Noeiaghdam S.   Sidorov D.  

A Novel Numerical Optimization technique to Control the Accuracy of Semi-Analytical Methods for Solving Volterra Integral Equations with Discontinuous Kernel

Докладчик: Noeiaghdam S.

The aim of this study is to discuss application of the CESTAC method and the
CADNA library to control the accuracy of the Adomian decomposition method and
the homotopy perturbation method to solve the linear and nonlinear Volterra integral
equations with discontinuous kernel. The importance of solving this problem is be-
cause of its applications in the load leveling problems, energy storage with renewable
and diesel generation, charge/discharge storages control and others [1].
In general, the mathematical methods for solving the mentioned problem are based
on
oating point arithmetic and the accuracy of the method has been discussed using
the traditional absolute error which depends on the exact solution and also a positive
small value ". But in real life problems we do not have the exact solution. Also, based
on this condition we will not be able to nd more accurate approximations because
we do not have information about optimal ". For small values of ", the numerical
algorithm can not be stopped and extra iterations will be produced without improving
the accuracy. For large values of ", the numerical algorithm will be stopped in initial
steps without producing enough iterations.
Because of the mentioned problems we apply a new termination criterion which
depends on two successive approximations. For this aim we apply the CESTAC
method and the CADNA library which are based on stochastic arithmetic. In this
condition, not only we do not need to have the exact solution but also we would be
able to identify the optimal approximation, optimal iteration and optimal error of
numerical procedure. Also, the CADNA library is applied as an important software
for this validation. The CADNA library should be done on the LINUX operating
system and its codes should be written using C, C++ or ADA codes [2].