Residual Power Series Approach for Solving Linear Fractional Swift-Hohenberg Problems

In this paper, the general state of quantum mechanics equations that can be typically expressed by nonlinear fractional Schrödinger models will be solved based on an attractive efficient analytical technique, namely the conformable residual power series (CRPS). The fractional derivative is considered in a conformable sense. The desired analytical solution is obtained using conformable Taylor series expansion through substituting a truncated conformable fractional series and minimizing its residual errors to extract a supportive approximate solution in a rapidly convergent fractional series. This adaptation can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics. The effectiveness and feasibility of the CRPS procedures are illustrated by verifying three realistic applications. The obtained numerical results and graphical consequences indicate that the suggested method is a convenient and remarkably powerful tool in solving different types of fractional partial differential models.

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Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

International Journal of Differential Equations ◽

10.1155/2018/8686502 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 9

Author(s):

Mohammad Alaroud ◽

Mohammed Al-Smadi ◽

Rokiah Rozita Ahmad ◽

Ummul Khair Salma Din

Keyword(s):

Power Series ◽

Differential Equations ◽

Fractional Differential Equations ◽

Optimization Technique ◽

Fractional Power ◽

Taylor Formula ◽

Residual Power Series ◽

Fractional Differential ◽

Residual Power ◽

Fuzzy Fractional Differential Equations

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1<γ≤2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

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Solving Boundary-Layer Problems by Residual-Power-Series Method

Journal of Mathematics Research ◽

10.5539/jmr.v8n3p68 ◽

2016 ◽

Vol 8 (3) ◽

pp. 68

Author(s):

Mohd Taib Shatnawi

Keyword(s):

Boundary Layer ◽

Power Series ◽

Nonlinear Boundary ◽

Power Series Method ◽

Boundary Layer Equations ◽

Residual Power Series Method ◽

Residual Power Series ◽

Linear And Nonlinear ◽

Boundary Layer Problems ◽

Residual Power

<p><span lang="EN-US">In this paper, the so-called residual-power-series (RPS) method is presented for solving nonlinear boundary-layer equations. The RPS method provides a single unified treatment for the linear and nonlinear terms in the equations. The accuracy and efficiency of the RPS method is demonstrated for both a single and a system of two coupled boundary-layer equations on an unbounded domain.</span></p>

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Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

Mathematical Problems in Engineering ◽

10.1155/2016/5492535 ◽

2016 ◽

Vol 2016 ◽

pp. 1-15 ◽

Cited By ~ 8

Author(s):

Fei Xu ◽

Yixian Gao ◽

Xue Yang ◽

He Zhang

Keyword(s):

Power Series ◽

Fractional Power ◽

Boussinesq Equations ◽

Power Series Method ◽

Residual Power Series Method ◽

Time Space ◽

Residual Power Series ◽

Fractional Differential ◽

Fractional Power Series ◽

Residual Power

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

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