The Order Classes of 2-Generator p

In order to classify a finite group using its elements orders, the order classes are defined. This partition determines the number of elements for each order. The aim of this paper is to find the order classes of 2-generator p-groups of class 2. The results obtained here are supported by Groups, Algorithm and Programming (GAP)

On Modified Integral Inequalities for a Generalized Class of Convexity and Applications

In this paper, we concentrate on and investigate the idea of a novel family of modified p-convex functions. We elaborate on some of this newly proposed idea’s attractive algebraic characteristics to support it. This is used to study some novel integral inequalities in the frame of the Hermite–Hadamard type. A unique equality is established for differentiable mappings. The Hermite–Hadamard inequality is extended and estimated in a number of new ways with the help of this equality to strengthen the findings. Finally, we investigate and explore some applications for some special functions. We think the approach examined in this work will further pique the interest of curious researchers

Modified Inequalities on Center-Radius Order Interval-Valued Functions Pertaining to Riemann–Liouville Fractional Integrals

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e., cr-h-preinvexity, are estimated. The fractional operator is used for the first time to prove such inequalities involving center–radius-ordered functions. Some numerical examples are also provided to validate the presented inequalities

Fejér-Type Midpoint and Trapezoidal Inequalities for the Operator <i>ω</i>₁,<i>ω</i>₂-Preinvex Functions

In this work, we obtain some new integral inequalities of the Hermite–Hadamard–Fejér type for operator ω1,ω2-preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator ω1,ω2-preinvex functions of the positive self-adjoint operator in the complex Hilbert spaces. We give the special cases to our results; thus, the established results are generalizations of earlier work. In the last section, we give applications for synchronous (asynchronous) functions

Heat Transfer of Buoyancy and Radiation on the Free Convection Boundary Layer MHD Flow across a Stretchable Porous Sheet

Theoretical influence of the buoyancy and thermal radiation effects on the MHD (magnetohydrodynamics) flow across a stretchable porous sheet were analyzed in the present study. The Darcy–Forchheimer model and laminar flow were considered for the flow problem that was investigated. The flow was taken to incorporate a temperature-dependent heat source or sink. The study also incorporated the influences of Brownian motion and thermophoresis. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. A favorable comparison with earlier published studies was achieved. Graphs were used to investigate and explain how different physical parameters affect the velocity, the temperature, and the concentration field. Additionally, tables are included in order to discuss the outcomes of the Sherwood number, the Nusselt number, and skin friction. The fundamental governing partial differential equations (PDEs), which are used in the modeling and analysis of the MHD flow problem, were transformed into a collection of ordinary differential equations (ODEs) by utilizing the similarity transformation. A semi-analytical approach homotopy analysis method (HAM) was applied for approximating the solutions of the modeled equations. The model finds several important applications, such as steel rolling, nuclear explosions, cooling of transmission lines, heating of the room by the use of a radiator, cooling the reactor core in nuclear power plants, design of fins, solar power technology, combustion chambers, astrophysical flow, electric transformers, and rectifiers. Among the various outcomes of the study, it was discovered that skin friction surges for 0.3 ≤F1≤ 0.6, 0.1 ≤k1≤ 0.4 and 0.3 ≤M≤ 1.0, snf declines for 1.0 ≤Gr≤ 4.0. Moreover, the Nusselt number augments for 0.5 ≤R≤ 1.5, 0.2 ≤Nt≤ 0.8 and 0.3 ≤Nb≤ 0.9, and declines for 2.5 ≤Pr≤ 5.5. The Sherwood number increases for 0.2 ≤Nt≤ 0.8 and 0.3 ≤Sc≤ 0.9, and decreases for 0.1 ≤Nb≤ 0.7

On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically

Abstract In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the Δ 2 $\Delta ^{2}$ , which will be useful to obtain the convexity results. We examine the correlation between the positivity of ( w 0 RL Δ α f ) ( t ) $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of ( 2 , 3 ) $(2,3)$ , H k , ϵ $\mathscr{H}_{\mathrm{k},\epsilon}$ and M k , ϵ $\mathscr{M}_{\mathrm{k},\epsilon}$ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of ( w 0 RL Δ α f ) ( t ) $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ and convexity of the function on a finite time set N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , … , P } $\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}$ for some P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , … } $\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}$ . The numerical part of the paper is dedicated to examinin the validity of the sets H k , ϵ $\mathscr{H}_{\mathrm{k},\epsilon}$ and M k , ϵ $\mathscr{M}_{\mathrm{k},\epsilon}$ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem

New Numerical Results on Existence of Volterra–Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type

Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach

Monotonicity and positivity analyses for two discrete fractional-order operator types with exponential and Mittag–Leffler kernels

The discrete analysed fractional operator technique was employed to demonstrate positive findings concerning the Atangana-Baleanu and discrete Caputo-Fabrizo fractional operators. Our tests utilized discrete fractional operators with orders between 1<φ<2, as well as between 1<φ<32. We employed the initial values of Mittag–Leffler functions and applied the principle of mathematical induction to ensure the positivity of the discrete fractional operators at each time step. As a result, we observed a significant impact of the positivity of these operators on ∇Q(τ) within Np0+1 according to the Riemann–Liouville interpretation. Furthermore, we established a correlation between the discrete fractional operators based on the Liouville-Caputo and Riemann–Liouville definitions. In addition, we emphasized the positivity of ∇Q(τ) in the Liouville-Caputo sense by utilizing this relationship. Two examples are presented to validate the results