Approximate Solutions of Nonlinear Fractional Kolmogorov—Petrovskii—Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

Abstract

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation

“…We can see from the obtained solution that the solution procedure of the proposed method is straightforward and simple to implement, whereas the solution obtained with the help of techniques presented in [24] is difficult, and it requires more computation in order to evaluate more terms in the series solution. The proposed technique provides us two parameters, namely, auxiliary parameter ( ) and embedding parameter q ∈ 0, 1 n (n ≥ 1), which helps to control and adjust the convergence region of the obtained solution.…”

“…There are many methods available in the literature to solve these equations. The KPP equation is studied through distinct techniques like the discrimination algorithm [36], the (G'/G)-expansion method [37], the homotopy perturbation method (HMP) [23], the generalized two-dimensional differential transform method [24], and many others [22,[38][39][40][41][42]. The rest of the paper is arranged as follows.…”

The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics

“…Example 2: Nonlinear fractional Kolmogorov-Petrovskii-Piskunov (KPP) equation has the form : . Song and Wang have implemented the differential transform method for the approximate solution of this equation (Song and Wang, 2012). When a ¼ 1 Eq.…”

Study of traveling soliton and fronts phenomena in fractional Kolmogorov-Petrovskii-Piskunov equation

“…These waves, which oscillate and include pulses, move between stable steady states. The NW equation is also relevant to Rayleigh-Benard convection, chemical processes, Faraday instability, and biological systems [32].The homotopy perturbation approach [33], discrimination algorithm [34], differential transform method [35], homotopy analysis method [36], and (G'/G)-expansion method [37] are only a few of the methods that have been devised to solve the KPP problem.KPP equations are often solved using numerical techniques or series solutions. The FIM (exact solutions by Fourier integral method) approach was used in this work to get precise solutions for the KPP, FHN, and NW equations.…”