Approximate Solutions and Error Bounds for Solving Matrix Differential Equations Without Increasing the Dimension of the Problem

In this paper we present a method for solving initial value problems related to second order matrix differential equations. This method is based on the existence of a solution of a certain algebraic matrix equation related to the problem, and it avoids the increase of the dimension of the problem for its resolution. Approximate solutions, and their error bounds in terms of error bounds for the approximate solutions of the algebraic problem, are given.     

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.     

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.     

An optimal filtering method for the Cauchy problem of the Helmholtz equation

In this paper, we consider a Cauchy problem for the Helmholtz equation in a rectangle. An optimal filtering method is presented for approximating the solution of this problem, and the Hölder type error estimate is obtained. Numerical illustration shows that the method works effectively.     

An adaptive finite element method for the sparse optimal control of fractional diffusion

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.     

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate 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 are presented to illustrate the validity and effectiveness of this method.     

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media. The flow equation is solved by the mixed finite element method, and the transport equation is approximated by a discontinuous Galerkin method. The scheme is continuous in time and a priori hp error estimates is presented.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

Discontinuous finite volume method for compressible miscible displacement problems in porous media

The compressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations: the pressure equation and the concentration equation are parabolic equation. In this article, we present discontinuous finite volume method for the concentration equation and the pressure equation. The optimal order error estimates for pressure and concentration are obtained in a mesh dependent norm.     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

无穷凹角区域各向异性问题的自然边界元法

本文研究无穷凹角区域上一类各向异性问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式和自然积分方程,给出了自然积分方程的数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.     

Epsilon-Trig Regularization Method for Bang-Bang Optimal Control Problems

Bang-bang control problems have numerical issues due to discontinuities in the control structure and require smoothing when using optimal control theory that relies on derivatives. Traditional smooth regularization introduces a small error into the original problem using error controls and an error parameter to enable the construction of accurate smoothed solutions. When path constraints are introduced into the problem, the traditional smooth regularization fails to bound the error controls involved. It also introduces a dimensional inconsistency related to the error parameter. Moreover, the traditional approach solves for the error controls separately, which makes the problem formulation complicated for a large number of error controls. The proposed Epsilon-Trig regularization method was developed to address these issues by using trigonometric functions to impose implicit bounds on the controls. The system of state equations is modified such that the smoothed control is expressed in sine form, and only one of the state equations contains an error control in cosine form. Since the Epsilon-Trig method has an error parameter only in one state equation, there is no dimensional inconsistency. Moreover, the Epsilon-Trig method only requires the solution to one control, which greatly simplifies the problem formulation. Its simplicity and improved capability over the traditional smooth regularization method for a wide variety of problems including the Goddard rocket problem have been discussed in this study.     

BOUNDARY ELEMENT ANALYSIS FOR A CLASS OF MULTI-DIMENSIONAL LINEAR PARABOLIC EQUATIONS

In this paper,by me as of beundary element method,we try to deal with the initial -boundary value problem for a class of linear parunolic equations,which is a linear heat conduction equation. We tresent a boundary integral equation for the solution to the problem and its variational formalation The well-posedness of the variational formulation is proved. And the error estimates for the approsutate solutions are provided. The results of this paper are more general than those of[1]     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method

In this paper, an improved Legendre collocation method is presented for a class of integro-differential equations which involves a population model. This improvement is made by using the residual function of the operator equation. The error differential equation, gained by residual function, has been solved by the Legendre collocation method (LCM). By summing the approximate solution of the error differential equation with the approximate solution of the problem, a better approximate solution is obtained. We give the illustrative examples to demonstrate the efficiency of the method. Also we compare our results with the results of the known some methods. In addition, an application of the population model is made.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

求解热源识别问题的修正吉洪诺夫正则化方法

该文研究了一个热源识别问题,通过引入修正吉洪诺夫方法来处理问题的不适定性,在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步验证了方法的有效性和稳定性.