A new approach on fractional calculus and probability density function

Chen, Shubo; Rashid, Saima; Noor, Muhammad Aslam; Ashraf, Rehana; Chu, Yu‐Ming

doi:10.3934/math.2020451

Cited by 65 publications

(13 citation statements)

References 27 publications

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“…(A 1 ) Theorem 3 and Corollary 2 deduce to Theorem 3.3 and Corollary 3.2 as in [34] with φ(t) = t; (A 2 ) Corollary 2 deduces to Theorem 2 as in [33] with α = 1 and φ(t) = t; (A 3 ) If we input ρ = 1, we have Theorem 3.5 and Corollary 3.6 in [40].…”

Section: Resultsmentioning

confidence: 93%

“…In 2019, the Hadamard fractional integral operator was used by Khellaf and co-workers [39] to construct several novel integral inequalities of the F.E and F.V functions of the C.R.V. In 2020, Chen and co-workers [40] analyzed numerous novel inequalities in the context of a generalized RLfractional integral operator with respect to another function via the C.R.V. The results are the generalizations and enhancements of the previously published papers in [41][42][43][44][45][46][47][48][49][50] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications

et al. 2022

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This study investigates a variety of novel estimations involving the expectation, variance, and moment functions of continuous random variables by applying a generalized proportional fractional integral operator. Additionally, a continuous random variable with a probability density function is presented in context of the proportional Riemann–Liouville fractional integral operator. We establish some interesting results of the proportional fractional expectation, variance, and moment functions. In addition, constructive examples are provided to support our conclusions. Meanwhile, we discuss a few specific examples that may be extrapolated from our primary results.

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Section: Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications

et al. 2022

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“…They reported that efficiency grows about 45% with fraction of 0.1. Greater productivity can be accessible with involving new techniques [81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99].…”

Section: Introductionmentioning

confidence: 99%

Modeling approach for nanomaterial convective migration with inclusion of Lorentz force

Rothan

2021

Z Angew Math Mech

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Hybrid nanomaterial migration through a permeable container has been simulated in current article with considering buoyancy effect. The inner surface absorbs constant heat flux and horizontal magnetohydrodynamic (MHD) is imposed. To employ the role of new fluid, homogeneous approach formulation was applied definition of stream function creates simpler form of equations. CVFEM was incorporated to model the equations. Entropy generations owing to various factors were examined with reporting contour plots. Variations of Be and Nu were reported by using 3D plots. Growing Ra, power of eddy rises and circular wall temperature decreases about 40%. Besides, Nu augments about 129.8% with growth of Ra while Be declines about 70.8%. As permeability augments, Nu enhances by 11.01% but Be decreases about 8.36%. Existence of Lorentz force make the hybrid nanofluid to move with lower velocity and Nu reduces about 24.5% while Be augments with impose of greater magnetic force.

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“…e DEs have ability to formulate many complex phenomena in various fields such as biology, fluid mechanics, plasma physics, fluid mechanics, and optics; many exact and numerical schemes have been being derived such as [9][10][11][12][13][14][15]. Differential equation of second order appears in models as well as in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations and quantum mechanics, biological, physical and chemical phenomena, optimization, mathematics of networks, and dynamical systems (see [16][17][18][19][20][21][22][23][24]).…”

Section: Introductionmentioning

confidence: 99%

Second-Order Differential Equation: Oscillation Theorems and Applications

Santra

Bazighifan

Ahmad

et al. 2020

Mathematical Problems in Engineering

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Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 , for y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

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A new approach on fractional calculus and probability density function

Cited by 65 publications

References 27 publications

Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications

Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications

Modeling approach for nanomaterial convective migration with inclusion of Lorentz force

Second-Order Differential Equation: Oscillation Theorems and Applications

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