Iterative and non-iterative methods for non-linear Volterra integro-differential equations

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Abstract

Iterative and non-iterative methods for the solution of nonlinear Volterra integro-differential equations are presented and their local convergence is proved. The iterative methods provide a sequence solution and make use of fixed-point theory, whereas the non-iterative ones result in series solutions and also make use of fixed-point principles. By means of integration by parts and use of certain integral identities, it is shown that the initial conditions that appear in the iterative methods presented here can be eliminated and the resulting iterative technique is identical to the variational iteration method which is derived here without making any use at all of Lagrange multipliers and constrained variations. It is also shown that the formulation presented here can be applied to initial-value problems in ordinary differential, Volterra’s integral and integro-differential, pantograph, and nonlinear and linear algebraic equations. A technique for improving/accelerating the convergence of the iterative methods presented here is also presented and results in a Lipschitz constant that may be varied as the iteration progresses. It is shown that this acceleration technique is related to preconditioning methods for the solution of linear algebraic equations. It is also argued that the non-iterative methods presented in this paper may not competitive with iterative ones because of possible cancellation errors, if implemented numerically. An analytical continuation procedure based on dividing the interval of integration into disjoint subintervals is also presented and its limitations are discussed.

Introduction

Nonlinear integro-differential equations arise in many fields, e.g., fluid dynamics, polymer science, population dynamics, thermoelasticity, chemical engineering, electrical engineering, etc. Although there is a solid theory for linear integral equations, e.g. [1], [2], [3], [4], [5], [6], [7], this is not the case for nonlinear integral or nonlinear integro-differential equations although considerable progress has been made [8]. As a consequence several numerical methods that use collocation [2] or quasilinearization [9] and analytical techniques based on Adomian’s decomposition, e.g., [10], [11], and He’s variational iteration method, e.g., [12], [13], [14], [15], [16], [17], have been employed recently to obtain approximate solutions of both linear and nonlinear Volterra integro-differential equations. The variational iteration method not only overcomes the difficulties associated with the calculation of Adomian’s polynomials [18], [19], [20], it also provides an iterative sequential approximation to the solution rather than the series solution provided by the decomposition technique.

For initial- and boundary-value problems of ordinary differential equations, the author [21] has shown that there are several manners to obtain the variational iteration method, e.g., integration by parts, Green’s functions, variation of parameters, adjoint operators, and weighted residuals, and that there is no need at all for introducing Lagrange multipliers and constrained variations as originally done by the proposer of this technique [12], [13] and frequently used in the literature [14], [15], [16], [17]. Furthermore, the author [21] has also shown that, for initial-value problems in ordinary differential equations, the variational iteration method can be derived from the well-known Picard (or fixed-point) iterative technique by elimination of the initial conditions, and both Picard’s iterative technique and the variational iteration method provide a convergent sequence of approximations (or iterates) to the solution, and the convergence of this technique is ensured provided that a Lipschitz-continuity condition is met. If the same Lipschitz-continuity is satisfied, one can obtain the solution of initial-value problems in ordinary differential equations as a convergent series [22]. In both the sequence [21] and the series [23] approaches, the ordinary differential equations were first reduced to Volterra integral equations.

The objective of this paper is several-fold. First, an iterative technique for nonlinear Volterra integro-differential equations is presented and its convergence is analyzed; this technique provides a sequence approximation to the solution and makes explicit use of the initial conditions. Second, by means of integration by parts, it is shown that this technique can be written as the variational iteration method without the need for Lagrange multipliers and constrained variations. Third, a non-iterative technique which provides a series solution is presented and its convergence analyzed. Finally, two acceleration methods are proposed to speed up the convergence of the sequence approximations and the connection between this acceleration method and preconditioning techniques of linear algebra is examined.

Although iterative techniques similar to the ones presented in this paper may be used to solve nonlinear partial differential equations of the evolutionary type by writing them as Volterra integral equations and, in fact, the author developed several iterative techniques for solving parabolic, hyperbolic and elliptic partial differential equations [21], the convergence of iterative methods for boundary-value problems is a more delicate issue than that for initial-value ones [8], [24], [25], [26] and will be addressed in the future. However, if the conditions of Theorem 1 presented in this paper are met in the whole spatial domain, then the iterative techniques presented herein can be applied to obtain the solution of evolutionary partial differential equations.

The paper has been organized as follows. Several iterative techniques for nonlinear Volterra integro-differential equations are presented and their convergence proved in Section 2 where the variational iteration method is derived from the formulation presented in this section. In Section 2, an acceleration method for improving the convergence of iterative methods is also presented and its connections with preconditioning techniques, and the solution of multi-pantograph equations are discussed. A non-iterative series solution method is presented and its convergence analyzed in Section 3, where another accelerating technique based on interval splitting and valid for both the iterative and the non-iterative methods presented in this paper are also discussed. A final section summarizes the most important findings of the paper.

Section snippets

Iterative methods

In this paper, we shall be mainly concerned with the following system of nonlinear Volterra integro-differential equations:A(t)u(n)(t)F(t,u(t))=f(t,u(t))+t0tg(s,u(s))ds,t0<t<,subject todjudtj(t0)u(j)(t0)=αj,0j(n-1),where ARN×N is a square invertible matrix, uRN,fRN,gRN,t0t< is the time variable, the superscript n denotes the nth order derivative with respect to t and αj with 0j(n-1) is a constant N×1 vector. In Eq. (1), t0 can be set without loss of generality to zero by means of

Noniterative methods

The iterative techniques presented in the previous section provide the solution as a sequence of iterates. In this section, we present a non-iterative technique that provides the solution as a series and prove that this series converges if the hypotheses made in Theorem 1 are met. Although the formulation presented in this section may be applied to Eqs. (6), (26), (30), (34), we shall only consider Eq. (6) which can be written asU(t)=U(t0)+H(t,U(t)),H(t,U(t))=t0tF(s,U(s))ds.We shall look for a

Conclusions

A variety of iterative and non-iterative methods for the solution of non-linear Volterra integro-differential equations, nonlinear Volterra integral equations, initial-value problems in ordinary differential equations (including pantograph equations), and algebraic equations has been presented and its local convergence has been proved. These iterative and noniterative methods provide the solution as a convergent sequence of iterates and a convergent series, respectively.

The iterative methods

Acknowledgements

The research reported in this paper was supported by Project FIS2005-03191 from the Ministerio de Eduación y Ciencia of Spain and fondos FEDER. The author is grateful to the referees for their careful reading of the manuscript and their suggestions.

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