第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

Spline collocation methods for nonlinear Volterra integral equations with unknown delay

In this paper we introduce and study polynomial spline collocation methods for systems of Volterra integral equations with unknown lower integral limit arising in mathematical economics. Their discretization leads to implicit Runge-Kutta-type methods. The global convergence and local superconvergence properties of these methods are proved, and the theory is illustrated by a numerical example arising in the application of such equations in certain mathematical models of liquidation.     

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(h ^2r )-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.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

L p solutions of backward stochastic Volterra integral equations

This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in L ^p (1 < p < 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in L ^p (p > 1) is also considered, which also generalizes the results in the existing literature.     

The iterative correction method for Volterra integral equations

We show that the (n – 1)-fold application of an iterative correction technique to the iterated collocation solution corresponding to the one-point Gauss collocation solution for a Volterra integral equation of the second kind l6eads to a significant improvement in the precision of these approximations: the resulting rate of (global) convergence is .The work of first author has been supported by the Natural Sciences and Engineering Research Council of Canada (Research Grant OGP0009406).     

On a class of backward stochastic Volterra integral equations

In this paper, under some restrictions of the time interval, we compare a class of backward stochastic Volterra integral equations with the corresponding simpler one; to be precise, we give the relations between their solutions under global and local Lipschitz conditions on their generator functions. Using these relations, it could be easier to study solutions of more complex equations, where coefficients in backward integrals could be treated as perturbations.     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

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.     

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

In this paper we analyse the local superconvergence propertiesof iterated piecewise polynomial collocation solutions for linearsecond-kind Volterra integral equations with (vanishing) proportionaldelays qt (0 < q < 1). It is shown that on suitable geometricmeshes depending on q, collocation at the Gauss points leadsto almost optimal superconvergence at the mesh points. Thiscontrasts with collocation on uniform meshes where the problemregarding the attainable order of local superconvergence remainsopen.     

Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces

In this paper, first a new fixed point theorem is established, and then, by the use of it, the existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces are investigated. The results obtained in this paper generalize and improve the results corresponding to those obtained by others.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

Legendre spectral Galerkin method for second-kind Volterra integral equations

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

A parameter choice strategy for (iterated) tikhonov regularization of iii-posed problems leading to superconvergence with optimal rates

For ordinary and iterated Tikhonov regularization of linear ill-posed problems, we propose a parameter choice strategy that leads to optimal (super-) convergence rates for certain linear functionals of the regularized solution. It is not necessary to know the smoothness index of the exact solution; approximate knowledge of the smoothness index for the linear functional suffices     

Spectral methods for weakly singular Volterra integral equations with smooth solutions

We propose and analyze a spectral Jacobi-collocation approximation for the linear Volterra integral equations (VIEs) of the second kind with weakly singular kernels. In this work, we consider the case when the underlying solutions of the VIEs are sufficiently smooth. In this case, we provide a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in the infinity norm and weighted Sobolev space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.