Superconvergence of collocation methods for Volterra integral equations of the first kind

Collocation methods for solving first-kind Volterra equations in the space of piecewise polynomials possessing finite (jump) discontinuities at their knots and having degreem≧0 are known to have global order of convergencep=m+1. It is shown that a careful choice of the collocation points (characterized by the Lobatto points in (0, 1]) yields convergence of order (m+2) at the corresponding Legendre points.     

Numerische Behandlung einer Volterraschen Integralgleichung

This paper deals with a generalized nonlinear Volterra integral equation, whose kernel contains the unknown function at two different arguments. The equation is solved by collocation with piecewise Hermite-polynomials. The method has order 2m,m ∈ ?, if polynomials of degree 2m?1 and appropriate integration formulas are used. The collocation points must be chosen in accordance with a certain stability condition.     

The approximate solution of initial-value problems for general Volterra integro-differential equations

We study the application of certain spline collocation methods to Volterra integro-differential equations of orderr where ther-th order derivative of the unknown solution occurs also in the kernel of the integral term. The analysis focuses on the question of the optimal discrete convergence order (at the knots of the approximating spline function).     

An <Emphasis Type="Italic">hp</Emphasis>-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels

In this paper, we present an hp-version Legendre–Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the \(H^1\)-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method.     

A note on collocation methods for Volterra integral equations of the first kind

It has been shown that if a Volterra integral equation of the first kind with continuous kernel is solved numerically in a given intervalI by collocation in the space of piecewise polynomials of degreem≧0 and possessing finite discontinuities at their knotsZ _N then a careful choice of the collocation points yields convergence of orderp=m+2 on a certain finite subset ofI (while the global convergence order ism+1; this subset does not contain the knotsZ _N . In this note it will be shown that superconvergence onZ _N can be attained only if some of the collocation points coalesce (Hermite-type collocation).     

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.     

Extended backward differentiation formulae in the numerical solution of general Volterra integro-differential equations

In this paper nonlinear Volterra integro-differential equations are considered with kernels of the formP(x,s,y(s)) andK(x,s,y(x),y(s)) and extended backward differentiation methods are applied as extended from their introduction for the solution of ordinary differential equations by Cash [4]. An error bound is obtained and a rate of convergence is found and validated by testing the method on some examples. The numerical results are compared with those obtained by applying standard backward differentiation and collocation methods.     

Metodi numerici per equazioni di volterra di seconda specie

The first part of this paper studies the stability and the order of a numerical method for the integration of the second kind of Volterra equations. This method is obtained by differentiatingm-times (m>-2) the Volterra equation and by integrating the «differential equations system» obtained via the trapezoidal rule. The proposed method has second order and an absolute stability region equal to the third quadrant of the plane (hλ,h ²μ). The second part of this paper, analyzes the stability properties and also the order of a class of numerical methods, obtained via the integration of the previous «differential equations system» though backward differentiation. It is shown that these methods have a high order and a very large stability region.     

Global stability analysis of the Runge-Kutta methods for volterra integral and integro-differential equations with degenerate kernels

We investigate the stability of the numerical solutions resulting from applying very general classes of Runge-Kutta methods to Volterra integral and integro-differential equations with degenerate kernels. The results are generalizations of previous results obtained by the authors for exact collocation methods for these equations.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments

Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.      

Moving least squares collocation method for Volterra integral equations with proportional delay

In this work, we apply the moving least squares (MLS) method for numerical solution of Volterra integral equations with proportional delay. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. An error bound is obtained to ensure the convergence and reliability of the method. Numerical results approve the efficiency and applicability of the proposed method.     

The collocation method for the numerical approximation of the periodic solutions of functional differential equations

A collocation method with trigonometric trial functions is presented form-order non-linear functional differential equations with periodicity boundary conditions. In general, uniform approximation of an isolated solution and of its firstm?1 derivatives is achieved, while them-derivative is approximated in mean square. In some special cases we have also the uniform approximation of them-derivative. The solution of then-th non-linear collocation equation may be approximated by Newton's iteration with an arbitrary starting point belonging to a suitable neighbourhood of an isolated solution, for alln>n ₀ withn ₀ large enough.     

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.     

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α,β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.     

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the mth-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

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.

     

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.