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Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions
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作者 徐承龙 郭本瑜 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2008年第3期311-331,共21页
The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are est... The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches. 展开更多
关键词 modified Laguerre orthogonal approximation and interpolation multiple dimensions spectral and pseudospectral methods nonlinear partial differential equations
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<i>L</i>-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations
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作者 Fidele Fouogang Ngwane Samuel Nemsefor Jator 《American Journal of Computational Mathematics》 2014年第2期87-92,共6页
An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic par... An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented. 展开更多
关键词 HYBRID Second DERIVATIVE Method Off-Step Point parabolic partial differential equations
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THE HIGH ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING PARABOLIC EQUATIONS 3-DIMENSION
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作者 孙鸿烈 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1999年第7期88-93,共6页
In this paper, an explicit three_level symmetrical differencing scheme with parameters for solving parabolic partial differential equation of three_dimension will be considered. The stability condition and local trunc... In this paper, an explicit three_level symmetrical differencing scheme with parameters for solving parabolic partial differential equation of three_dimension will be considered. The stability condition and local truncation error for the scheme are r<1/2 and O( Δ t 2+ Δ x 4+ Δ y 4+ Δ z 4) ,respectively. 展开更多
关键词 parabolic partial differential equation of three_dimension implicit difference scheme explicit difference scheme local truncation error absolutely stable condition stable
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A SINGULAR PERTURBATION PROBLEM FOR PERIODIC BOUNDARY PARTIAL DIFFERENTIAL EQUATION
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作者 林鹏程 江本铦 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1991年第3期281-290,共10页
In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular ... In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, we prove that the scheme constructed by this paper converges uniformly to the solution of its original problem with O(r+h2). 展开更多
关键词 elliptic-parabolic partial differential equation singular perturbation problem periodic boundary difference scheme uniform convergence
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Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems
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作者 Paula Chen Jerome Darbon Tingwei Meng 《Communications on Applied Mathematics and Computation》 EI 2024年第2期1428-1471,共44页
Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we p... Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Additionally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time. 展开更多
关键词 Optimal control Hamilton-Jacobi partial differential equations Grid-free numerical methods High dimensions Field-programmable gate arrays(FPGAs)
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Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations
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作者 Ernie Tsybulnik Xiaozhi Zhu Yong-Tao Zhang 《Communications on Applied Mathematics and Computation》 EI 2023年第4期1339-1364,共26页
High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th... High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids. 展开更多
关键词 Weighted essentially non-oscillatory(WENO)schemes Multi-resolution WENO schemes Sparse grids High spatial dimensions Hyperbolic partial differential equations(PDEs)
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A POSTERIORI ERROR ESTIMATE FOR BOUNDARY CONTROL PROBLEMS GOVERNED BY THE PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS 被引量:3
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作者 Wei Gong Ningning Yan 《Journal of Computational Mathematics》 SCIE CSCD 2009年第1期68-88,共21页
In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori er... In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state, the boundary state and the final state. It is proven that these estimators are reliable bounds of the finite element approximation errors, which can be used as the indicators of the mesh refinement in adaptive finite element methods. 展开更多
关键词 Boundary control problems Finite element method A posteriori error estimate parabolic partial differential equations.
