Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data

ABSTRACT

Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented.     

On Multistep Methods for Differential Equations of Fractional Order

This paper concerns with numerical methods for the treatment of differential equations of fractional order. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which order conditions and stability properties are investigated. Dedicated to the memory of Professor Aldo Cossu     

解第一类线性算子方程的多层次迭代法(英文)

In this paper we develop two multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for linear ill-posed problems. The two algorithms and their convergence analyses are presented in an abstract framework.     

On One Class of Nonclassical Linear Volterra Integral Equations of the First Kind

Ukrainian Mathematical Journal - On the basis of a new approach, we prove the uniqueness theorem and construct Lavrent’ev’s regularizing operators for the solution of nonclassical...     

On Boundary Integral Equations of the First Kind

A large class of elliptic boundary value problems in elasticity and fluid mechanics can be reduced to systems of boundary integral equations of the first kind. This paper summarizes some of the basic concepts and results concerning the mathematical foundation of boundary element methods for treating such a class of boundary integral equations.     

Iterative Method for Solving Integral Equations of the First Kind

A new iterative method is proposed for solving integral equations of the first kind. The efficiency of the method is demonstrated using examples of typical integral kernels.     

First Kind Cordial Volterra Integral Equations 1

For (pure) cordial Volterra integral equations of the first kind, we establish stability estimates of the solution in the scales of Banach spaces of C ^m -smooth functions on (0, T] and on [0, T]. This enables to estimate the error of polynomial or generalized polynomial approximate solutions which are easy to be determined.     

First Kind Cordial Volterra Integral Equations 2

We reduce a cordial Volterra integral equation of the first kind to a cordial integral equation of the second kind and establish a unique solvability of those in the scales of Banach spaces of C ^m -smooth functions on [0, T] and (0, T]. Stability estimates w.r.t. perturbation of data are derived. Polynomial and spline collocation methods are discussed for the solving the equation.     

非线性延迟积分微分方程线性多步法的渐近稳定性

本文研究求解非线性延迟积分微分方程的线性多步法的渐近稳定性,其中积分部分采用复化梯形公式计算,结果表明:在问题真解渐近稳定的条件下,A-稳定的线性多步法也是渐近稳定的.     

A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations

We investigate the stability properties of numerical methodsfor weakly singular Volterra integral equations of the secondkind. Our theory extends the stability theory of linear multistepmethods for ordinary differential equations. We introduce theconcepts of A-stability, A()-stability etc. for Abel-Volterraequations. The stability region is characterized in terms ofthe weights of the method. It is shown that the order of anA-stable convolution quadrature cannot exceed 2. Further westudy the stability properties of implicit Adam methods, withparticular emphasis on the question of A()-stability.     

线性随机微分方程多步法的稳定性

研究了多步法用于求解线性随机微分方程的稳定性,利用维纳过程的增量服从正态分布的性质,得到了在乘性噪声情况下,多步法用于线性随机微分方程的均方稳定性的条件,并用MATLAB对实际算例进行了数值模拟.     

Discrete Galerkin Methods for Fredholm Integral Equations of the Second Kind

The discrete Galerkin and discrete iterated Galerkin methodsarise when the integrals required in the Galerkin and iteratedGalerkin methods are calculated using numerical integration.In this paper, prolongation and restriction operators are usedto give an error analysis for these two discrete Galerkin methods.From this analysis, we can then give conditions on the quadratureerrors, under which the two discrete Galerkin solutions havethe same order of convergence as their exact counterparts.     

多时滞微分方程数值稳定性

考虑了时滞微分方程的初值问题,分析了用线性多步法求解一类滞后型微分系统数值解的稳定性,在一定的Lagrange插值条件下,给出并证明了求解滞后型微分系统的线性多步法数值稳定的充分必要条件.     

求解广义中立型系统的线性多步方法的NGP_G-稳定性

建立了广义中立型延迟系统理论解渐近稳定的充分条件 ,分析了用线性多步方法求解广义中立型延迟系统数值解的稳定性 ,在一定的Lagrange插值条件下 ,证明了数值求解广义中立型系统的线性多步方法NGPG_稳定的充分必要条件是线性多步方法是A_稳定的·     

Fredholm Integral Equations of the First Kind and Topological Information Theory

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in terms of eigenfunction expansions. One of the possible methods of regularization consists in truncating this formal expansion after restricting the class of admissible solutions through a-priori global bounds. In this paper we reconsider various possible methods of truncation from the viewpoint of the ${\varepsilon}$ -coverings of compact sets.     

Further Studies of the Application of Constrained Minimization Methods to Fredholm Integral Equations of the First Kind

Permanent address: Department of Mathematics, University of Queensland, Australia. Following earlier work of Babolian & Delves (J. Inst. MathsApplics (1979) 24, 157–174) the Galerkin equations forintegral equations of the first kind are stablized by imposingasympotic decay rates on the expansion coefficients. Results for the formulation in the l₂ norm are compared withresults of Babolian & Delves where the l₁ norm was used. The importance of the choice of the constants which specifythe decay rates is also considered. Theoretical results andcomputational experiments show that previously used automaticselection of these constants needs to be safeguarded by monitoringthe residuals of the Galerkin equations.     

求解第一类Fredholm积分方程的一种新的正则化算法

应用一种新的正则化方法建立了一类新的求解第一类Fredholm积分方程的正则化算法, 并借助Matlab软件给出了数值算例.数值结果与理论分析基本一致,而且表明文中建立的正则化比通常的Tikhonov正则化更精确.     

On the Numerical Solution of Fredholm Integral Equations of the First Kind

Two previously given methods for the numerical solution of Fredholmintegral equations of the first kind are investigated by theuse of polynomial spline functions. As a byproduct a new methodis presented for obtaining the eigensolutions of the kernelfunction. From numerical experiments zero order regularizationappears to give more accurate results than higher orders. Acomparison is made as to the relative accuracy and speed ofthe two methods. A scheme is presented to enable a choice tobe made, from the set of truncated solutions, as to which memberis closest to the solution of the integral equation.     

Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (à la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm. Accepted 1 April 1997     

Multistep Discretization of Linear Hyperbolic Equations

Order and stability of multistep finite-difference discretizationsof the first-order linear hyperbolic equation u₁ = a(x)u_x areconsidered. We prove that if a stable method uses s upwind andrdownwind points and the coefficients depend only on the Courantnumber and on a(x) and its derivatives at the underlying gridpoint, then the order may not exceed r + s. This bound on orderis exactly half the bound of Strang & Iserles (1983) forconstant a. Furthermore, we prove that if r = s and a(x) isboth uniformly bounded and uniformly positive for x R thenthe new order barrier is attainable for every s 1.