,Boca Raton,2000,192–199]to the fractional p-Laplacian setting.
