On some comparison of multistep second derivative methods with the multistep hybrid methods and their application to solve integro-differiential equations

Abstract

Abstract As is known the necessity to solve the initial-value problem for the Volterra integro-differential equations arises in investigation of the residual knowledge of some objects. Volterra by using the integro-differential equations has studied the memory of land and many other important problems which have been arisen in the study of some phenomena from the different industries of natural sciences. By using a similar form of the initial-value problem for both Volterra integro-differential equation and ODEs have established some relation. Here, by continuing this investigation receives the direct relation between ODE and Volterra integro-differential equation. By using this relation to solve Volterra integro-differential equation have applied the methods which are used in solving of the initial-value problem for ODEs. For this aim have proposed to use the general form of the multistep second derivative hybrid methods. Demonstrated the advantages of this method and constructed one-step methods with order of exactness p ≤ 10. The constructed here methods have compared with the known methods. And also by using the model problem have been illustrated the advantages of the received here results.