Boundary value methods for Volterra integral and integro-differential equations

New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.     

Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations

In this paper, we construct a class of extended block boundary value methods (B₂VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B₂VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B₂VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B₂VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.     

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.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

Periodic boundary value problems for second order impulsive integro-differential equations of volterra type

The existence of solutions of periodic boundary value problems for second order impulsive integro-differential equations of Volterra type is investigated. By using the method of upper and lower solutions, it is proved that the problem in whi h impulses occur at fixed times has a solution. Some impulsive integro-differential inequalities related to such problem are also established.     

On an initial-value method for quickly solving Volterra integral equations: A review

A method of converting nonlinear Volterra equations to systems of ordinary differential equations is compared with a standard technique, themethod of moments, for linear Fredholm equations. The method amounts to constructing a Galerkin approximation when the kernel is either finitely decomposable or approximated by a certain Fourier sum. Numerical experiments from recent work by Bownds and Wood serve to compare several standard approximation methods as they apply to smooth kernels. It is shown that, if the original kernel decomposes exactly, then the method produces a numerical solution which is as accurate as the method used to solve the corresponding differential system. If the kernel requires an approximation, the error is greater, but in examples seems to be around 0.5% for a reasonably small number of approximating terms. In any case, the problem of excessive kernel evaluations is circumvented by the conversion to the system of ordinary differential equations.     

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.     

Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations

This paper presents a computational method for solving a class of system of nonlinear singular fractional Volterra integro-differential equations. First, existences of a unique solution for under studying problem is proved. Then, shifted Chebyshev polynomials and their properties are employed to derive a general procedure for forming the operational matrix of fractional derivative for Chebyshev wavelets. The application of this operational matrix for solving mentioned problem is explained. In the next step, the error analysis of the proposed method is investigated. Finally, some examples are included for demonstrating the efficiency of the proposed method.     

An approximate method for solving a class of weakly-singular Volterra integro-differential equations

In this paper, we present a new approach to resolve linear and nonlinear weakly-singular Volterra integro-differential equations of first- or second-order by first removing the singularity using Taylor’s approximation and then transforming the given first- or second-order integro-differential equations into an ordinary differential equation such as the well-known Legendre, degenerate hypergeometric, Euler or Abel equations in such a manner that Adomian’s asymptotic decomposition method can be applied, which permits convenient resolution of these equations. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained demonstrate this approach is indeed practical and efficient.     

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.     

B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations

A numerical method based on quintic B-spline has been developed to solve the linear and nonlinear Fredholm and Volterra integro-differential equations up to order 4. The solution and its derivatives are collocated by quintic B-spline and then the integral equation is approximated by the 4-points Gauss–Turán quadrature formula with respect to the weight function Legendre. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods which show that our method is accurate.     

Block by block method for the systems of nonlinear Volterra integral equations

The approach given in this paper leads to numerical methods for solving system of Volterra integral equations which avoid the need for special starting procedures. The method has also the advantages of simplicity of application and at least four order of convergence which is easy to achieve. Also, at each step we get four unknowns simultaneously. A convergence theorem is proved for the described method. Finally numerical examples presented to certify convergence and accuracy of the method.     

Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations

We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to form a posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.     

Impulsive anti-periodic boundary value problem of first-order integro-differential equations

This paper is concerned with the anti-periodic boundary value problem of first-order nonlinear impulsive integro-differential equations. We first establish a new comparison principle, and then obtain the existence of extremal solutions by upper-lower solution and monotone iterative techniques. Some examples are presented to illustrate the main results.     

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.     

Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations

In this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost. AMS subject classification (2000)  65R20, 45D05, 44A35, 44A10     

Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations

In this paper, the nonlinear Volterra-Fredholm integro-differential equations are solved by using the homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of a series which its components are computed easily . The existence and uniqueness of the solution and the convergence of the proposed method are proved. A numerical example is studied to demonstrate the accuracy of the presented method.     

Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays

This paper is concerned with the study of the stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. We are interested in the comparison between the analytical and numerical stability regions. First, we focus on scalar equations with real coefficients. It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution. Then, we consider the multidimensional case. A new stability condition for the stability of the analytical solution is given. Under this condition, the asymptotic stability of Gauss-Pouzet methods is investigated.      

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.     

Discretization scheme in Volterra integro-differential equations that preserves stability and boundedness

A nonstandard discretization scheme is applied to continuous Volterra integro-differential equations. We will show that under our discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms and z-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved.