Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Youssri, Y. H.

doi:10.3390/fractalfract5030100

Cited by 37 publications

(9 citation statements)

References 28 publications

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“…A fractional generalization of the Heisenberg uncertainty relation has been found [50]. In [51], the authors developed and analyzed a new fractal-fractional operational matrix for orthonormal normalized ultraspherical polynomials that have significant importance in physics. At the end, we illustrated the obtained topological results with the fractal example, that mathematically describes the nature of matter and space on quantum level.…”

Section: Discussionmentioning

confidence: 99%

Mathematical Approach to Distant Correlations of Physical Observables and Its Fractal Generalisation

et al. 2022

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In this paper, the new mathematical correlation of two quantum systems that were initially allowed to interact and then separated is being formulated and analyzed. These correlations are illustrated by many examples and are also connected with fractals at a certain level. The main idea of the paper arises from the EPR paradox, the paradox of Einstein, Podolsky, and Rosen that occurs when the measurement of a physical observable performed on one system has an immediate effect on the other separate system being entangled with it. That is a physical phenomenon, especially when the particles are separated by a large distance. In this paper, we define distant correlations as the advanced method for the exact interpretation of strong connection and influence among those particles even when they are widely separated. On the given topological space (X,τ), we define a notion of τ−metric such that the set X is a τ−metric space and we prove some properties of these spaces. By using this new proposed model, we nullify the contradiction that appears in the EPR paradox. An illustrative example involving fractals is given. This innovative mathematical approach to this physical phenomenon may be attractive for future research in the field of quantum physics.

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Section: Discussionmentioning

confidence: 99%

Mathematical Approach to Distant Correlations of Physical Observables and Its Fractal Generalisation

et al. 2022

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“…Standard spectral methods offer high accuracy when the solution is smooth [19][20][21][22][23][24][25], yet they may not be of much use when the solution is not smooth. For this reason, one may select appropriate basis functions that can effectively approximate the underlying nonsmooth solutions.…”

Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.

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“…For recent relevant work on generalized fractional derivatives, one may see refs. [36][37][38]. In ref.…”

Section: Introductionmentioning

confidence: 99%

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

Yang

Bai

Yang

2022

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In this paper, we investigate the controllability of a class of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive ψ-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results.

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Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Cited by 37 publications

References 28 publications

Mathematical Approach to Distant Correlations of Physical Observables and Its Fractal Generalisation

Mathematical Approach to Distant Correlations of Physical Observables and Its Fractal Generalisation

Novel spectral schemes to fractional problems with nonsmooth solutions

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

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