Abstract
The researchers aimed to study the nonlinear fractional order model of malaria infection based on the Caputo-Fabrizio fractional derivative. The homotopy analysis transform method (HATM) is applied based on the floating-point arithmetic (FPA) and the discrete stochastic arithmetic (DSA). In the FPA, to show the accuracy of the method we use the absolute error which depends on the exact solution and a positive value ε. Because in real life problems we do not have the exact solution and the optimal value of ε, we need to introduce a new condition and arithmetic to show the efficiency of the method. Thus the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. The CESTAC method is based on the DSA. Also, a new termination criterion is used which is based on two successive approximations. Using the CESTAC method we can find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorem of the CESTAC method is proved to show that the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs of the exact and approximate solutions. Plotting several graphs, the regions of convergence are demonstrated for different number of iterations k = 5, 10. The numerical results based on the simulated data show the advantages of the DSA in comparison with the FPA.
Highlights
Malaria is a life-threatening disease caused by a parasite
Dynamical validation of fuzzy integral equations [27], control of accuracy on the load leveling problem [28], solving the reverse osmosis system [29,30], validation of collocation method for various problems [31], validation of mathematical models [32] and finding the optimal regularized parameter of the regularization method [18] are some applications of the CESTAC method and the CADNA library
Assume that the series (13) are the approximate solutions of the mathematical model of Malaria infection (1) which are produced by the homotopy analysis transform method (HATM), the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs between the exact and approximate solutions
Summary
Malaria is a life-threatening disease caused by a parasite. The causative agent of this disease to humans is the bite of infected female Anopheles mosquitoes. We will have one or two iterations for large values of ε without producing the accurate results and large number of iterations for small values of ε without improving the accuracy This is the big fault of the mathematical methods based on the FPA. Dynamical validation of fuzzy integral equations [27], control of accuracy on the load leveling problem [28], solving the reverse osmosis system [29,30], validation of collocation method for various problems [31], validation of mathematical models [32] and finding the optimal regularized parameter of the regularization method [18] are some applications of the CESTAC method and the CADNA library. The advantages of the DSA than the FPA to solve the fractional order of malaria infection can be found
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