The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

Numerical solution of Volterra integral equations with singularities

The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

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.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

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.     

Numerical solution of multiple nonlinear Volterra integral equations

We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation. We prove existence and uniqueness of solutions, and convergence and estimates of the order of convergence for the numerical methods of solution.     

Spectral methods for weakly singular Volterra integral equations with pantograph delays

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L^∞-norm and the weighted L²-norm. Numerical examples are presented to complement the theoretical convergence results.     

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.     

Differential transform method for solving Volterra integral equation with separable kernels

In this paper, Volterra integral equations with separable kerenels are solved using the differential transform method. The approximate solution of this equation is calculated in the form of a series with easily computable terms. Exact solutions of linear and nonlinear integral equations have been investigated and the results illustrate the reliability and the performance of the differential transform method.     

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.     

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.     

A stable numerical method for Volterra integral equations with discontinuous kernel

Numerical methods for Volterra integral equations with discontinuous kernel need to be tuned to their peculiar form. Here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, we first investigate about the behavior of the solution of a suitable (basic) test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem.     

On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1?K*f were obtained as |f(t)|?1, |f(t)|?2 and |f(t)|?4 in (J. Math. Anal. Appl. 1978; 64 :381–397; Int. J. Math. Math. Sci. 1982; 5 (1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|?2ⁿ in (Integr. Equ. Oper. Theory 2002; 43 :466–479), where t∈[0, ∞) and n is a natural number such that n?2. In (Math. Comp. 2006; 75 :1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43 :466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43 :466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f⁽ⁿ⁺³⁾, (n∈?) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1?K*f. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.     

On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation

The existence of solutions of a nonlinear quadratic Volterra integral equation is studied. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. Such an approach allows us to obtain a result on the existence of solutions of an equation in question which are uniformly locally attractive or asymptotically stable.