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OSCILLATION THEOREM TO SYSTEMS OF IMPULSIVE NEUTRAL DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS 被引量:5
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作者 Luo Liping Ouyang Zigen 《Annals of Differential Equations》 2007年第3期297-303,共7页
In this paper, we study the oscillation of solutions to the systems of impulsive neutral delay parabolic partial differential equations. Under two different boundary conditions, we obtain some sufficient conditions fo... In this paper, we study the oscillation of solutions to the systems of impulsive neutral delay parabolic partial differential equations. Under two different boundary conditions, we obtain some sufficient conditions for oscillation of all solutions to the systems. 展开更多
关键词 IMPULSE neutral type DELAY system of parabolic partial differential equations OSCILLATION
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A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory
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作者 R.Nageshwar Rao P.Pramod Chakravarthy 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2014年第1期23-40,共18页
In this paper,a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable.Similar boundary v... In this paper,a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable.Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory.Here,the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach.When the shift parameter is smaller than the perturbation parameter,the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed.The proposed finite difference scheme is unconditionally stable.When the shift parameter is larger than the perturbation parameter,a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed.This scheme is also unconditionally stable.The applicability of the proposed methods is demonstrated by means of two examples. 展开更多
关键词 Singular perturbations parabolic partial differential equation exponentially fitted method differential-difference equations
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ODE-Based Multistep Schemes for Backward Stochastic Differential Equations
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作者 Shuixin Fang Weidong Zhao 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE CSCD 2023年第4期1053-1086,共34页
In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of referen... In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems. 展开更多
关键词 Backward stochastic differential equation parabolic partial differential equation strong stability preserving linear multistep scheme high order discretization
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OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS 被引量:9
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作者 Luo Lipng Peng Baiyu Yang Liu 《Annals of Differential Equations》 2007年第4期470-476,共7页
In this paper, oscillation of solutions to a class of impulsive delay parabolic partial differential equations system with higher order Laplace operator is studied. Under two different boundary value conditions, we es... In this paper, oscillation of solutions to a class of impulsive delay parabolic partial differential equations system with higher order Laplace operator is studied. Under two different boundary value conditions, we establish some sufficient criteria with respect to the oscillations of such systems, employing first-order impulsive delay differential inequalities. The results fully reflect the influence action of impulsive and delay in oscillation. 展开更多
关键词 IMPULSE DELAY system of parabolic partial differential equations OSCILLATION higher order Laplace operator
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EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR A NONLINEAR PARABOLIC EQUATION RELATED TO IMAGE ANALYSIS 被引量:1
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作者 Wang Lihe Zhou Shulin 《Journal of Partial Differential Equations》 2006年第2期97-112,共16页
In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a nonlinear parabolic partial differential equation, which is related to the Malik-Perona model in im... In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a nonlinear parabolic partial differential equation, which is related to the Malik-Perona model in image analysis. 展开更多
关键词 EXISTENCE UNIQUENESS nonlinear parabolic partial differential equations.
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A QUASI-NEWTON METHOD IN INFINITE-DIMENSIONAL SPACES AND ITS APPLICATION FOR SOLVING A PARABOLIC INVERSE PROBLEM
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作者 Wen-huan Yu(Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China.) 《Journal of Computational Mathematics》 SCIE CSCD 1998年第4期305-318,共14页
A Quasi-Newton method in Infinite-dimensional Spaces (QNIS) for solving operator equations is presellted and the convergence of a sequence generated by QNIS is also proved in the paper. Next, we suggest a finite-dimen... A Quasi-Newton method in Infinite-dimensional Spaces (QNIS) for solving operator equations is presellted and the convergence of a sequence generated by QNIS is also proved in the paper. Next, we suggest a finite-dimensional implementation of QNIS and prove that the sequence defined by the finite-dimensional algorithm converges to the root of the original operator equation providing that the later exists and that the Frechet derivative of the governing operator is invertible. Finally, we apply QNIS to an inverse problem for a parabolic differential equation to illustrate the efficiency of the finite-dimensional algorithm. 展开更多
关键词 Quasi-Newton method parabolic differential equation inverse problems in partial differential equations linear and Q-superlinear rates of convergence
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On the speed of convergence of Picard iterations of backward stochastic differential equations
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作者 Martin Hutzenthaler Thomas Kruse Tuan Anh Nguyen 《Probability, Uncertainty and Quantitative Risk》 2022年第2期133-150,共18页
It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to... It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution.In this paper we prove that this convergence is in fact at least square-root factorially fast.We show for one example that no higher convergence speed is possible in general.Moreover,if the nonlinearity is zindependent,then the convergence is even factorially fast.Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations. 展开更多
关键词 Backward stochastic differential equation Picard iteration A priori estimate Semilinear parabolic partial differential equation
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An efficient technique for solving fractional-order diffusion equations arising in oil pollution
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作者 Hardik Patel Trushit Patel Dhiren Pandit 《Journal of Ocean Engineering and Science》 SCIE 2023年第3期217-225,共9页
In this article,non-linear time-fractional diffusion equations are considered to describe oil pollution in the water.The latest technique,fractional reduced differential transform method(FRDTM),is used to ac-quire app... In this article,non-linear time-fractional diffusion equations are considered to describe oil pollution in the water.The latest technique,fractional reduced differential transform method(FRDTM),is used to ac-quire approximate solutions of the time fractional-order diffusion equation and two cases of Allen-Cahn equations.The acquired results are collated with the exact solutions and other results from literature for integer-orderα,which reveal that the proposed method is effective.Hence,FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science. 展开更多
关键词 FRDTM Time-fractional nonlinear partial differential equation Diffusion equation Allen-Cahn(AC)equation parabolic equations
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具连续分布滞量的非线性中立型抛物偏泛函微分方程的振动性 被引量:13
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作者 罗李平 高正晖 欧阳自根 《应用数学》 CSCD 北大核心 2006年第3期651-655,共5页
考虑一类具连续分布滞量的非线性中立型抛物偏泛函微分方程解的振动性,借助Green定理和时滞微分不等式获得了这类方程在Robin,Dirichlet边值条件下所有解振动的若干充分条件.
