Abstract: Many problems in natural science are reduced to solving integral equations, among which the Volterra integral equation occupies a significant place. As is known, the Volterra integral equations are represented in two forms, which are called the Volterra integral equations of the first and second kind. Given that there is a transition from one kind of integral equation to another, here consider to investigation of the Volterra integral equation of the second kind. There are many approximate methods for solving Volterra integral equations of the second kind. Here, have studied the numerical solution of the integral equation of the Volterra type. For this aim, here is a recommendation using the multistep methods with constant coefficients, which are very popular in solving initial-value problems for ODEs of the first order. Some authors for the construction of multistep methods have used the quadrature methods which are used to calculate definite integrals. Here by showing the disadvantages of the quadrature methods, suggested constructing the multistep methods for solving the Volterra integral equations of the second kind. For the construction of multistep methods with the best properties, suggest using advanced multistep and multistep second derivative methods. By defining the maximum order of accuracy for the stable methods of the above-mentioned types, here have recommended some optimal methods and by using them have constructed an optimal algorithm for solving the Volterra integral equations.

Keywords: Initial-value problem (IVP), Ordinary Differential Equations (ODEs) of the 1st -2nd orders, Stability and Degree, Multistep Multiderivative Methods, Bilateral Methods

Pages: 486-496