Volterra Integral Equations Of Second Kind Research Articles

Superconvergence in collocation methods for Volterra integral equations with vanishing delays

In this paper, we investigate the optimal (global and local) convergence orders of the (iterated) collocation solutions for second-kind Volterra integral equations with vanishing delays on quasi-geometric meshes. It turns out that the classical global convergence results still hold under certain regularity conditions of the given functions. In particular, it is shown that the optimal local superconvergence order p = 2 m can be attained if collocation is at the m Gauss(–Legendre) points, which contrasts with collocations both on uniform meshes and on geometric meshes. Numerical experiments are performed to confirm our theoretical results.

Jacobi spectral method for the second-kind Volterra integral equations with a weakly singular kernel

The Jacobi spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ω α , β 2 -norm and the L ∞ -norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate theoretical results.

Second-Kind Linear Volterra Integral Equations with Noncompact Operators

In this article, we consider second-kind linear Volterra integral equations (VIEs) with noncompact operators, that is, μu = f + V ϕ u, where with the core ϕ ∈L 1(0, 1). We present some different properties of noncompact operators V ϕ from compact operators, such as eigenvalues, eigenfunctions, null spaces, and ranges. In many applications, the cores belong to L p (0, 1) for some p > 1. In this case, we completely describe the eigenvalues of V ϕ and the null space and the range of μI − V ϕ. In addition, a necessary and sufficient condition is given such that μI − V ϕ is a Fredholm operator. In the end, we discuss the regularity of solutions.

Features of behavior of numerical methods for solving Volterra integral equations of the second kind

Systems of second-kind Volterra integral equations with stiff and oscillating components are considered. An implicit noniterative method of the second order is proposed for the numerical solution of such problems. The efficiency of the method is demonstrated using several examples.

Jacobi spectral Galerkin method for second-kind Volterra integral equations with a weakly singular kernel

Jacobi spectral Galerkin method for second-kind Volterra integral equations with a weakly singular kernel

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt $(0<q<1)$ . This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$ , the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$ , where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.

Integral-Algebraic Equations: Theory of Collocation Methods I

Our analysis of collocation solutions for general systems of linear integral-algebraic equations (IAEs) is based on the notions of the tractability index and the $\nu$-smoothing property of a Volterra integral operator. These are used to decouple the given IAE system into the inherent system of regular second-kind Volterra integral equations (VIEs) and a system of first-kind VIEs. This decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions. Numerical examples illustrate the theoretical results.

Convergence Analysis of the Spectral Method for Second-Kind Volterra Integral Equations with a Weakly Singular Kernel

The Legendre spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 -norm and the L∞ -norm ) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

Prediction Method of Long-Term Mechanical Behavior of Largely Deformed Sand Asphalt with Constant Loading Creep Tests

Because of large pronounced deformation in the compressive creep process of soft sand asphalt, it is necessary that both the true stress and true strain are measured. Accordingly, instead of fictitious creep compliance (FCC), true creep compliance (TCC) relating the true strain to the true stress is used to characterize the viscoelastic properties of sand asphalt in this paper. The relationship between TCC and FCC is described by a second-kind Volterra integral equation (VIE). A prediction method of long-term mechanical behavior of largely deformed sand asphalt with constant loading creep tests is proposed. In this method, the FCC master curves are constructed based on the time-temperature superposition principle, and then the linear VIE from the FCC to the TCC is solved with the collocation method, and the nonlinear VIE from the TCC to the FCC is solved with an iterative formula. Unconstrained compressive creep tests for 3,600 s were conducted on sand asphalt mixture samples in various temperature and nominal stress conditions. As an application example, the long-term creep behavior of sand asphalt at the given reference temperature is predicted with the proposed method.

Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels

Abstract Within the class of second-kind Volterra equations, an important subclass is the second-kind convolution equations with non-negative kernels k , with ‖ k ‖ 1 1 , and positive forcing terms. Sharper results about the effect of kernel perturbations are obtained if attention is focussed on this subclass. In the standard perturbation and stability analysis for second-kind Volterra integral equations, it is the effect on the solution of perturbations in the forcing term and the stability of numerical methods for their solution that are examined. In applications, where only approximations for the kernel are available, such as arise in rheology, risk analysis, and renewal theory, the effect on the solution of perturbations in the kernel is equally important. In this paper, estimates are derived for this situation. The resolvent kernel framework is used to establish conditions such that, with respect to relative error perturbations in the kernel, the corresponding relative error perturbations in the solution are bounded. Defect renewal equations form a subset of the convolution Volterra integral equations examined.

Order conditions for Volterra Runge–Kutta methods

We consider Runge-Kutta methods for second-kind Volterra Integral Equations with weakly singular kernel. Order conditions, whose number and structure depend on the singularity of the equation, are derived in a recursive manner using an approach originally devised by P. Albrecht for Ordinary Differential Equations. Order conditions are hence generated, in an automatic way, by means of a symbolic algorithm and some numerical experiments are presented.

Geometric meshes in collocation methods for Volterra integral equations with proportional delays

In this paper we analyse the local superconvergence properties of iterated piecewise polynomial collocation solutions for linear second-kind Volterra integral equations with (vanishing) proportional delays qt (0 < q < 1). It is shown that on suitable geometric meshes depending on q, collocation at the Gauss points leads to almost optimal superconvergence at the mesh points. This contrasts with collocation on uniform meshes where the problem regarding the attainable order of local superconvergence remains open.

Numerical Solutions for second-kind Volterra integral equations by Galerkin methods

In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the O(h2r)-convergence rate in the piecewise-polynomial space of degree not exceeding (r−1). As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

Higher Accuracy Methods for Second-Kind Volterra Integral Equations Based on Asymptotic Expansions of Iterated Galerkin Methods

On the basis of asymptotic expansions, we study the Richardson extrapolation method and two defect correction schemes by an interpolation post-processing technique, namely, interpolation correction and iterative correction for the numerical solution of a Volterra integral equation by iterated finite element methods. These schemes are of higher accuracy than the postprocessing method and analyzed in a recent paper [5] by Brunner, Q. Lin and N. Yan. Moreover, we give a positive answer to a conjecture in [5].

Discrete-time waveform relaxation Volterra-Runge-Kutta methods: Convergence analysis

The discrete-time relaxation methods based on Volterra-Runge-Kutta methods for solving large system of second-kind Volterra integral equations are proposed. Convergence of the discrete-time iteration process with particular attention to parallel methods is investigated.

On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations

It is well known that, in contrast to Fredholm integral equations, iterated collocation solutions (based on collocation at the Gauss points) to Volterra integral equations of the second kind exhibit optimal discrete superconvergence only at the mesh points. Here, we show that some degree of global superconvergence is possible on the entire interval.

Gravitational waves as a source for the polarization of the Cosmic Microwave Background

The polarization of the Cosmic Microwave Background (CMB) induced by gravitational waves (GWs) is studied by solving in a semi-analytical way the Chandrasekhar radiative transfer equation; following the Polnarev approach, the equation is written as a second-kind Volterra integral equation and its kernel is handled by performing a series expansion of the trigonometric functions it contains. In this way, a recursive calculation of the Volterra equation gets possible and the polarizing effect of the gravitational waves can be brought out.

Collocation methods for second-kind Volterra integral equations with weakly singular kernels

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

Stability of parallel Volterra-Runge-Kutta methods

In this paper, we analyse parallel iteration of Volterra-Runge-Kutta methods (PIVRK methods) for solving second-kind Volterra integral equations on parallel computers. We focus on the determination of the region of convergence C and on the stability region S m of the iterated method obtained after m iterations. Results are presented for the convolution test equation. It turns out that the stability region S m does not necessarily converge to the stability region S of the corrector. However, for finite m, S m need not to be contained in C or S and may be much larger than C .