G.Yu. Mehdiyeva, V.R. Ibrahimov, M.N. Imanova
On the application of the hybrid methods to the solving of the Volterra integro-differential equations


As is known, in the early 20th century, for solving of some problems in the field of natural sciences, Vito Volterra faced the necessity for solving integro-differential equations with variable boundaries. Several authors studied integro-differential equations in junction of differential and integral equations. Therefore, they used quadrature method or its modification. Among these, stable methods that have a high order of accuracy was of theoretical and practical interest presented. In this paper, hybrid methods are proposed for the construction of numerical methods with this property. The authors construct concrete second derivative multistep methods with the orders of accuracy of p=6 and p=8 using information pertaining to the solution of the considered problem with one and two mesh points, respectively. Also here, any algorithm can be formulated for using the constructed methods. A comparison of the hybrid methods with the Gauss method is given.

Keywords:
Volterra integro-differential equations, Hybrid method, Degree and stability, Necessary and sufficient conditions Initial-value problem
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