Application of the Bilateral Hybrid Methods to Solving Initial -Value Problems for the Volterra Integro-Differential Equations

Abstract

The many problems of natural sciences are reduced to solving integro-differential equations with variable boundaries. It is known that Vito Volterra, for the study of the memory of Earth, has constructed the integro-differential equations. As is known, there is a class of analytical and numerical methods for solving the Volterra integro-differential equation. Among them, the numerical methods are the most popular. For solving this equation Volterra himself used the quadrature methods. How known in solving the initial-value problem for the Volterra integro-differential equations, increases the volume of calculations, when moving from one point to another, which is the main disadvantage of the quadrature methods. Here the method is exempt from the specified shortcomings and has found the maximum value for the order of accuracy and also the necessary conditions imposed on the coefficients of the constructed methods. The results received here are the development of Dahlquist’s results. Using Dahlquist’s theory in solving initial-value problem for the Volterra integro-differential equation engaged the known scientists as P.Linz, J.R.Sobka, A.Feldstein, A.A.Makroglou, V.R.Ibrahimov, M.N.Imanova, O.S.Budnikova, M.V.Bulatova, I.G.Buova and ets. The scientists taking into account the direct connection between the initial value problem for both ODEs and the Volterra integrodifferential equations, the scientists tried to modify methods, that are used in solving ODEs and applied them to solve Integro-differential equations. Here, proved that some modifications of the methods, which are usually applied to solve initial-value problems for ODEs, can be adapted for solving the Volterra integro-differential equations. Here, for this aim, it is suggested to use a multistep method with the new properties. In this case, a question arises, how one can determine the validity of calculated values. For this purpose, it is proposed here to use bilateral methods. As is known for the calculation of the validity values of the solution of investigated problems, usually have used the predictor-corrector method or to use some bounders for the step-size. And to define the value of the boundaries, one can use the stability region using numerical methods. As was noted above, for this aim proposed to use bilateral methods. For the illustration advantage of bilateral methods is the use of very simple methods, which are called Euler’s explicit and implicit methods. In the construction of the bilateral methods it often becomes necessary to define the sign for some coefficients. By taking this into account, here have defined the sign for some coefficients.