Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order

Abstract

The Newell–Whitehead–Segel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion, and convection system. In this analysis, a recent numeric-analytic technique, called the fractional residual power series (FRPS) approach, was successfully employed in obtaining effective approximate solutions to the Newell–Whitehead–Segel equation of the fractional sense. The proposed algorithm relies on a generalized classical power series under the Caputo sense and the concept of an error function that systematically produces an analytical solution in a convergent fractional power series form with accurately computable structures, without the need for any unphysical restrictive assumptions. Meanwhile, two illustrative applications are included to show the efficiency, reliability, and performance of the proposed technique. Plotted and numerical results indicated the compatibility between the exact and approximate solution obtained by the proposed technique. Furthermore, the solution behavior indicates that increasing the fractional parameter changes the nature of the solution with a smooth sense symmetrical to the integer-order state.

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

“…Meanwhile, the same procedure can be performed for other cases. To do that, let p and D 1 1 p are (1)-differentiable, that is, D 1 n p(t) and D 2 n,m p(t) exists, therefore, according to the RPS approach [25][26][27][28][29], the solutions of the converted crisp system (5) at t 0 = 0 can be given by the following forms:…”

“…By applying the same procedure until an arbitrary order n, the desirable unknown coefficients a n and b n of the series solutions (11) can be obtained, and then the n th -truncated series solutions p n,1r (t) and p n,2r (t) of (1,1)-system are also given. On the other hand, more iterations of n lead to more accurate solutions [28][29][30].…”

“…It has been successfully used to establish reliable approximate solutions of many physical and engineering problems, including crisp initial value problems, differential algebraic equations system, singular initial value problems of nonlinear systems, and a fractional stiff system [20][21][22][23]. This approach aims to construct series solutions expansion, by minimizing the residual functions in computing the desirable unknown coefficients of these solutions, which typically produces the solutions in rapidly convergent series forms with no need linearization or any limitation on the nature of the problem and its classification [24][25][26][27][28][29][30][31][32][33].…”

Adaptation of Conformable Residual Power Series Scheme in Solving Nonlinear Fractional Quantum Mechanics Problems

“…In the light of showing the agreement between the exact solutions and CRPS solutions at = 1 of Equations (20) and (21), the absolute and relative errors are listed in Table A1 (Appendix A) for = 9 and at some selected grid points ( , ) in the domain [0, 1] × [0, 1] with step-size 0.1 for time and space directions. From the table, it can be noted that the CRPS approximate solutions are in good agreement with the exact solutions over the domain of interest.…”

“…Unlike the classical calculus, which has unique definitions and clear geometrical and physical interpretations, there are numerous definitions for the operations of differentiation and integration of fractional order. Riemann-Liouville, Riesz, Grünwald-Letnikov, and Caputo are some examples of these definitions [21][22][23][24]. In this light, a novel definition of fractional derivative, the conformable fractional concept, was proposed in 2014 by Khalil et al [25].…”