Axioms

Research

28 pages, 370 KiB

Open AccessArticle

A Comprehensive Analysis of Hermite–Hadamard Type Inequalities via Generalized Preinvex Functions

by Muhammad Tariq, Hijaz Ahmad, Hüseyin Budak, Soubhagya Kumar Sahoo, Thanin Sitthiwirattham and Jiraporn Reunsumrit

Axioms 2021, 10(4), 328; https://doi.org/10.3390/axioms10040328 - 30 Nov 2021

Cited by 5 | Viewed by 1870

Abstract

The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite–Hadamard inequality in the mode [...] Read more.

The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite–Hadamard inequality in the mode of the newly introduced concept. To add more insight into the newly introduced concept, we have discussed some algebraic properties and examples as well. Besides, we discuss a few new exceptional cases for the derived results, which make us realize that the results of this paper are the speculations and expansions of some recently known outcomes. The immeasurable concepts and chasmic tools of this paper may invigorate and revitalize additional research in this mesmerizing and absorbing field. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

10 pages, 4386 KiB

Open AccessArticle

A Simple Frequency Formulation for the Tangent Oscillator

by Ji-Huan He, Qian Yang, Chun-Hui He and Yasir Khan

Axioms 2021, 10(4), 320; https://doi.org/10.3390/axioms10040320 - 26 Nov 2021

Cited by 69 | Viewed by 2468

Abstract

The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary [...] Read more.

The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary initial conditions. The tangent oscillator, the hyperbolic tangent oscillator, a singular oscillator, and a MEMS oscillator are chosen to elucidate the simple solving process. The results, when compared with those obtained by the homotopy perturbation method, exhibit a good agreement. This paper introduces a very convenient procedure for attaining quick and accurate insight into the vibration property of a nonlinear vibration system. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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15 pages, 290 KiB

Open AccessArticle

Nabla Fractional Derivative and Fractional Integral on Time Scales

by Bikash Gogoi, Utpal Kumar Saha, Bipan Hazarika, Delfim F. M. Torres and Hijaz Ahmad

Axioms 2021, 10(4), 317; https://doi.org/10.3390/axioms10040317 - 24 Nov 2021

Cited by 6 | Viewed by 2535

Abstract

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann–Liouville sense. We also introduce the nabla fractional derivative in Grünwald–Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed. [...] Read more.

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann–Liouville sense. We also introduce the nabla fractional derivative in Grünwald–Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

21 pages, 339 KiB

Open AccessArticle

New Integral Inequalities via Generalized Preinvex Functions

by Muhammad Tariq, Asif Ali Shaikh, Soubhagya Kumar Sahoo, Hijaz Ahmad, Thanin Sitthiwirattham and Jiraporn Reunsumrit

Axioms 2021, 10(4), 296; https://doi.org/10.3390/axioms10040296 - 07 Nov 2021

Cited by 6 | Viewed by 1312

Abstract

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function [...] Read more.

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

6 pages, 272 KiB

Open AccessArticle

On the Finite Dimensionality of Closed Subspaces in L_p(M, dμ) ∩ L_q(M, dν)

by Alexander A. Balinsky and Anatolij K. Prykarpatski

Axioms 2021, 10(4), 275; https://doi.org/10.3390/axioms10040275 - 25 Oct 2021

Viewed by 1104

Abstract

Finding effective finite-dimensional criteria for closed subspaces in

L_{p},

endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces,

S_{p} \subset L_{p},

[...] Read more.

Finding effective finite-dimensional criteria for closed subspaces in

L_{p},

endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces,

S_{p} \subset L_{p},

whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace

S_{p} \subset L_{p}

is constructed by means of some additional conditions and constraints on

L_{p}

with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace,

S_{p}^{(q)}

, of the functional Banach space,

L_{p} (M, d μ)

, and, moreover, one assumes that additionally,

S_{p}^{(q)} \subset L_{q} (M, d ν)

is subject to a probability measure

ν

on

M .

Then, we show that closed subspaces of

L_{p} (M, d μ) \cap L_{q} (M, d ν)

for

q > max {1, p}, p > 0

are finite dimensional. The finite dimensionality result concerning the case when

q > p > 0

is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to

L_{p} (M, d μ) \cap L_{q} (M, d ν) .

Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

26 pages, 63790 KiB

Open AccessArticle

Nonlinear EHD Instability of Two-Superposed Walters’ B Fluids Moving through Porous Media

by Ji-Huan He, Galal M. Moatimid and Aya Sayed

Axioms 2021, 10(4), 258; https://doi.org/10.3390/axioms10040258 - 18 Oct 2021

Cited by 19 | Viewed by 1669

Abstract

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across [...] Read more.

