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

Shqair, Mohammed; Al‐Smadi, Mohammed; Momani, Shaher; El‐Zahar, Essam R.

doi:10.3390/app10030890

Cited by 20 publications

(11 citation statements)

References 40 publications

(58 reference statements)

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“…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:…”

Section: The Rps Methods For Fuzzy Duffing Oscillatormentioning

confidence: 99%

“…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%

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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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Section: The Rps Methods For Fuzzy Duffing Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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“…In this section we test the proposed scheme on some examples to investigate the efficiency and accuracy of the RPS method. We consider the system of coupled partial differential equations (22) with different parameters and regimes. All computations were carry out by using of Mathematica 10 software package.…”

Section: Numerical Investigationmentioning

confidence: 99%

“…or, by direct substitution into Equation (22). The surface plots of the option prices are presented in Fig.…”

Section: Numerical Investigationmentioning

confidence: 99%

“…The RPSM was first developed for solving first-order fuzzy differential equations. Later, it has been successfully applied to find numerical solutions for other equations, including ordinary and partial differential equations, nonlinear systems of singular initial value problems, pantograph delay differential equation, fractional differential equations, fuzzy fractional differential models [1,2,3,4,5,12,13,16,17,18,20,21,22].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Modeling of Coupled Partial Differential Equations Using Residual Error Functions

Komashynska¹

2020

Int. J. of Appl. Math.

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In this paper, the residual power series method is developed to solve a class of coupled partial differential equations. This approach improves solutions by reducing the residual error functions to create a rapidly convergent series. The description of the proposed method is presented to approximate the solution by highlighting all the steps necessary to implement the algorithm. Meanwhile, the scheme is tested on several cases of examples arising in the field of finance. Numerical results obtained justify that the proposed method is effective, accurate and simple in application.

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Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches

Hasan

Hadid

Al‐Smadi

et al. 2020

Forum for Interdisciplinary Mathematics

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In this chapter, we are interested in the fractional release of the Verhulst model according to Caputo’s sense, which is popular in applying environmental, biological, chemical and social studies describing the population growth model. Such a model, which is sometimes called logistic growth model related to systems in which the rate of change depends on their previous memory. In the light of this, three advanced numerical and analytical algorithms are presented to obtain approximate solutions for different classes of logistical growth problems, including reproducing kernel algorithm, fractional residual series algorithm and successive substitutions algorithm. The first technique relies on the reproducing property that characterises a specific function of building a complete orthogonal system at desired Hilbert spaces. The RPS technique relies on residual error function and generalised Taylor series to reduce residual errors and generate a converging power series, while the last technique converts the fractional logistic model to Volterra integral equation based on Riemann-Liouville integral operator. To demonstrate consistency with the theoretical framework, some realistic applications are tested to show the accuracy and efficiency of the proposed schemes. Numerical results are displayed in tables and figures for different fractional orders to illustrate the effect of the fractional parameter on population growth behaviour. The results confirm that the proposed schemes are very convenient, effective and do not require long-term calculations.

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

Cited by 20 publications

References 40 publications

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Numerical Modeling of Coupled Partial Differential Equations Using Residual Error Functions

Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches

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