Variational Regularization of the Inverse Problem of a Class of Nonlinear Time-Fractional Diffusion Equations北大核心CSCD

考虑了一类二维非线性时间分数阶扩散方程,并从最终位置获取的测量数据来反演物质在u(0, y, t)处的物理信息。这个问题是严重不适定的,即问题的解并不连续依赖于测量数据,因此提出了变分型正则化方法来稳定求解该问题。给出了精确解与正则近似解之间的误差估计,数值算例验证了该方法的有效性。     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

一种迭代正则化方法求解一类同时带有两个扰动数据的反向问题北大核心CSCD

该文考虑了一类带有扰动扩散系数和扰动终值数据的空间分数阶扩散方程反向问题,从终值时刻的测量数据来反演初始时刻数据.该问题是严重不适定的,因此该文提出了一种迭代正则化方法来处理该反向问题,并利用先验正则化参数选取规则得到了正则化解和精确解之间的误差估计,最后进行了一些数值模拟,验证了方法的有效性.     

A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms北大核心CSCD

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式（WB-HLL）,并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.     

识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法

该文研究具有Riemann-Liouville时间分数阶导数的Rayleigh-Stokes方程未知源识别问题.首先证明这个问题是不适定的,并应用分数阶Landweber正则化方法求解此反问题.基于条件稳定性结果,在先验和后验正则化参数选取规则下,分别给出精确解与正则解之间的误差估计.最后通过数值例子说明此方法求解此类...     

时间分数次扩散方程反演源项问题的迭代正则化方法

时间分数次扩散方程中反演源项问题是一类经典不适定问题.本文构造了一种新的迭代格式作为正则化方法,给出了先验和后验参数选取下相应的收敛性分析.数值算例验证该方法的有效性.     

On the Integration by Parts and Characterization of a Fractional Diffusion Process北大核心CSCD

本文研究分数扩散过程和其分部积分公式的关系.首先利用Bismut方法给出拉回公式,进而得到分数扩散过程的分部积分公式。反过来,证明了分数扩散过程可由其分部积分公式唯一刻画.     

The Cubic B-Spline Method for a Class of Caputo-Fabrizio Fractional Differential Equations北大核心CSCD

基于分数阶微积分基本定理和三次B样条理论,构造了求解线性Caputo-Fabrizio型分数阶微分方程数值解的三次B样条方法,利用分数阶微积分基本定理将初值问题转化为关于解函数的表达式,再使用三次B样条函数逼近表达式中积分项的被积函数,进而计算了一类Caputo-Fabrizio型分数阶微分方程的数值解.给出了所构造的三次B样条方法的误差估计、收敛性和稳定性的理论证明.数值实验表明,该文数值方法在求解一类Caputo-Fabrizio型分数阶微分方程数值解时具有一定的可行性和有效性,且计算精度和计算效率优于现有的两种数值方法.     

Asymptotic of the Solutions of the Initial Boundary Value Problem for the Diffusion Equations for Semiconductors（Ⅱ）

The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.     

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations北大核心CSCD

针对三维非稳态对流扩散反应方程,构造了一种高精度紧致有限差分格式,对空间的离散采用四阶紧致差分方法,对时间的离散采用Taylor级数展开和余项修正技术,所提格式在时间上的精度为二阶、在空间上的精度为四阶。利用Fourier稳定性分析法证明了该格式是无条件稳定的。最后给出数值算例验证了理论结果。     

含源反应扩散方程的两类修正差分格式

本文针对含扰动参数ε的含源反应扩散方程,采用待定系数法,在三点模板的中心点处进行泰勒展开,对泰勒展式中的高阶导数项充分利用原微分方程进行"降阶",然后分别从"横向"和"纵向"两个角度进行修正,得到了两类差分格式,其中横向系列差分格式(HDS)的精度分别达到二阶、四阶和六阶.数值实验与参考格式比对效果较好,且横向差分格式...     

Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter $gamma$, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When $gamma$ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of $gamma$. However, when $gamma$ is greater than zero, the optimal convergence rate depends on the value of $gamma$ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

Modified High-order Upwind Method for Convection Diffusion Equation

Abstract In this paper, we study the high-order upwind finite difference method for steady convection-diffusionproblems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference schemeon nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is inconversation form, satisfies maximum value principle and has second-order error estimates in discrete H~1 norm.To illustrate our conclusion, several numerical examples are given.     

Modified High-order Upwing Method for Convection Diffusion Equation

Abstract   In this paper, we study the high-order upwind finite difference method for steady convection-diffusion problems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference scheme on nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is in conversation form, satisfies maximum value principle and has second-order error estimates in discrete H ¹ norm. To illustrate our conclusion, several numerical examples are given. Supported by the Research Fund for Doctoral Program of High Education of China State Education Commission.     

具有振动边值的扩散方程Dirichlet问题

本文以上（下）连续函数作为扩散方程u_t=1/2Δu+cu 在Ｄ内的Ｄｉｒｉｃｈｌｅｔ问题边值函数，讨论了振动边值的Ｄｉｒｉｃｈｌｅｔ问题，并用概率方法证明解的存在性、唯一性和稳定性，把古典Ｄｉｒｉｃｈｌｅｔ问题边值条件减弱到最一般情形     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.     

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.     

m-拉普拉斯型抛物方程解的局部化新证明

采用De Giorgi迭代技巧,给出m-拉普拉斯型抛物方程解的局部化新证明,并得到一些先验估计。