Some refinement of the notion of symmetry for the Volterra integral equations and the construction of symmetrical methods to solve them

The theory of integral calculations is employed in most fields of the natural sciences for computing the volumes of rotating bodies, areas with different shapes, distances between objects, and other applications. They can be used to explore energy signals to study earthquakes, the distribution of telecommunications signals, and other uses. In particular, many problems formulated as mathematical models use integral equations with variable boundaries. The most interesting applications use integrals where the boundaries are symmetric functions. In the present study, to solve the Volterra integral equations with variable boundaries, we propose multi-step methods such as hybrids, and we construct some simple algorithms for their application where the order of accuracy is p⩽8(k=1).