In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

A new method called the modification of step variational iteration method (MoSVIM) is introduced and used to solve the fractional biochemical reaction model. The MoSVIM uses general Lagrange multipliers for construction of the correction functional for the problems, and it runs by step approach, which is to divide the interval into subintervals with time step, and the solutions are obtained at each subinterval as well adopting a nonzero auxiliary parameter ℏ to control the convergence region of series′ solutions. The MoSVIM yields an analytical solution of a rapidly convergent infinite power series with easily computable terms and produces a good approximate solution on enlarged intervals for solving the fractional biochemical reaction model. The accuracy of the results obtained is in a excellent agreement with the Adam Bashforth Moulton method (ABMM).

In this paper, a vaccination susceptible exposed infected recovered (VSEIR) model f o rtuberculosis in North Sumatera is discussed. The VSEIR model is formed by a system of nonlinear differentialequations. The approximate solution of this model is obtained by using the step variational iteration method(SVIM) and variational iteration method (VIM). VIM used the general Lagrange multiplier in the correctionfunctional and runs iteratively. SVIM also used the general Lagrange multipliers for construction of the correctionfunctional for the problems, and runs by step approach, where the computation is done b y d i v i d i n g theinterval into subintervals with time steps. The two methods are alternative methods to obtain the approximatesolutions of the VSEIR model. Comparison is made with the conventional numerical method, t h e fourth-orderRunge–Kutta method (RK4). From the results, the SVIM solution is more accurate than the VIM solution for longtime interval when compared to RK4. Keywords: VSEIR model, General Lagrange Multiplier, Variational Iteration Method, Step Variational Iteration Method,Fourth Order Runge Kutta