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.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

指数有界双连续n阶α次积分C半群与高阶抽象Cauchy问题研究

算子半群理论在求解Cauchy问题等领域具有很大的应用价值,并且其在泛函分析理论等各方面的研究中同样有着重要意义。此次研究在Banach空间上,基于泛函分析、算子半群理论对柯西问题进行探讨。并基于n阶α次积分C半群的生成定理,对指数有界双连续n阶α次积分C半群的生成定理与其Laplace逆变换表达式进行推算。由结果可知,L∞范数误差在分数阶n为1.47时达到最低值,为0.04,这表明其与实际值极为接近。在T=0.2,0.5两种时刻下,中精确解和正则解的变化趋势基本一致,并且二者的误差在x∈0.1,0.9区间内接近于0。研究有效验证了指数有界双连续n阶α次积分C半群解决高阶抽象Cauchy问题具有适用性与可靠性。

一类拟线性方程Cauchy问题的可解性

根据一类拟线性方程的积分曲面和特征曲线之间的关系,研究了一类拟线性方程Cauchy问题的可解性。给出了特征曲线的求法,并将积分曲面推广到n元函数形式。

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

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.

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.

ON THE STABILITY OF THE POSITIVE RADIAL STEADY STATES FOR A SEMILINEAR CAUCHY PROBLEM INVOLVING CRITICAL EXPONENTS

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 solutions of △u ＋ K（｜x｜）u^p =0, is obtained when p is critical for general K（｜x｜）.

ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.

GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.

GLOBALLY SMOOTH SOLUTIONS TO CAUCHY PROBLEM OF A QUASILINEAR HYPERBOLIC SYSTEM ARISING IN BIOLOGY

This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.

INITIAL-BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM FOR A QUASILINEAR EVOLUTION EQUATION

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.

Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation

A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas.Based 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[J.Differential Equations,2018,265(5):2076-2120].Our results also improve upon the results obtained in Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].

A NOTE ON "THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS"

In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations…… admits a unique global generalized solution in ……and a unique global classical solution in…… the sufficient conditions of the blow up of the solution in finite time are given, and also two examples are given.

L^p-L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.

PERIODIC BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM OF THE GENERALIZED CUBIC DOUBLE DISPERSION EQUATION

In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.

A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.

Cauchy Problems for KdV-type Equations and Higher-Order Conditional Symmetries

<Abstract>We develop the generalized conditional symmetry (GCS) approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations.We characterize these equations that admit certain higher- order GCSs and show the main reduction procedure by some examples.The obtained reductions cannot be derived within the framework of the standard Lie approach.

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

这份报纸被奉献给为承认某些概括有条件的对称(GCS ) 的第四顺序的伪线性的寓言的方程学习 Cauchy 问题的对称减小。完全的组分类结果被介绍，并且一些例子被给显示出主要减小过程。