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The initial-boundary-Value Problems for the Hirota Equation on the Half-Line

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Chinese Annals of Mathematics,Series B 2020年第1期41卷 117-132页

作者： Lin HUANG School of Science Hangzhou Dianzi UniversityHangzhou 310018China School of Mathematical Sciences Fudan UniversityShanghai 200433China

An initial boundary-value problem for the Hirota equation on the half-line,00, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert(RH) problem in the complex k-plane. This RH problem has explicit(x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x,0) = q0(x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g0(t), qx(0, t) = g1(t) and qxx(0, t) = g2(t). The spectral functions satisfy an algebraic global relation which characterizes, say, g2(t) in terms of {q0(x), g0(t), g1(t)}.The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

关键词： Hirota equation Riemann-Hilbert problem initial-boundary value problem Global relation

initial-boundary VALUE PROBLEM AND CAUCHY PROBLEM FOR A QUASILINEAR EVOLUTION EQUATION

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Acta Mathematica Scientia 1999年第S1期19卷 487-496页

作者：杨志坚 Department of Mathematics Zhengzhou University Zhengzhou 450002 China Dept. of Math. and Mech. Zhengzhou Univ. of Tech. Zhengzhou 450002 China

In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.

关键词： initial-boundary value problem Cauchy problem quasilinear evolution equation local solution blowup

Construction of Global Weak Entropy Solution of initial-boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

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American Journal of Computational Mathematics 2017年第4期7卷 451-468页

作者： Yihong Dai Jing Zhang Department of Mathematics Jinan University Guangzhou China Department of Optoelectronic Engineering Jinan University Guangzhou China

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

关键词： Scalar Conservation Laws with Weak Discontinuous Flux initial-boundary Value Problem Elementary Wave Interaction Structure of Global Weak Entropy Solution

Numerical Treatment of initial-boundary Value Problems with Mixed boundary Conditions

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American Journal of Computational Mathematics 2018年第2期8卷 153-174页

作者： Nawal Abdullah Alzaid Huda Omar Bakodah Department of Mathematics King Abdulaziz University Jeddah KSA

In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.

关键词： Decomposition Method Modified Adomian Decomposition Method Linear and Nonlinear Partial Differential Equations Mixed boundary Conditions initial-boundary Value Problem

Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

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Science China Mathematics 2019年第1期62卷 33-62页

作者： Haibo Cui Haiyan Yin Changjiang Zhu Limei Zhu School of Mathematical Sciences Huaqiao University School of Mathematics South China University of Technology

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant(x,t)∈R^+×R^+,with the null-Dirichlet boundary condition or the null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends timeasymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang(1999), and Jiang and Zhu(2009, Discrete Contin Dyn Syst), we obtain a general result when the initial perturbation belongs to the same space. In addition,our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.

关键词： Euler equations with time-depending damping nonlinear diffusion waves initial-boundary value problem decay estimates

QUENCHING PHENOMENA FOR SYSTEMS OF SEMILINEAR PARABOLIC EQUATIONS I

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Systems Science and Mathematical Sciences 1997年第4期10卷 361-371页

作者： DAI Qiuyi(Department of Basic Science, Xiangtan Mining Institute, Hunan Xiangtan 411201, China)GU Yonggeng(Institute of Systems Science, Academia Sinica, Beijing 100080, China) 湘潭矿业学院基础科学系湖南湘潭中科院系统科学所北京

initial-boundary value problems for a class of systems of semilinear parabolicequations with singular terms are investigated, conditions for the existence and nonexistence of the global solutions to the above problems... 详细信息

关键词： Quenching critical length quenching time initial-boundary value problem systems of parabolic equations

Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

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American Journal of Computational Mathematics 2018年第4期8卷 326-342页

作者： Xiaoqian Li Jing Zhang Department of Mathematics Jinan University Guangzhou China Department of Optoelectronic Engineering Jinan University Guangzhou China

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

关键词： Nonlinear Conservation Laws with Weak Discontinuous Flux initial-boundary Value Problem Shock Wave Rarefaction Wave Contact Discontinuity Interaction Structure of Global Weak Entropy Solution

The <I>G</I>-functions Series Method Adapted to the Numerical Integration of Parabolic PDE

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Journal of Applied Mathematics and Physics 2018年第1期6卷 161-173页

作者： Mónica Cortés-Molina José Antonio Reyes Fernando García-Alonso Department of Applied Mathematics University of Alicante S. Vicente del Raspeig Spain

The Method Of Lines (MOL) and Scheifele’s G-functions in the design of algorithms adapted for the numeric integration of parabolic Partial Differential Equations (PDE) in one space dimension are applied. The semi-discrete system of ordinary differential equations in the time direction, obtained by applying the MOL to PDE, is solved with the use of a method of Adapted Series, based on Scheifele’s G-functions. This method integrates exactly unperturbed linear systems of ordinary differential equations, with only one G-function. An implementation of this algorithm is used to approximate the solution of two test problems proposed by various authors. The results obtained by the Dufort-Frankel, Crank-Nicholson and the methods of Adapted Series versus the analytical solution, show the results of mistakes made.

关键词： Series Method Numerical Solutions Parabolic initial-boundary Value Problems Method of Lines