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

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