Abstract
In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.
Highlights
Most mathematical models used in many problems of physics, biology, chemistry, engineering, and in other areas are based on integral equations like the linear Volterra integral equation of the second kind:t x t y0 t k t, s x s ds t ∈ a, b, where k : a, b × a, b → R and y0 : a, b → R are two known continuous functions and x : a, b → R is the unknown function to be determined.Many authors have paid attention to the study of linear Volterra integral equation of the second kind from the viewpoint of their theoretical properties, numerical treatment, as well as its applications see e.g., 1–4 and the references therein
For m 1, the result is clearly true. Suppose that it holds for p ≤ m − 1
First we show that for all t ∈ 0, 1, wj t ≤ y0 j tn k n, n 1 n!
Summary
Most mathematical models used in many problems of physics, biology, chemistry, engineering, and in other areas are based on integral equations like the linear Volterra integral equation of the second kind:. Many authors have paid attention to the study of linear Volterra integral equation of the second kind from the viewpoint of their theoretical properties, numerical treatment, as well as its applications see e.g., 1–4 and the references therein. In this paper a new technique for solving this linear Volterra integral equation is shown. Schauder bases in adequate Banach spaces have been used in other numerical methods for solving some integral, differential, or integrodifferential equations see 5–10 , in each problem the analytical techniques are quite different, as fixed point theorems, duality mapping in a Banach space, and generalized least-squares methods. Among the main advantages that our method presents over the classical ones, as collocation or quadrature see 11 , we can point out that it is not necessary to solve linear equations systems.
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