A new analytic numeric method solution for fractional modified epidemiological model for computer viruses

Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider the modified epidemiological model for computer viruses (SAIR) proposed by J. R. C. Piqueira and V. O. Araujo. This model includes an antidotal population compartment (A) representing nodes of the network equipped with fully effective anti-virus programs. The multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MSGDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that this method is very effective. Mathematica 9 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution

Solution of the SIR models of epidemics using MSGDTM

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we use the multi-step generalized differential transform method (MSGDTM) to approximate the numerical solution of the SIR model and numerical simulations are presented graphically

Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient

Analytical solution of fractional Burgers–Huxley equations via residual power series method

This paper is aimed at constructing fractional power series (FPS) solutions of fractional Burgers–Huxley equations using residual power series method (RPSM).RPSM is combining Taylor's formula series with residual error function. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme are reliable and powerful in finding the numerical solutions of fractional Burgers–Huxley equations. The numerical results reveal that the RPSM is very effective, convenient and quite accurate to time dependence kind of nonlinear equations. It is predicted that the RPSM can be found widely applicable in engineering

Application of Multistep Generalized Differential Transform Method for the Solutions of the Fractional-Order Chua's System

We numerically investigate the dynamical behavior of the fractional-order Chua's system. By utilizing the multistep generalized differential transform method (MSGDTM), we find that the fractional-order Chua's system with “effective dimension” less than three can exhibit chaos as well as other nonlinear behavior. Numerical results are presented graphically and reveal that the multistep generalized differential transform method is an effective and convenient method to solve similar nonlinear problems in fractional calculus

Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems

A new reliable algorithm based on an adaptation of the standard generalized differential transform method (GDTM) is presented. The GDTM is treated as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of fractional-order Rössler chaotic and hyperchaotic systems. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus

Approximation of Solution of Time Fractional Order Three-Dimensional Heat Conduction Problems with Jacobi Polynomials

In this paper, we extend the idea of pseudo spectral method to approximate solution of time fractional order three-dimensional heat conduction equations on a cubic domain. We study shifted Jacobi polynomials and provide a simple scheme to approximate function of multi variables in terms of these polynomials. We develop new operational matrices for arbitrary order integrations as well as for arbitrary order derivatives. Based on these new matrices, we develop simple technique to obtain numerical solution of fractional order heat conduction equations. The new scheme is simple and can be easily simulated with any computational software. We develop codes for our results using MatLab. The results are displayed graphically.&nbsp