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The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with nonlocal boundary conditions

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Advances in Applied Mathematics and Mechanics 2013年第4期5卷 442-460页

作者： B.Bialecki G.Fairweather J.C.Lopez-Marcos Department of AppliedMathematics and Statistics Colorado School of MinesGoldenCO 80401USA Mathematical Reviews American Mathematical Society416 Fourth StreetAnn ArborMI 48103USA Departamento de Matem´atica Aplicada Universidad de ValladolidValladolidSpain

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial *** an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in *** discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear *** present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

关键词： Heat equation nonlocal boundary conditions orthogonal spline collocation Hermite cubic splines convergence analysis superconvergence

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A Quasilinear Parabolic System with nonlocal Sources and Weighted nonlocal boundary conditions

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Journal of Mathematical Research and Exposition 2011年第5期31卷 761-769页

作者： Cheng Yuan QU Rui Hong JI Si Ning ZHENG School of Mathematical Science Dalian University of Technology Liaoning 116024 P. R. China Geomathematics Key Laboratory of Sichuan Province Chengdu University of Technology Sichuan 610059 P. R. China

In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.

关键词： quasilinear parabolic system nonlocal boundary conditions critical exponent blow-up rate weight functions.

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Highly Efficient Method for Solving Parabolic PDE with nonlocal boundary conditions

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Applied Mathematics 2022年第2期13卷 101-119页

作者： Mohamed El-Gamel Galal I. El-Baghdady Mahmoud Abd El-Hady Department of Mathematical Sciences Faculty of Engineering Mansoura University Egypt

In this work, a highly efficient algorithm is developed for solving the parabolic partial differential equation (PDE) with the nonlocal condition. For this purpose, we employ orthogonal Chelyshkov polynomials as the basis. The convergence analysis of the proposed scheme is derived. Numerical experiments are carried out to explain the efficiency and precision of the proposed scheme. Furthermore, the reliability of the scheme is verified by comparisons with assured existing methods.

关键词： Chelyshkov Collocation Method Parabolic nonlocal boundary conditions

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带有非线性非局部流量边界条件反应扩散方程的有限差分格式

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高等学校计算数学学报 2008年第4期30卷 310-324页

作者：刘剑明徐州师范大学数学系

1引言本文考虑如下带有非线性非局部流量边界条件的拟线性抛物型方程

关键词： reaction-diffusion robin type nonlinear nonlocal boundary conditions finite difference solvability convergence.

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Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries

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Science China Mathematics 2007年第9期50卷 1251-1266页

作者： Ling-hua KONG~(1+) Ming-xin WANG~(2,3) 1 Department of Applied Mathematics,Dalian University of Technology,Dalian 116024,China 2 Department of Mathematics,Southeast University,Nanjing 210018,China 3 Department of Mathematics,Xuzhou Normal University,Xuzhou 221116,China Department of Applied Mathematics Dalian University of Technology Dalian 116024 China Department of Mathematics Southeast University Nanjing 210018 China Department of Mathematics Xuzhou Normal University Xuzhou 221116 China

This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.

关键词： parabolic system nonlocal sources nonlocal boundary conditions blow-up set simultaneous blow-up uniform blow-up profile 35K15 35K65

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Growing Sandpile Problem with Local and Non-Local Boundaries conditions

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Journal of Applied Mathematics and Physics 2021年第10期9卷 2414-2429页

作者： Urbain Traoré Laboratoire de Mathématiques et d’Informatique (LAMI) UFR Sciences Exactes et Appliquées Université Ouaga I Pr Joseph

In this paper, we study a class of Prigozhin equation for growing sandpile problem subject to local and a non-local boundary condition. The problem is a generalized model for a growing sandpile problem with Neumann boundary condition (see [1]). By the semi-group theory, we prove the existence and uniqueness of the solution for the model and thanks to a duality method we do the numerical analysis of the problem. We finish our work by doing numerical simulations to validate our theoretical results.

关键词： Sandpile Collapsing Avalanche Nonlinear Semi-Group nonlocal boundary conditions Euler Discretization in Time Time Dependent Gradient Constraints

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