Research of Collocation and Collocation-Iterate Methods for Solution of one Type of Integro-Functional Equations with Small Nonlinearity

Different types of differential, integral, integro-differential, differential-functional, integro-functional equations and their systems are mathematical models of many problems of natural science and technology. In the study of mathematical models, qualitative and analytical methods of the theory of differential equations and methods of computational mathematics are widely used. Among the large number of approximate methods that have a wide range of application in solving various applied problems, a common drawback is observed, the characteristic feature of which is power-law convergence and computational instability. In addition, finding sufficiently accurate approximations using projection methods is often associated with the need to solve systems of high-order equations, which turns out to be quite a difficult task. The desire to simplify cumbersome computational schemes leads to the development and increasingly widespread use of the collocation method and the collocation-iterative method. Integro-functional equations are quite widely used in various fields of science, in particular, such equations are reduced to equations with a deviation of the argument as a neutral type and with a delay. The article considers one type of integro-functional equation with small nonlinearity. The process of transforming such an equation into an equation of a much simpler structure is described in detail. It is shown that under certain conditions, the initial equation and the equation obtained after simplification are equivalent. The existence theorem of the solution of the above-mentioned equation is formulated, the ideas of applying collocation and collocation-iterative methods to one type of integro-functional equation with small nonlinearity are described.