Chapter 15 - Application of the stochastic arithmetic to validate the results of nonlinear fractional model of HIV infection for CD8+T-cells

Abstract

In this chapter, the mathematical model of HIV infection for CD8+T cells is illustrated. At first, we will start modelizing the mentioned phenomenon in the fractional form using the Caputo-Fabrizio fractional derivative. After that the existence of solution using the Picard-Lindelof approach and the Banach fixed point theorem is studied and the fixed point theorem is used to prove the stability analysis of the method. Combining the homotopy analysis method and the Laplace transformation we will present the homotopy analysis transform method (HATM) to solve the obtained fractional problem. This method is among flexible methods because of the auxiliary parameters and functions specially a parameter to control the convergence region. The main aim of this study is to apply the CESTAC method and the CADNA library to validate the results of the HATM. In the CESTAC method instead of the absolute error we apply a novel termination criterion which depends on two successive approximations. Using the CESTAC method we will be able to find the optimal solution, the optimal iteration and the optimal error of the HATM. For this aim, instead of applying the usual mathematical packages we use the CADNA library. This library should be done on LINUX operating system and all CADNA codes should be written by C/C++, FORTRAN, or ADA codes. The main theorem of the CESTAC method is proved. This theorem will support us analytically. Applying this theorem we will show that the number of common significant digits for exact and approximate solutions are almost equal to the number of common significant digits for two successive approximations. Thus, we will be able to apply the new termination criterion instead of traditional absolute error. Finally, after finding the optimal results using the CESTAC method, we will be able to forecast the models better and more accurate than other methods.