Adaptation of residual power series method to solve Fredholm fuzzy integro-differential equations

Shammari, Mohammad Al; Al‐Smadi, Mohammed; Arqub, Omar Abu; Hashim, Ishak; Alias, Mohd Almie

doi:10.1063/1.5111209

Cited by 10 publications

(9 citation statements)

References 19 publications

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“…As the classical power series [36][37][38][39][40][41], it is clear that all terms of the multiple fractional PS (Equation (7)) vanish as soon as = 0 , except the first term, which means the multiple fractional PS is convergent when = 0 . Furthermore, for ≥ 0 , this multiple fractional series is definitely convergent for | − 0| < 1/ , ( > 0), where 1/ is the radius of convergence of the series.…”

Section: Notations and Preliminariesmentioning

confidence: 99%

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

et al. 2020

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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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Section: Notations and Preliminariesmentioning

confidence: 99%

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

et al. 2020

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“…According to the RPSM algorithm, the solution details for each case of FVIPs (21) and (22) can be found in Appendix A. However, since this model does not have an exact solution, so to illustrate the efficiency and accuracy of RPS algorithm, the following residual error is defined E(t; r) = p n (t; r) + p n (t; r) + (p n (t; r) ) 3 .…”

Section: (25)mentioning

confidence: 99%

“…In 2013, the RPS method was first proposed and developed by Jordanian mathematician Abu Arqub [19] as an efficient and accurate analytical-numerical method in solving first and second FIVPs. 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].…”

Section: Introductionmentioning

confidence: 99%

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at r = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter r = 1. The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.

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“…The objective of this work is to apply an advanced algorithm, called the fractional residual power series (FRPS) algorithm, for solving the time-FNWSE. The FRPS is a novel numeric-analytic technique for dealing with both linear and nonlinear issues, which enables us to obtain analytical and approximate solutions in convergent fractional power series (FPS) by combining Taylor's fractional series formula and residual error functions without requiring any constrained assumptions [19][20][21][22][23]. The outline of this work is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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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.

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Adaptation of residual power series method to solve Fredholm fuzzy integro-differential equations

Cited by 10 publications

References 19 publications

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

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

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

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

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