An Effective Meshless Approach for Inverse Cauchy Problems in 2D and 3D Electroelastic Piezoelectric Structures

In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.

Symmetry Reductions of Cauchy Problems for Fourth-Order Quasi-Linear Parabolic Equations

This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.

Fast Solving the Cauchy Problems of Poisson Equation in an Arbitrary Three-Dimensional Domain

In this paper we propose a novel two-stage method to solve the threedimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary.The solution is decomposed into a particular solution and a homogeneous solution.In the first stage a multiple-scale polynomial method(MSPM)is used to approximate the forcing term and then the formula of Tsai et al.[Tsai,Cheng,and Chen(2009)]is used to obtain the corresponding closed-form solution for each polynomial term.Then in the second stage we use a multiple/scale/direction Trefftz method(MSDTM)to find the solution of Laplace equation,of which the directions are uniformly distributed on a unit circle 1,and the scales are determined a priori by the collocation points on boundary.Two examples of 3D data interpolation,and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.

Determination of pollution point source in parabolic system model

This paper considers an inverse problem for a partial differential equation to identify a pollution point source in a watershed. The mathematical model of the problem is a weakly coupled system of two linear parabolic equations for the concentrations u（x, t） and v（x, t） with an unknown point source F（x, t） = A（ t）δ（x- s） related to the concentration u（x, t）, where s is the point source location and A（t） is the amplitude of the pollution point source. Assuming that source F becomes inactive after time T＊, it is proved that it can be uniquely determined by the indirect measurements { v（0, t）, v（ a, t）, v（ b, t）, v（ l, t）, 0 〈 t ≤ T, T＊ 〈 T}, and, thus, the local Lipschitz stability for this inverse source problem is obtained. Based on the proof of its uniqueness, an inversion scheme is presented to determine the point source. Finally, two numerical examples are given to show the feasibility of the inversion scheme.

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

The uniqueness and existence of BV-solutions for Cauchy problem of the form are proved.

INVERSE COEFFICIENT PROBLEMS FOR PARABOLIC HEMIVARIATIONAL INEQUALITIES

This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.

The Quasi-reversibility Method to Solve the Cauchy Problems for Parabolic Equations

In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of QT with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ε that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.

Hlder Stability Estimate for an Inverse Parabolic Problem

This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distribution at any intermediate moment from the heat distribution measurements in arbitrary accessible subdomain ωΩ at some time interval. Our main result is the Hlder stability estimate in the inverse problem and the proof is completed with a Carleman estimate and a eigenfunction expansion for parabolic equations.

A Local Meshless Method for Two Classes of Parabolic Inverse Problems

A local meshless method is applied to find the numerical solutions of two classes of inverse problems in parabolic equations. The problem is reconstructing the source term using a solution specified at some internal points;one class is that the source term is time dependent, and the other class is that the source term is time and space dependent. Some numerical experiments are presented and discussed.

Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations

The uniqueness of solutions for Cauchy problem of the formis studied.It is proved that if u ∈BVx and A(u) is strictly increasing,the solution is unique.

Coefficient Determination in Parabolic Equations Solved as a Moment Problem Two-Dimensional in a Rectangular Domain

The problem is to considerer a parabolic equation depending on a coefficient a (t), and find the solution of the equation and the coefficient. The objective is to solve the problem as an application of the inverse moment problem. An approximate solution and limits will be found for the error of the estimated solution using the techniques of inverse problem moments. In addition, the method is illustrated with several examples.

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

The inverse problem for a class of nonlinear evolution equations of dispersive type type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution it,as given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin.

Reconstructing the Time-Dependent Thermal Coefcient in 2D Free Boundary Problems

The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefcients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is,for the rst time,numerically investigated.This inverse problem appears extensively in the modelling of various phenomena in engineering and physics.For instance,steel annealing,vacuum-arc welding,fusion welding,continuous casting,metallurgy,aircraft,oil and gas production during drilling and operation of wells.From literature we already know that this inverse problem has a unique solution.However,the problem is still ill-posed by being unstable to noise in the input data.For the numerical realization,we apply the alternating direction explicit method along with the Tikhonov regularization to nd a stable and accurate numerical solution of nite differences.The root mean square error(rmse)values for various noise levels p for both smooth and non-smooth continuous time-dependent coef-cients Examples are compared.The resulting nonlinear minimization problem is solved numerically using the MATLAB subroutine lsqnonlin.Both exact and numerically simulated noisy input data are inverted.Numerical results presented for two examples show the efciency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

基于终端观测条件反演非线性抛物型方程的辐射系数

研究了利用终端观测数据重构非线性热传导方程辐射系数的反演问题。与一般的抛物型方程不同的是,非线性热传导方程不仅具有非线性源项,而且边界也是非线性的。主要研究了一维逆问题的理论分析。对于多维情况,同样是适用的。基于最优控制框架,考虑到原问题的不适定性,首先将原问题转化为一个优化问题,在对原问题做估计的同时,证明了控制泛函极小值的存在性以及极值存在的必要条件。其次由于最优控制问题是非凸的,一般不存在全局唯一性,该解只在一些特殊条件下是局部唯一的。最后简单给出了解的稳定性分析。需要指出的是,一般情况下,参数识别是一个非线性反问题,即使其基本方程是一个线性方程。模型复杂性的增大,一方面使得该模型可以刻画更多的物理现象,另一方面也会给相应的分析带来更大的困难。因此,逆问题P的非线性程度,在某种意义上,要高于那些由线性方程控制的非线性程度,在复杂性方面也是如此。

单边退化抛物方程反问题的收敛性分析

考虑利用已知终端观测数据进行单边退化抛物型方程辐射系数反演的收敛性分析问题.首先,对单边退化抛物型方程,需满足Fichera条件以确保问题的可解性;其次,将原反问题转化为一个最优控制问题,通过最优控制方法找到辐射系数的最优解;最后,结合Gateaux导数并引入新的源条件,证明最优解的收敛性.

多维拟线性退化抛物方程Cauchy问题解的唯一性和稳定性

本文证明了拟线性退化抛物方程■的Cauchy问题BV解的唯一性和稳定性.

推广线性二阶抛物型方程Cauchy问题解的Feynman-Kac定理

在金融数学中 ,用跳跃 -扩散型随机微分方程模型描述证券价格过程更为符合实际 ,讨论了由高维 Poisson过程和 Brown运动共同驱动的随机微分方程的 Feynman- Kac定理 .首先建立了高维 Poisson过程的两个基本性质 ,在此基础上 ,导出了推广的向后热传导方程 Cauchy问题解的Feynman- Kac定理 .其次 ,利用 Burkholder不等式建立了跳跃 -扩散型随机过程的矩不等式 ,并由此建立了推广的二阶线性抛物型方程 Cauchy问题解的 Feynman-

积分型Cauchy中值定理的逆问题及中间点的渐近性

讨论了积分型Cauchy中值定理的逆问题,并就此积分型Cauchy中值定理讨论了在积分区间长度趋于零时“中间点”ξ的渐近性.

再论Cauchy微分中值定理的逆问题

文[1]给出了"Cauchy微分中值定理"中值点唯一的条件,并得到了其逆定理的较弱表述.继续文[1]的工作,利用闭区间上连续函数的性质及函数的单调性,解决了其逆定理中端点的唯一性.