Abstract
There are many classes’ methods for finding of the approximately solution of Volterra integral equations of the second kind. Recently, the numerical methods have been developed for solving the integral equations of Volterra type, which is associated with the using of computers. Volterra himself suggested quadrature formula for finding the numerical solution of integral equation with the variable bounders. By using some disadvantages of mentioned methods here proposed to use some modifications of the quadrature formula which have called as the multistep methods with the fractional step-size. This method has comprised with the known methods and found some relation between constructed here methods with the hybrid methods. And also, the advantages of these methods are shown. Constructed some simple methods with the fractional step-size, which have the degree p≤4 of the receiving results. Here is applied one of suggested methods to solve some model problem and receive results, which are corresponding to theoretical results
Highlights
As is known the construction and application of integral equations usually associated with the name as Abel (see for example Polushuk (1977)
Recent time there is some modification of this method as the multistep advanced method or multistep hybrid methods
The numerical methods with fractional steps have investigated by academician Yanenko
Summary
As is known the construction and application of integral equations usually associated with the name as Abel (see for example Polushuk (1977). Extensive information about the emergence of integral equations with the variable bounders, happened by the intensive work of Vito Volttera (see for example Polushuk (1977), Volterra (1982), Verlan and Sizikov (1986), Verjibitskiy (2001), Hairier et al (1990), Imanova (2020). For finding the numerical solution of these equations, Volterra proposed to use quadrature methods and some of modification. Here consider the construction and application multistep methods with the new properties to solve Volterra integral equations and give some comparison constructed here methods with the known. Let us to consider the following equation. Yy(xx) = ff(xx) + ∫xxxx00 KK xx, ss, yy(ss) dddd , xx00 ≤ ss ≤ xx ≤ XX.
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