In this paper, based on the natural boundary reduction suggested by Feng and Yu, an overlapping domain decomposition method with its discretization is discussed. This method is very effective especially for problems o...In this paper, based on the natural boundary reduction suggested by Feng and Yu, an overlapping domain decomposition method with its discretization is discussed. This method is very effective especially for problems over unbounded domains. The geometric convergency of both continuous and discrete problems is proved. The theoretical results as well as the numerical examples show that the convergence rate of this discrete Schwarz iteration is independent of the finite element mesh size, but dependent on the frequency of the exact solution and the overlapping degree of the subdomains.展开更多
In this paper, we are concerned with the following class of elliptic problems:where 2 = 2N/（N-2） is the critical Sobolev exponent, 2 【 q 【 2 , 0≤μ 【 μˉ=（N-2）2<sub>4</sub> , a（x）, k（x） ∈ C（...In this paper, we are concerned with the following class of elliptic problems:where 2 = 2N/（N-2） is the critical Sobolev exponent, 2 【 q 【 2 , 0≤μ 【 μˉ=（N-2）2<sub>4</sub> , a（x）, k（x） ∈ C（RN ）. Through a compactness analysis of the functional corresponding to the problems , we obtain the existence of positive solutions for this problem under certain assumptions on a（x） and k（x）.展开更多
This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinea...This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.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
In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state...In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.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
This paper presents a novel approach called the boundary integrated neural networks(BINNs)for analyzing acoustic radiation and scattering.The method introduces fundamental solutions of the time-harmonic wave equation ...This paper presents a novel approach called the boundary integrated neural networks(BINNs)for analyzing acoustic radiation and scattering.The method introduces fundamental solutions of the time-harmonic wave equation to encode the boundary integral equations(BIEs)within the neural networks,replacing the conventional use of the governing equation in physics-informed neural networks(PINNs).This approach offers several advantages.First,the input data for the neural networks in the BINNs only require the coordinates of“boundary”collocation points,making it highly suitable for analyzing acoustic fields in unbounded domains.Second,the loss function of the BINNs is not a composite form and has a fast convergence.Third,the BINNs achieve comparable precision to the PINNs using fewer collocation points and hidden layers/neurons.Finally,the semianalytic characteristic of the BIEs contributes to the higher precision of the BINNs.Numerical examples are presented to demonstrate the performance of the proposed method,and a MATLAB code implementation is provided as supplementary material.展开更多
文摘In this paper, based on the natural boundary reduction suggested by Feng and Yu, an overlapping domain decomposition method with its discretization is discussed. This method is very effective especially for problems over unbounded domains. The geometric convergency of both continuous and discrete problems is proved. The theoretical results as well as the numerical examples show that the convergence rate of this discrete Schwarz iteration is independent of the finite element mesh size, but dependent on the frequency of the exact solution and the overlapping degree of the subdomains.
基金supported by National Natural Science Foundation of China (Grant Nos.10631030, 10871075)the PhD Specialized Grant of the Ministry of Education of China (Grant No. 20060511001)the Natural Science Foundation of Guangdong Province, China (Grant No. 9451064201003736)
文摘In this paper, we are concerned with the following class of elliptic problems:where 2 = 2N/（N-2） is the critical Sobolev exponent, 2 【 q 【 2 , 0≤μ 【 μˉ=（N-2）2<sub>4</sub> , a（x）, k（x） ∈ C（RN ）. Through a compactness analysis of the functional corresponding to the problems , we obtain the existence of positive solutions for this problem under certain assumptions on a（x） and k（x）.
基金Research is supported in part by National Natural Science Foundation of China （No. 10471073） and RGC of Hong Kong and in part by RGC of Hong Kong and FRG of Hong Kong Baptist University.
文摘This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.
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.
文摘In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.
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.
基金Natural Science Foundation of Shandong Province of China,Grant/Award Numbers:ZR2022YQ06,ZR2021JQ02Development Plan of Youth Innovation Team in Colleges and Universities of Shandong Province,Grant/Award Number:2022KJ140+2 种基金National Natural Science Foundation of China,Grant/Award Number:12372199Fund of the Key Laboratory of Road Construction Technology and Equipment,Chang'an University,Grant/Award Number:300102253502Water Affairs Technology Project of Nanjing,Grant/Award Number:202203。
文摘This paper presents a novel approach called the boundary integrated neural networks(BINNs)for analyzing acoustic radiation and scattering.The method introduces fundamental solutions of the time-harmonic wave equation to encode the boundary integral equations(BIEs)within the neural networks,replacing the conventional use of the governing equation in physics-informed neural networks(PINNs).This approach offers several advantages.First,the input data for the neural networks in the BINNs only require the coordinates of“boundary”collocation points,making it highly suitable for analyzing acoustic fields in unbounded domains.Second,the loss function of the BINNs is not a composite form and has a fast convergence.Third,the BINNs achieve comparable precision to the PINNs using fewer collocation points and hidden layers/neurons.Finally,the semianalytic characteristic of the BIEs contributes to the higher precision of the BINNs.Numerical examples are presented to demonstrate the performance of the proposed method,and a MATLAB code implementation is provided as supplementary material.
