A New Algorithm of Residual Power Series (RPS) Technique

Arafa, A. A. M.

doi:10.1007/s40819-020-00812-7

Cited by 8 publications

(7 citation statements)

References 32 publications

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“…i by utilizing Taylor series expansion. Such a result is fully consistent with the results obtained by homotopy analysis method [57], and Hirota bilinearisation method [37].…”

supporting

confidence: 91%

“…In this regard, the CFD of any constant is zero and the initial conditions can be specified for fractional models similar to those in the classical integer ones. Thereafter, numerous mathematicians and physicists studied the properties of conformable operators [24,[52][53][54][55][56][57][58][59][60]. In this section, we outline the CFD and some essential propositions for enabling us to follow the pattern of periodic and solitary wave solutions for nonlinear TFSEs and nonlinear coupled TFSEs.…”

Section: Mathematical Conformable Conceptsmentioning

confidence: 99%

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Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense

2020

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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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“…i by utilizing Taylor series expansion. Such a result is fully consistent with the results obtained by homotopy analysis method [57], and Hirota bilinearisation method [37].…”

supporting

confidence: 91%

Section: Mathematical Conformable Conceptsmentioning

confidence: 99%

Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense

2020

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“…Modified versions of the RPS technique are developed to acquire better approximate solutions of FDEs such as the solutions of STFDE in [29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

Bayrak

Demir

Ozbilge

2022

Advances in Mathematical Physics

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The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

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“…Qayyum et al extended this technique to higher order BVPs [36]. Other researchers also used RPSM to various problems arise in science and engineering [37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

Qayyum

Khan

2022

Journal of Mathematics

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The construction of mathematical models for different phenomena, and developing their solutions, are critical issues in science and engineering. Among many, the Buckmaster and Korteweg-de Vries (KdV) models are very important due to their ability of capturing different physical situations such as thin film flows and waves on shallow water surfaces. In this manuscript, a new approach based on the generalized Taylor series and residual function is proposed to predict and analyze Buckmaster and KdV type models. This algorithm estimates convergent series with an easy-to-use way of finding solution components through symbolic computation. The proposed algorithm is tested against the Buckmaster and KdV equations, and the results are compared with available solutions in the literature. At first, proposed algorithm is applied to Buckmaster-type linear and nonlinear equations, and attained the closed-form solutions. In the next phase, the proposed algorithm is applied to highly nonlinear KdV equations (namely, classical, modified, and generalized KdV) and approximate solutions are obtained. Simulations of the test problems clearly reassert the dominance and capability of the proposed methodology in terms of accuracy. Analysis reveals that the projected scheme is reliable, and hence, can be utilized for more complex problems in engineering and the sciences.

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A New Algorithm of Residual Power Series (RPS) Technique

Cited by 8 publications

References 32 publications

Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense

Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense

An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

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