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across the interface. Additionally, the structure is pervaded via a uniform, normal electrical field in the absence of superficial charges. The nonlinear scheme basically depends on analyzing the linear principal equation of motion, and then applying the appropriate nonlinear boundary-conditions. The current organization creates a nonlinear characteristic equation describing the amplitude performance of the surface waves. The classical Routh–Hrutwitz theory is employed to judge the linear stability criteria. Once more, the implication of the multiple time scale with the aid of Taylor theory yields a Ginzburg–Landau equation, which controls the nonlinear stability criteria. Furthermore, the Poincaré–Lindstedt technique is implemented to achieve an analytic estimated bounded solution for the surface deflection. Many special cases draw upon appropriate data selections. Finally, all theoretical findings are numerically confirmed in such a way that ensures the effectiveness of various physical parameters. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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20 pages, 328 KiB

Open AccessArticle

Near-Common Fixed Point Result in Cone Interval b-Metric Spaces over Banach Algebras

by Muhammad Sarwar, Ziaul Islam, Hijaz Ahmad, Hüseyin Işık and Samad Noeiaghdam

Axioms 2021, 10(4), 251; https://doi.org/10.3390/axioms10040251 - 09 Oct 2021

Cited by 3 | Viewed by 1233

Abstract

In this article, we proposed the concept of cone interval b-metric space over Banach algebras. Furthermore, some near-fixed point and near-common fixed point results are proved in the context of cone interval b-metric space and normed interval spaces for self-mappings under [...] Read more.

In this article, we proposed the concept of cone interval b-metric space over Banach algebras. Furthermore, some near-fixed point and near-common fixed point results are proved in the context of cone interval b-metric space and normed interval spaces for self-mappings under different types of generalized contractions. An example is presented to validate our main outcome. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

24 pages, 906 KiB

Open AccessArticle

Mathematical Modeling and Forecasting of COVID-19 in Saudi Arabia under Fractal-Fractional Derivative in Caputo Sense with Power-Law

by Mdi Begum Jeelani, Abeer S. Alnahdi, Mohammed S. Abdo, Mansour A. Abdulwasaa, Kamal Shah and Hanan A. Wahash

Axioms 2021, 10(3), 228; https://doi.org/10.3390/axioms10030228 - 15 Sep 2021

Cited by 17 | Viewed by 2385

Abstract

This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order

μ

and fractal dimension

χ

. We give some detailed analysis on the existence and uniqueness of the [...] Read more.

This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order

μ

and fractal dimension

χ

. We give some detailed analysis on the existence and uniqueness of the solution to the proposed problem. Furthermore, some results regarding basic reproduction number and stability are given. For the proposed theoretical analysis, we use fixed point theory while for numerical analysis fractional Adams–Bashforth iterative techniques are utilized. Using our numerical scheme is verified by using some real values of the parameters to plot the approximate solution to the considered model. Graphical presentations corresponding to different values of fractional order and fractal dimensions are given. Moreover, we provide some information regarding the real data of Saudi Arabia from 1 March 2020 till 22 April 2021, then calculated the fatality rates by utilizing the SPSS, Eviews and Expert Modeler procedure. We also built forecasts of infection for the period 23 April 2021 to 30 May 2021, with 95% confidence. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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9 pages, 246 KiB

Open AccessArticle

Inequalities on General L_p-Mixed Chord Integral Difference

by Hongying Xiao, Weidong Wang and Zhaofeng Li

Axioms 2021, 10(3), 220; https://doi.org/10.3390/axioms10030220 - 10 Sep 2021

Viewed by 1283

Abstract

In this article, we introduce the concept of general

L_{p}

-mixed chord integral difference of star bodies. Further, we establish the Brunn–Minkowski type, Aleksandrov–Fenchel type and cyclic inequalities for the

L_{p}

-mixed chord integral difference. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

25 pages, 2508 KiB

Open AccessArticle

Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

by Ji-Huan He, Mahmoud H. Taha, Mohamed A. Ramadan and Galal M. Moatimid

Axioms 2021, 10(3), 200; https://doi.org/10.3390/axioms10030200 - 24 Aug 2021

Cited by 14 | Viewed by 1672

Abstract

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the [...] Read more.

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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15 pages, 3106 KiB

Open AccessArticle

An Improved Tikhonov-Regularized Variable Projection Algorithm for Separable Nonlinear Least Squares

by Hua Guo, Guolin Liu and Luyao Wang

Axioms 2021, 10(3), 196; https://doi.org/10.3390/axioms10030196 - 22 Aug 2021

Cited by 1 | Viewed by 1590

Abstract

In this work, we investigate the ill-conditioned problem of a separable, nonlinear least squares model by using the variable projection method. Based on the truncated singular value decomposition method and the Tikhonov regularization method, we propose an improved Tikhonov regularization method, which neither [...] Read more.

In this work, we investigate the ill-conditioned problem of a separable, nonlinear least squares model by using the variable projection method. Based on the truncated singular value decomposition method and the Tikhonov regularization method, we propose an improved Tikhonov regularization method, which neither discards small singular values, nor treats all singular value corrections. By fitting the Mackey–Glass time series in an exponential model, we compare the three regularization methods, and the numerically simulated results indicate that the improved regularization method is more effective at reducing the mean square error of the solution and increasing the accuracy of unknowns. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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16 pages, 2286 KiB

Open AccessArticle

Periodic Property and Instability of a Rotating Pendulum System

by Ji-Huan He, Tarek S. Amer, Shimaa Elnaggar and Abdallah A. Galal

Axioms 2021, 10(3), 191; https://doi.org/10.3390/axioms10030191 - 18 Aug 2021

Cited by 64 | Viewed by 2931

Abstract

The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic [...] Read more.

The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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