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The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

a b s t r a c tIn this paper, we present an efficient algorithm for solving a fractional oscillator using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of a fractional oscillator. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.

In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.

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Mathematical modeling of fractional resonant Schrödinger equations is an extremely significant topic in the classical of quantum mechanics, chromodynamics, astronomy, and anomalous diffusion systems. Based on conformable residual power series, a novel effective analytical approach is considered to solve classes of nonlinear time-fractional resonant Schrödinger equation and nonlinear coupled fractional Schrödinger equations under conformable fractional derivatives. The solution methodology lies in generating an infinite conformable series solution with reliable wave pattern by minimizing the residual error functions. The main motivation for using this approach is high accuracy convergence and low computational cost compared to other existing methods. In this orientation, the competency and capacity of the proposed method are examined by implementing several numerical applications. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features in feasibility, stability, and suitability for dealing with many fractional models emerging in physics and optics using the new conformable derivative.

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