The Way to Construct Innovative Methods for Solving Initial-Value Problem of the Volterra Integro-Differential Equation

Mathematical models for many problems in the natural sciences are often simplified to solving initial-value problems (IVPs) for the Volterra integro-differential equations (VIDE). Numerical methods of a multistep type are typically used to solve these problems. It is known that in some cases, the multi-step method (MSM) is applied to solving the IVPs of both ordinary differential equations (ODEs) and VIDE encountered in solving some problems in mathematical biology. Here, to solve such problems by combining different methods, some modifications of established methods were developed, and it was demonstrated that these methods outperform the existing ones. As is known, one of the main issues in solving the aforementioned problems is determining the reliability of calculating values using the known mathematicalstatistical models (MSMs). In this regard, some experts utilize the predictor-corrector method. Having highlighted the disadvantages of this method, the proposal is to develop an innovative approach and assess the errors that may arise when applying this method to solve various problems. Here, the IVPs for the VIDE of the first order are primarily investigated. To illustrate the benefits of the innovative methods proposed here, we discuss the use of simple numerical methods to solve some common examples.