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Zbl 1202.65179
Ramos, J.I.
Iterative and non-iterative methods for non-linear Volterra integro-differential equations. (English)
[J] Appl. Math. Comput. 214, No. 1, 287-296 (2009). ISSN 0096-3003

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 {\it A. B. Vasilieva} and {\it 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.
[Vladimir Gorbunov (Ul'yanovsk)]

MSC 2000:: *65R20 Integral equations (numerical methods)
45J05 Integro-ordinary differential equations

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

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