Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions

Arqub, Omar Abu; Al‐Smadi, Mohammed

doi:10.1007/s00500-020-04687-0

Cited by 170 publications

(67 citation statements)

References 48 publications

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“…Consequently, by operating on both sides of Equation (37), using the fact that = 0 for > at = 0 and equating the resulting equations to zero at = 0 yields As the former, the fourth unknown coefficients 4 ( ) and 4 ( ) can be obtained using the same procedure of CRPS with the help of the fact that Hence, the first few CRPS solutions for Equations (30) and (31) are given as follows: + sin( ) 3 3! 3 + cos( ) 4 4!…”

Section: Example 2 Consider the Following Linear Fractional Schrödinmentioning

confidence: 99%

“…To show the geometric behaviors of the fourth CFPS approximations of Equations (30) and (31), the numerical comparison of exact and approximate solutions at different values of such that ∈ {1, 0.9, 0.8, 0.7} is presented in Table A2 for each value of in [0, 1] with step-size 0.2 when = 0.1 and = 5. From these numerical comparisons, it can be concluded that the results obtained by the proposed method are excellent in comparison with the exact solution.…”

Section: Example 2 Consider the Following Linear Fractional Schrödinmentioning

confidence: 99%

“…Since then, several studies have appeared in the literature. For more details see [26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

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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: Example 2 Consider the Following Linear Fractional Schrödinmentioning

confidence: 99%

Section: Example 2 Consider the Following Linear Fractional Schrödinmentioning

confidence: 99%

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

et al. 2020

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“…Intensive work on extending classic approaches to develop models involving uncertainty can be also seen in construction of algorithms for solving differential equations. For example, in the works [96][97][98][99], authors propose algorithms that operate on fuzzy numbers and use the theory of kernel reproduction. Processing and modeling of imprecision involving similar approaches seems to be an interesting approach in analyses related to human behavior.…”

Section: Theoretical Advancementsmentioning

confidence: 99%

Modeling human thinking about similarities by neuromatrices in the perspective of fuzzy logic

Grobelny

Michalski

Weber

2020

Neural Comput & Applic

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In this work, we propose a new method for modeling human reasoning about objects’ similarities. We assume that similarity depends on perceived intensities of objects’ attributes expressed by natural language expressions such as low, medium, and high. We show how to find the underlying structure of the matrix with intensities of objects’ similarities in the factor-analysis-like manner. The demonstrated approach is based on fuzzy logic and set theory principles, and it uses only maximum and minimum operators. Similarly to classic eigenvector decomposition, we aim at representing the initial linguistic ordinal-scale (LOS) matrix as a max–min product of other LOS matrix and its transpose. We call this reconstructing matrix a neuromatrix because we assume that such a process takes place at the neural level in our brain. We show and discuss on simple, illustrative examples, how the presented way of modeling grasps natural way of reasoning about similarities. The unique characteristics of our approach are treating smaller attribute intensities as less important in making decisions about similarities. This feature is consistent with how the human brain is functioning at a biological level. A neuron fires and passes information further only if input signals are strong enough. The proposal of the heuristic algorithm for finding the decomposition in practice is also introduced and applied to exemplary data from classic psychological studies on perceived similarities between colors and between nations. Finally, we perform a series of simulation experiments showing the effectiveness of the proposed heuristic.

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“…In this section, necessary definitions and main results concerning the theory of fuzzy calculus are presented briefly. For more details, refer to [34][35][36][37][38][39]. The set R F , stands to the set of fuzzy numbers are defined on R, which is a convex, normal, completely supported, and upper semi-continuous variable.…”

Section: Overview Of Fuzzy Calculus Theorymentioning

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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Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions

Cited by 170 publications

References 48 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

Modeling human thinking about similarities by neuromatrices in the perspective of fuzzy logic

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

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