关键词 非线性 中立型 抛物型偏泛函微分方程 振动性 连续分布滞量
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脉冲向量中立型抛物方程解的H-振动性 被引量:9
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作者 罗李平 杨柳 曾云辉 《高校应用数学学报(A辑)》 CSCD 北大核心 2010年第4期463-468,共6页
研究一类脉冲向量中立型抛物偏微分方程边值问题解的振动性,利用Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维脉冲中立型微分不等式正解的不存在性问题,并借助于一阶脉冲中立型微分不等式,给出了该类边值问题所有... 研究一类脉冲向量中立型抛物偏微分方程边值问题解的振动性,利用Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维脉冲中立型微分不等式正解的不存在性问题,并借助于一阶脉冲中立型微分不等式,给出了该类边值问题所有解H-振动的若干充分性判据,这里H是R^M中的单位向量. 展开更多
关键词 脉冲 向量 中立型 抛物型偏微分方程 H-振动
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具连续偏差变元的向量抛物型方程的H-振动性 被引量:10
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作者 罗李平 俞元洪 《浙江大学学报(理学版)》 CAS CSCD 北大核心 2009年第2期137-139,共3页
考虑一类具连续偏差变元的向量抛物型偏微分方程的振动性,利用Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维泛函微分不等式不存在最终正解的问题,给出了该类方程在Robin边值条件下所有解H-振动的若干充分条件,这... 考虑一类具连续偏差变元的向量抛物型偏微分方程的振动性,利用Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维泛函微分不等式不存在最终正解的问题,给出了该类方程在Robin边值条件下所有解H-振动的若干充分条件,这里H是RM中的单位向量. 展开更多
关键词 向量 抛物型偏微分方程 H-振动性 连续偏差变元
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具拟线性扩散系数的脉冲中立型抛物系统的(强)振动性 被引量:14
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作者 罗李平 俞元洪 《振动与冲击》 EI CSCD 北大核心 2011年第8期183-186,共4页
研究一类具拟线性扩散系数的脉冲中立型抛物偏微分系统解的(强)振动性,直接利用振动的定义、Green公式和Neumann边值条件将这类脉冲中立型抛物系统的振动问题转化为脉冲中立型微分不等式不存在最终正解的问题,并利用最终正解的定义和脉... 研究一类具拟线性扩散系数的脉冲中立型抛物偏微分系统解的(强)振动性,直接利用振动的定义、Green公式和Neumann边值条件将这类脉冲中立型抛物系统的振动问题转化为脉冲中立型微分不等式不存在最终正解的问题,并利用最终正解的定义和脉冲中立型微分不等式,获得了该类系统(强)振动的充分判据.所得结果充分反映了脉冲和时滞在振动中的作用。 展开更多
关键词 拟线性扩散系数 脉冲 中立型 抛物偏微分系统 (强)振动
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非线性脉冲中立型时滞抛物偏微分方程的振动性 被引量:7
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作者 罗李平 欧阳自根 《吉林大学学报(理学版)》 CAS CSCD 北大核心 2007年第1期23-28,共6页
研究一类非线性脉冲中立型时滞抛物偏微分方程解的振动性,借助一阶脉冲中立型微分不等式,获得了该类方程在两类不同边值条件下振动的若干新的充分性判据.所得结果改进了已有的结果,且充分反映了脉冲和时滞在振动中的影响作用.
关键词 脉冲 非线性中立型 时滞抛物偏微分方程 振动
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