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

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.