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.     

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 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.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

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.     

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.      

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.     

The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations

In this work, a combined form of the Laplace transform method with the Adomian decomposition method is developed for analytic treatment of the nonlinear Volterra integro-differential equations. The combined method is capable of handling both equations of the first and second kind. Illustrative examples will be examined to support the proposed analysis.     

A novel numerical method and its convergence for nonlinear delay Volterra integro-differential equations

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

Convergence of one-leg methods for nonlinear neutral delay integro-differential equations

Some convergence results of one-leg methods for nonlinear neutral delay integro-differential equations (NDIDEs) are obtained. It is proved that a one-leg method is E (or EB) -convergent of order p for nonlinear NDIDEs if and only if it is A-stable and consistent of order p in classical sense for ODEs, where p = 1, 2. A numerical example that confirms the theoretical results is given in the end of this paper. This work was supported by National Natural Science Foundation of China (Grant No. 10871164), the Natural Science Foundation of Hunan Province (Grant No. 08JJ6002), and the Scientific Research Fund of Changsha University of Science and Technology (Grant No. 1004259)     

A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations

^** Email: jingtang{at}lsec.cc.ac.cn^*** Email: hermann{at}math.mun.ca In this paper we establish a posteriori error estimates forthe discontinuous Galerkin (DG) method applied to linear, semilinearand non-standard (non-linear) Volterra integro-differentialequations. We also present an analysis of the DG method withquadrature for the memory term. Numerical experiments basedon three integro-differential equations are used to illustratevarious aspects of the error analysis.     

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n-1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

Picard’s iterative method for nonlinear advection-reaction-diffusion equations

Picard’s iterative method for the solution of nonlinear advection-reaction-diffusion equations is formulated and its convergence proved. The method is based on the introduction of a complete metric space and makes uses of a contractive mapping and Banach’s fixed-point theory. From Picard’s iterative method, the variational iteration method is derived without making any use at all of Lagrange multipliers and constrained variations. Some examples that illustrate the advantages and shortcomings of the iterative procedure presented here are shown.     

Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations

In this paper, we present a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Compared with Chebyshev–Halley methods, the new methods require one additional evaluation of the function. The numerical results presented show that the new methods compete with Chebyshev–Halley methods.     

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816-1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14-19] are the special cases of the new improvements.     

Finite element and DG analysis for neutral-type Volterra integro-differential equations with weakly singular kernels

In this paper we analyze the numerical solution of Volterra integro-differential equations of neutral type with weakly singular kernels. We establish a priori error estimations for the finite-element-method semi-discretization of the given problem by defining a suitable Ritz-Volterra projection operator: here, the key point in the proof is the fact that the definition of the Ritz-Volterra projection operator that is not related to the neutral term. We then discuss the discontinuous Galerkin time-stepping method for the semi-discretized equation, together with a fully discretized form.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.     

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.