Toward computational algorithm for time-fractional Fokker–Planck models

Freihet, Asad; Hasan, Shatha; Alaroud, Mohammad; Al‐Smadi, Mohammed; Ahmad, Rokiah Rozita; Din, Ummul Khair Salma

doi:10.1177/1687814019881039

Cited by 33 publications

(23 citation statements)

References 24 publications

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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%

“…To demonstrate the effectiveness of the RPS method for solving Equations (21) and (22), the lower and upper bounds of 10 th -RPS approximate solutions of (1,1)-system, with their residual errors, are computed and listed in Table 5 for t = 0.1 and r ∈ [0, 1] with step size 0.25, which can illustrate the efficiency of the proposed method. Symmetry is generally found in many mathematical works involving the fuzzy systems and fuzzy logic, if not most of them.…”

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%

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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: (25)mentioning

confidence: 99%

Section: (25)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

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“…Thus, using the procedures of the RPS algorithm [25][26][27][28], the 4th RPS approximate solution of FBTEs (13) and (14) can be given by ω 4 (t) = 2t 4α Γ(4α+1) . Consequently, the RPS solution at α = 1/2 will be ω(t) = t 2 , which is fully compatible with the exact solution investigated earlier in [32].…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Later, RPSM has been used in generating a fractional power series (FPS) solutions for strongly nonlinear FDEs in the form of a rapidly convergent with a minimum size of calculations without any restrictive hypotheses. Thus, this adaptive can be used as an alternative technique in solving several nonlinear problems arising in engineering and physics [24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

et al. 2019

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Featured Application: Fractional differential equations play a significant role in modeling certain dynamical systems arising in many fields of applied sciences and engineering. In this paper, the authors develop an attractive analytic-numeric technique, residual power series (RPS) method, for solving fractional Bagley-Torvik equations with a source term involving Caputo fractional derivative. In regard to its simplicity, the method can be applicable to a wide class of fractional partial differential equations, fractional fuzzy differential equations, fractional oscillator equations, and so on.Abstract: Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley-Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

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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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Toward computational algorithm for time-fractional Fokker–Planck models

Cited by 33 publications

References 24 publications

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches

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