Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations

Abstract

A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.

Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

“…Step 3: Substituting the resulting equation of Equation 16into Equation (6) with the aid of Equations (9) and (11), the function P in Equation (6) can be transformed into a polynomial in φ and ψ, in which the degree of ψ is not larger than one. Equating each coefficient of the resulting polynomial to zero, we obtain a system of algebraic equations, which can be solved using symbolic computational packages, such as Maple or Mathematica, for the following unknowns: a 0 , a j , b j (j = 1, 2, .…”

“…Step 5: Similarly to Step 3, substituting the resulting equation of Equation 16into Equation (6) with the aid of Equations (9) and (15) for the case λ = 0, we can obtain the traveling wave solutions of Equation (4) by using the transformation in Equation 5. The resulting exact solutions are expressed as rational functions.…”

“…If λ > 0, we substitute Equation (50) into Equation (48), along with the use of Equations (9) and (13). Then, the left-hand side of Equation (48) becomes a polynomial in φ(ξ) and ψ(ξ).…”

“…Over the last few decades, many kinds of solutions of NLEEs, including exact solutions, analytical approximate solutions, and numerical solutions, have been successfully obtained using various and efficient methods. Examples of the methods for obtaining analytical approximate solutions of NLEEs are the Adomian decomposition method (ADM) [7,8], the revised variational iteration method (RVIM) [9], the reduced differential transform method [10], and the homotopy perturbation method (HPM) [11,12]. Useful methods for solving NLEEs numerically are those such as the finite element method [13], the finite volume method [14], and the finite-difference predictor-corrector method [15].…”

Numerical Solutions of Fractional Order Autocatalytic Chemical Reaction Model

“…The researchers have been developed various schemes for solving fractional differential equations (FDE) in two significant categories: theoretical and numerical approaches. For the sake of completeness we refer to some of them: the nonstandard finite difference method (NSFD) [33], the Adomian decomposition method [30,31], the variational iteration method [37], the homotopy perturbation method [30], the operational method [23], the differential transform method [1,13,14], the Adams-Moulton method [16,18], the predictorcorrector methods [7,8,18] and the product integration rules [20,38].…”