In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains.We show that the use of the Hermite basis can de-convolute the troublesome convolutional ope...In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains.We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian.As a result,the“stiffness”matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N^(2))arithmetic operations.Moreover,the singular factor in a typical kernel function can be fully absorbed by the basis.With the aid of Fourier analysis,we can prove the convergence of the scheme.We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions.We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.展开更多
We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically ...We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data.We show that for small values of the parameter the corresponding solutions decay to O,while for large values the related solutions converge to 1 uniformly on compacts.Moreover,we prove that the transition from extinction(converging to O)to propagation(converging to 1)is sharp.Numerical results are provided to verify the theoretical results.展开更多
基金supported in part by the National Natural Science Foundation of China(Grant Nos.11871145,11971016,12131005)The research of L.-L.Wang is partially supported by Singapore MOE AcRF Tier 1(Grant RG 15/21)R.Liu would like to thank Nanyang Technological University for hosting the visit where this research topic was initialised.
文摘In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains.We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian.As a result,the“stiffness”matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N^(2))arithmetic operations.Moreover,the singular factor in a typical kernel function can be fully absorbed by the basis.With the aid of Fourier analysis,we can prove the convergence of the scheme.We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions.We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.
基金supported in part by NSFC(Grant Nos.12071175,11171132,11571065)National Research Program of China(Grant No.2013CB834100)+1 种基金by the Natural Science Foundation of jilin Province(Grant Nos.20200201253JC,201902013020JC)by the Project of Science and Technology Development of Jilin Province,China(Grant No.2017C028-1).
文摘We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data.We show that for small values of the parameter the corresponding solutions decay to O,while for large values the related solutions converge to 1 uniformly on compacts.Moreover,we prove that the transition from extinction(converging to O)to propagation(converging to 1)is sharp.Numerical results are provided to verify the theoretical results.
