Spline Collocation for Cordial Volterra Integral Equations

We study the convergence and convergence speed of two versions of spline collocation methods on the uniform grids for linear Volterra integral equations of the second kind with noncompact operators.     

Convergence of spline collocation for Volterra integral equations

Estimates for step-by-step interpolation projections are established. Depending on the spectrum of the transfer matrix these estimates allow to obtain the pointwise convergence of the projectors to the identity operator or, in some limit cases, to prove stable convergence of the corresponding approximate operators of integral equations. This, via general convergence theorems for operator equations, permits to get the convergence of collocation method for Volterra integral equations of the second kind in spaces of continuous or certain times continuously differentiable functions. Applications in the case of the most practical types of splines are analyzed.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

Some approximation results for Durrmeyer operators

In the present paper, we obtain a sequence of positive linear operators which has a better rate of convergence than the Szász-Mirakian Durrmeyer and Baskakov Durrmeyer operators and their Voronovskaya type results.     

On the convergence of the product of independent random operators

In this paper, we study infinite products of independent random operators. Conditions for the existence a.s. and the convergence in mean of order p of such infinite products are established.     

Sobolev and strict continuity of general hysteresis operators

The most natural and important topologies connected with hysteresis operators are those induced by uniform convergence, W^{1, 1}‐convergence, and strict convergence. Indeed the supremum norm and the variation are invariant under reparameterization. We prove a general result that implies that if a hysteresis operator is continuous with respect to the topology of W^{1, 1}, then it is continuous with respect to the strict topology. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations

In this paper, the nonlinear Volterra-Fredholm integro-differential equations are solved by using the homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of a series which its components are computed easily . The existence and uniqueness of the solution and the convergence of the proposed method are proved. A numerical example is studied to demonstrate the accuracy of the presented method.     

Abstract Volterra operators

We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A.N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.     

非线性弱奇性Volterra积分方程的谱配置法

基于已有文献的研究成果及前期工作,我们考察了非线性弱奇性Volterra积分方程（VIE）的谱配置法,并对该方法进行了收敛性分析.得到的结论是数值误差呈谱收敛.误差收敛阶与配置点个数及方程解的正则性相关.数值实验也证实了这一结论.本文的方法解决了已有文献中类似数值方法（Allaei（2016）,Sohrabi（2017））存在的问题.     

Homogenization for conductive thin layers of pre-fractal type

The aim of this work is to obtain convergence results for the solutions of transmission problems across highly conductive layers of pre-fractal type from the point of view of homogenization. We prove the M-convergence of the energy functionals to an energy functional which incorporates a singular term, supported within the layer. From the convergence of the energy functionals we deduce the convergence of the approximating solutions to the limit solution in a suitable sense.     

A note on bounds for non-linear multivalued homogenized operators

In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis.     

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.

     

Invariant and Hyperinvariant Subspaces of the Operator J α in Sobolev Spaces W p s [0,1]

Mathematical Notes -     

Solvability of convolution type integro-differential equations on ℝ

The paper looks for the solutions of integro-differential equations of the form

$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $

in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L ₁ (?), 0 ≤ k ∈ L ₁ (?), ∫_? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.

     

On a graded mesh method for a class of weakly singular Volterra integral equations

In this paper a class of weakly singular Volterra integral equations with an infinite set of solutions is investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation. The aim of this paper is to improve the convergence rates by a graded mesh method. The convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings.     

On the integrated Baskakov type operators

The present paper deals with the study of the rate of convergence of the integrated Baskakov-Durrmeyer type operators in simultaneous approximation. We also mention some of the improvements on the recent paper of Deo [N. Deo, Direct results on exponential type operators, Applied Mathematics and Computation 204(1) (2008) 109-115].     

Construction of a new family of Bernstein‐Kantorovich operators

In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors.     

Voronovskaya‐type theorems for Urysohn type nonlinear Bernstein operators

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators.