Cubic Spline Collocation for Volterra Integral Equations

In the standard step-by-step cubic spline collocation method for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval. Such a method is stable in the same region of collocation parameter as in the step-by-step implementation with linear splines. The results about stability and convergence are based on the uniform boundedness of corresponding cubic spline interpolation projections. The numerical tests given at the end completely support the theoretical analysis. Received: January 15, 2002; revised July 27, 2002 Published online: December 19, 2002     

Volterra积分方程组所描述的系统的能控性

本文给出关于一类Volterra积分方程组所描述的系统的完全能控性结果.     

Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations

This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Galerkin methods, a rigorous error analysis in both the infinity and weighted norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.     

Smoothing Transformation and Piecewise Polynomial Collocation for Weakly Singular Volterra Integral Equations

We discuss a possibility to construct high-order numerical algorithms on uniform or mildly graded grids for solving linear Volterra integral equations of the second kind with weakly singular or other nonsmooth kernels. We first regularize the solution of integral equation by introducing a suitable new independent variable and then solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid.     

Discrete Time Relaxation Based on Direct Quadrature Methods for Volterra Integral Equations

Discrete-time relaxation methods based on direct quadrature methods for the numerical solution of second kind Volterra systems are proposed. The convergence of the discrete-time iterations is analyzed.     

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where \(n+1\) denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order \(\mathcal{O}(n^{-m}\log n)\) in the infinite norm and \(\mathcal{O}(n^{-m})\) in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is \(\mathcal{O}(\log ^2 n)\) and its spectral condition number is \(\mathcal{O}(1)\). Numerical examples are presented to demonstrate the effectiveness of the proposed method.     

Multipoint Necessary Optimality Conditions for Singular Controls in Processes Described by the System of Volterra Integral Equations

We consider the optimal control problem described by the system of Volterra nonlinear integral equations. The necessary optimality conditions for controls that are singular in the sense of the Pontryagin maximum principle are obtained.     

On the Test Volterra Equations of the First Kind in the Integral Models of Developing Systems

The paper was devoted to analysis of the test Volterra equations of the first kind enabling one to study the specificity of important classes of integral equations in the mathematical models of developing system. Along with theoretical results, presented were numerical calculations for the model examples.     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

Method of Successive Approximations for Solving Integral Equations of the Theory of Risk Processes

A model of a classical risk process describing the evolution of an insurance company's capital is generalized. Integral equations for the bankruptcy probability are derived. The method of successive approximations is used to solve these equations.     

Fixed point method for solving nonlinear quadratic Volterra integral equations

We consider a paper of Bana? and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of a quadratic Volterra integral equation. In this paper, we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic Volterra integral equation. Then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results.     

Legendre Spectral Collocation Methods for Pantograph Volterra Delay-Integro-Differential Equations

This paper is concerned with the convergence properties of the Legendre spectral collocation methods when used to approximate smooth solutions of Volterra integro-differential equations with proportional (vanishing) delays. We provide a vigorous error analysis for the proposed methods. Furthermore, we prove that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in L ²-norm and L ^??-norm. Some numerical experiments are given to confirm the theoretical results.     

Least squares support vector regression for solving Volterra integral equations

In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method.

     

Multilinear Volterra Equations of the First Kind

The Lambert function is used to derive unimprovable estimates for the solutions of nonlinear integral inequalities that play a pivotal role in the study of multilinear Volterra equations of the first kind.     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

Artificial neural networks based modeling for solving Volterra integral equations system

Properly designing an artificial neural network is very important for achieving the optimal performance. This study aims to utilize an architecture of these networks together with the Taylor polynomials, to achieve the approximate solution of second kind linear Volterra integral equations system. For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system. Then we apply the suggested neural net for adjusting the numerical coefficients of given expansions in resulting system. Consequently, the reported architecture using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. The proposed method was illustrated by several examples with computer simulations. Subsequently, performance comparisons with other developed methods was made. The comparative experimental results showed that this approach is more effective and robust.