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A Modified Tikhonov Regularization Method for a cauchy problem of the Biharmonic Equation

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Journal of Mathematical Research with Applications 2024年第3期44卷 359-386页

作者： Fan YANG Jianming XU Xiaoxiao LI School of Science Lanzhou University of Technology

In this paper, the cauchy problem of biharmonic equation is considered. This problem is ill-posed, i.e., the solution(if exists) does not depend on the measurable data. Firstly, we give the conditional stability result under the a priori bound assumption for the exact ***, a modified Tikhonov regularization method is used to solve this ill-posed *** the a priori and the a posteriori regularization parameter choice rule, the error estimates between the regularization solutions and the exact solution are obtained. Finally, some numerical examples are presented to verify that our method is effective.

关键词： Biharmonic equations inverse problem cauchy problem Tikhonov regularization method

cauchy problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure

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Acta Mathematica Sinica,English Series 2004年第4期20卷 663-682页

作者： HuiJunFAN InstituteofMathematics AcademyofMathematicsandSystemSciencesChineseAcademyofSciencesBeijing100080P.R.China

This paper discusses the cauchy problem of the equation u_t=▽.(|▽u^m|^(P-2)▽u^m)-u^q with initial datum a *** the assumption of the parameters,one proves the existence and non-existence of the non-negative generali... 详细信息

关键词： cauchy problem Degenerate Existence Measure

cauchy　problem　FOR　A　NONLINEAR　SINGULAR　INTEGRAL－DIFFERENTIAL　EVOLUTION　SYSTEM

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Applied Mathematics(A Journal of Chinese Universities) 1994年第1期9卷 45-53页

作者： ZHANG LINGHAI(Inst.of Appl.Phys.and Comp.Math.,Beijing 100088) Inst. of Appl. Phys. and Comp. Math. Beijing

We study the cauchy problem for a nonlinear evolution system with singularintegral differential *** means of some a priori estimates of the solution and theLeray-Schauder's fixed point theorem, we prove the existence ... 详细信息

关键词： cauchy problem Nonlinear Evolution System Singular integral-differentialTerm Integral Estimate.

Variational regularization method of solving the cauchy problem for Laplace's equation: Innovation of the Grad–Shafranov(GS) reconstruction

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Chinese Physics B 2014年第10期23卷 650-655页

作者：颜冰黄思训 Institute of Meteorology and Oceanography PLA University of Science and Technology State Key Laboratory of Satellite Ocean Environment Dynamics Second Institute of Oceanography State Oceanic Administration

The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.

关键词： Grad-Shafranov reconstruction variational regularization method cauchy problem

Lipschitz Continuous Solutions to the cauchy problem for Quasi-linear Hyperbolic Systems

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Chinese Annals of Mathematics,Series B 2012年第4期33卷 521-536页

作者： Xiang CHEN School of Mathematical Sciences Fudan University Shanghai 200433 China.

Lipschitz continuous solutions to the cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions o... 详细信息

关键词： First order quasi-linear hyperbolic systems Lipschitz continuous solution cauchy problem Existence and uniqueness

ON THE STABILITY OF THE POSITIVE RADIAL STEADY STATES FOR A SEMILINEAR cauchy problem INVOLVING CRITICAL EXPONENTS

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Acta Mathematica Scientia 2008年第2期28卷 348-354页

作者：邓引斌杨芬 Department of Mathematics Huazhong Normal University Department of Information and Computation Science College of Science Wuhan University of Science and Technology

This article is contributed to the cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive so... 详细信息

关键词： Stability cauchy problem asymptotic stability

BLOW UP OF SOLUTIONS TO THE cauchy problem FOR NONLINEAR WAVE EQUATIONS

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Chinese Annals of Mathematics,Series B 2001年第3期22卷 275-280页

作者： ZHOU YI Institute of Mathematics, Fudan University, Shanghai 200433, China Institute of Mathematics Fudan University Shanghai 200433 China

The author proves blow up of solutions to the cauchy problem of certain nonlinear equations and, also, estimates the time when the blow up occurs.

关键词： cauchy problem Nonlinear wave equation Blow up

The cauchy problem of the Nonlinear Schr■dinger Equations in R^(1+1)

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Acta Mathematicae Applicatae Sinica 2007年第4期23卷 593-610页

作者： Gui-xiang Xu Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088 China

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.

关键词： Schrodinger equation cauchy problem besov space littlewood-paley decomposition self-similar solution

REMARKS ON THE cauchy problem OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS

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Acta Mathematica Scientia 2020年第4期40卷 1020-1034页

作者： Yongkai LIAO Institute of Applied Physics and Computational Mathematics Beijing 100088China

This paper is concerned with the large-time behavior of solutions to the cauchy problem of a one-dimensional viscous radiative and reactive *** on the elaborate energy estimates,we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t)and W(t)introduced by Liao and Zhao[*** Equations,2018,265(5):2076-2120].Our results also improve upon the results obtained in Liao and Zhao[*** Equations,2018,265(5):2076-2120].

关键词： large-time behavior viscous radiative and reactive gas cauchy problem large initial data