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Journal of Applied Mathematics and Physics 2024年第4期12卷 1364-1382页

作者： Jake L. Nkeck Department of Mathematics University of Buea Buea Cameroon

The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.

关键词： Singularities Finite Element Methods heat equation Predictor-Corrector Algorithm

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

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Chinese Physics B 2014年第4期23卷 76-82页

作者：于欣任志刚许超 Ningbo Institute of Technology Zhejiang University The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control Zhejiang University

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

关键词： boundary optimal control heat equation variable domain finite element method control parame-terization method

APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL heat equation

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Journal of Computational Mathematics 2007年第6期25卷 730-745页

作者： Chunxiong Zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China Institut für Numerische Mathematik Universitaet Miinster Germany

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

关键词： heat equation Artificial boundary conditions Fast evaluation Unbounded domains.

Controllability of the heat equation with a Control Acting on a Measurable Set

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Chinese Annals of Mathematics,Series B 2012年第1期33卷 149-160页

作者： Hang YU School of Mathematical Sciences Fudan UniversityShanghai 200433China

The paper deals with the controllability of a heat equation. It is well-known that the heat equation yt - △y = uxE in （0, T） × Ω with homogeneous Dirichlet boundary conditions is null controllable for any T 〉 0 and any open nonempty subset E of Ω. In this note, the author studies the case that E is an arbitrary measurable set with positive measure.

关键词： heat equation Measurable sets Null controllability Observabilityinequality

TIME ANALYTICITY FOR THE heat equation ON GRADIENT SHRINKING RICCI SOLITONS

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Acta Mathematica Scientia 2022年第4期42卷 1690-1700页

作者： Jiayong WU Department of Mathematics Shanghai UniversityShanghai200444China

On a complete non-compact gradient shrinking Ricci soliton,we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space *** growth condition is *** an application,we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

关键词： Gradient shrinking Ricci soliton heat equation time analyticity

THE heat equation IN R WITH ANTI-PERIODICBOUNDARY CONDITION

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Acta Mathematica Scientia 1999年第4期19卷 391-401页

作者：汪更生刘昌良 Department of Mathematics Huazhong Normal University Wuhan 430079 China

This paper is concerned with some nonlinear heat equations with initial condition and anti-periodic boundary condition. Also some two-point value nonlinear heat equations with anti-periodic boundary condition are discussed. The existence and uniqueness of the solutions are given. Some asymptotic behaviors of the solutions are studied.

关键词： semigroup m-dissipative heat equation antiperiodic solutions

ON STABILITY AND REGULARIZATION FOR BACKWARD heat equation

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Chinese Annals of Mathematics,Series B 2003年第1期24卷 35-44页

作者： LIU JIJUN LUO DINGJUN Department of Mathematics Nanjing Normal University Nanjing 210097 China and Department of Mathematics Southeast University Nanjing 210096 China.Department of Mathematics Nanjing Normal University Nanjing 210097 China.

Consider a backward heat equation in a bounded domain Ω (?) R2 with the noisy data in the initial time geometry. The aim is to find the temperature for 0 < ε < t < T. For this ill-posed problem, the authors give a continuous dependence estimate of the solution. Moreover, the convergence rate of the approximate solution is also given.

关键词： heat equation Inverse problem Stability Regularization

FINITE DIFFERENCE METHODS FOR THE heat equation WITH A NONLOCAL BOUNDARY CONDITION

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Journal of Computational Mathematics 2015年第1期33卷 17-32页

作者： V. Thomee A.S. Vasudeva Murthy Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GSteborg Sweden TIFR Centre for Appl. Math Yelahanka New Town Bangalore India

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the 0-method for 0 〈 θ ≤ 1, in both cases in maximum-norm, showing O（h2 ＋ k） error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ= 1/2, the Crank-Nicolson method, using energy arguments, yielding a O（h2 ＋ k3/2） error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.

关键词： heat equation Artificial boundary conditions unbounded domains productquadrature.

A new fourth order finite difference scheme for the heat equation

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Communications in Nonlinear Science and Numerical Simulation 2000年第4期5卷 151-157页

作者： Yuehong QIAN, Hudong CHEN, Raoyang ZHANG and Shiyi CHEN (Exa Corporation, 450 Bedford Street, Lexington, MA 02420, USA e-mail.. qean＠*** Deportment of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21280, USA) (Received December 1, Exa Corporation 450 Bedford Street Lexington MA 02420 United States Department of Mechanical Engineering Johns Hopkins University Baltimore MD 21280 United States

We propose and investigate a new simple fourth order finite difference scheme for the heat equation. Numerical simulations do confirm theoretical analysis of accuracy and stability condition.

关键词： heat equation CFL condition stability high-order scheme

Zero Extension Problem for the heat equation

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Acta Mathematica Sinica,English Series 2022年第11期38卷 1981-1997页

作者： Ge Yang DU Qiang XU Shu Lin ZHOU School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China School of Mathematics and Statistics Lanzhou UniversityLanzhou 730000P.R.China LMAM School of Mathematical SciencesPeking UniversityBeijing 100871P.R.China

In this paper we present a necessary and sufficient condition to guarantee that the zeroextended function of the solution for the heat equation in a smaller cylinder is still the solution of the corresponding extension problem in a larger *** prove the results under the frameworks of classical solutions,strong solutions and weak ***,we generalize these results to uniformly parabolic equations of divergence form.

关键词： heat equation zero extension initial-boundary value problem