Iterative and non-iterative methods for non-linear Volterra integro-differential equations. (English) Zbl 1202.65179

Ramos, J. I.

Appl. Math. Comput. 214, No. 1, 287-296 (2009).

The author considers the problem of numerical solving the initial problem for the equation \[ A(t)u^{(n)}(t)=f(t,u(t)) + \int_{t_0}^{t}g(s,u(s))ds, \; t_0<t< \infty, \] subject to \(u^{(j)}(t_0)=\alpha_j, \; 0<j\leq(n-1)\), where \(A(t)\) are invertible square matrices of the order \(N\) and \(u^{(j)}\) denotes the \(j\)-th order derivative of the unknown \(N\)-dimensional function \(u(t)\). This problem is deep theoretically investigated, and the local existence theorem 1 presented and proved in the paper is presented (with non-essential simplification \(A(t)=I\)) in the text-book by A. B. Vasilieva and A. N. Tikhonov [Integralnye Uravnenia (Russian). Izdat. Moskovskogo Universiteta, Moscow (1986)]. The author presents and discusses a few variants of iterative algorithms of Picard kind, and a series method of solving the initial problem. The latter is the presentation of an approximation of the required solution as a finite functional sum. The first summand is the given initial value \(\alpha\), and the each following summand is an integral iteration of one or two predecessors. Convergence of the sum to the required solution is proved.

Reviewer: Vladimir Gorbunov (Ul’yanovsk)

MSC:

65R20	Numerical methods for integral equations
45J05	Integro-ordinary differential equations

Keywords:

Volterra-type integro-differential equations; iterative methods; Picard iterations; series method

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