Search

In this study, the solutions of 2 + 1 -dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Polya-Szego, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function $ \varpi $ and proportionality index $ \varphi. $ It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.

At first, we recall the q-operators in the context of q-calculus and by examining these operators we will introduce new definitions of the partial q-operators. Then, we investigate some new refinements inequalities of Hermite–Hadamard ($H-H$ H − H ) type on the coordinated convex functions involving the new defined partial q-operators. From our main results, we establish several specific inequalities and we point out the existing results which had already been obtained in the literature.

In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Gruss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

In this paper, we propose a new family of interval valued ($ \mathrm{IV} $) convex functions termed as generalized modified $ (p, h) $-convex $ \mathrm{IV} $ functions. We obtain the counterpart of Hermite-Hadamard $ H\cdot H $ inequality by extending the $ \mathrm{IV} $ fractional integral to the $ \mathrm{IV} \; \psi_{k} $-Riemann-Liouville ($ \psi_{k}-RL $) fractional integrals. Also, several inequalities using extended operations on the newly defined class of convex $ \mathrm{IV} $ functions are given.

In this paper, we introduced response analysis and controlling the nonlinear vibration of van der pol duffing oscillator subject to parametric and external excitations by connecting the nonlinear integral positive position feedback (NIPPF). The controller combines the characteristics of the positive position feedback controller (PPF) and integral resonant control (IRC) and these controllers are viewed in a blocked scheme for the closed-loop system (system under control). The analytical solution and all resonance cases are obtained by applying the multiple time scales perturbation method (MSPM). The stability analysis is studied using the frequency response equations at the supposed worst resonance case. For numerical simulation, the system is examined before and after connecting the NIPPF controller using Matlab software program (package ode45). The influences of different parameters are examined numerically for the vibrating system and controller. The approximate analysis is confirmed with numerical simulation outcomes. Finally, making a comparison with previously related published work.

In this paper, new intuitionistic fuzzy fixed point results for sequence of intuitionistic fuzzy set-valued maps in the structure of $ b $-metric spaces are examined. A few nontrivial comparative examples are constructed to keep up the hypotheses and generality of our obtained results. Following the fact that most existing concepts of Ulam-Hyers type stabilities are concerned with crisp mappings, we introduce the notion of stability and well-posedness of functional inclusions involving intuitionistic fuzzy set-valued maps. It is a familiar fact that solution of every functional inclusion is a subset of an appropriate space. In this direction, intuitionistic fuzzy fixed point problem involving $ (\alpha, \beta) $-level set of an intuitionistic fuzzy set-valued map is initiated. Moreover, novel sufficient criteria for existence of solutions to an integral inclusion are investigated to indicate a possible application of the ideas presented herein.

The proposed work focuses on improving the performance of the traditional nonlinear saturation controller (NSC) algorithm. In this work, a tuning mechanism can be implemented by providing the NSC control unit with the measured speed $\Omega $ of the rotor from a shaft encoder device in order to introduce a tuned NSC (TNSC). In other words, we are going to tune the NSC’s natural frequency ${\omega _{c}}$ at one half of $\Omega $ such that $\Omega =2{\omega _{c}}$ . This TNSC is adopted to reduce the vibratory amplitudes of a 16-pole constant-stiffness rotor-Active magnetic bearings AMBs) system for a wide range of speeds and eccentricities. The whole controlled system is studied mathematically to seek its approximate solutions via the multiple scales technique. Different relations between the rotor’s amplitudes and its parameters are plotted so as to verify the proposed tuning mechanism and its crucial role in improving the NSC’s performance.

This paper focuses on controlling the dynamics of a mass-damper-spring model via a servo-controlled linear actuator (SCLA). The displacements of the mass are sensed and measured by a linear variable differential transformer (LVDT) whose output electrical signal is proportional with the position of the mass. Furthermore, a positive position feedback (PPF) controller is then provided with this signal to be processed and to generate the suitable control signal. The control signal itself is applied to the actuator which in turn controls the mass position. The whole system dynamical equations for the amplitudes and phases are derived with the aid of Krylov-Bogoliubov averaging perturbation method. 2D and 3D visualizations are included to give the reader a wider aspect of the system behavior before and after control.

In this paper, the authors determine the number of infected cells and concentration of infected (viral) particles in plasma during HIV-1 (human immunodeficiency virus type one) infections using Shehu transformation. For this, the authors first defined some useful properties of Shehu transformation with proof and then applied Shehu transformation on the mathematical representation of the HIV-1 infection model. The mathematical model of HIV-1 infections contains a system of two simultaneous ordinary linear differential equations with initial conditions. Results depict that Shehu transformation is very effective integral transformation for determining the number of infected cells and concentration of viral particles in plasma during HIV-1 infections.

In this paper, we introduce a nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces. By adding the nonlinear proportional derivative controller (NPD) to the vibrating motion of the vertical vibration conveyor, the energy was transferred between uncontrolled and controlled system. We calculate the approximate solutions of the vibrating system utilizing the method of multiple scales. In addition, we investigate the stability at worst resonance cases using phase plane technique, equations of frequency response, and averaging method. The vertical vibration conveyor behavior was studied numerically at the values of its different parameters. The results exhibit the efficiency of NPD control unit to avoid the oscillations of the vertical conveyor system. Numerical simulations have been carried out using MAPLE and MATLAB software's to ensure the fidelity of our results. Comparisons are made between analytical and numerical results. Also, findings of the present work are discussed in details and compared with published works.

The nonlinear dynamics control of a contact-mode atomic force microscopy (AFM) system with multi forces (harmonic and parametric excitation force) utilizing the time delay proportional derivative (PD) controller is investigated. The perturbation method is utilized to calculate the first-order approximate solutions for the AFM system. The stability of the AFM system is investigated at the worst resonance case by Lyapunov's first method. We also show the bifurcation diagrams of response curves using frequency response equations before and after control are performed. Furthermore, we focus on the effect of the time-delayed control and returns signals gain on the vibration amplitude and the stability analysis of the controlled system in primary, sub-harmonic resonance for different parameter variations. In addition, the numerical results are procured using MATLAB program. Eventually, validation curves are presented to estimate the nearness degree between the analytical predictions also numerical simulation. The obtained results exhibited that, the time-delayed PD control efficiency to put down the nonlinear oscillations of the system.

The impact of Newtonian heating on a time-dependent fractional magnetohydrodynamic (MHD) Maxwell fluid over an unbounded upright plate is investigated. The equations for heat, mass and momentum are established in terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu (ABC) fractional derivatives. The solutions are evaluated by employing Laplace transforms. The change in the momentum profile due to variability in the values of parameters is graphically illustrated for all three C, CF and ABC models. The ABC model has proficiently revealed a memory effect.

A number of mathematical methods have been developed to determine the complex rheological behavior of fluid’s models. Such mathematical models are investigated using statistical, empirical, analytical, and iterative (numerical) methods. Due to this fact, this manuscript proposes an analytical analysis and comparison between Sumudu and Laplace transforms for the prediction of unsteady convective flow of magnetized second grade fluid. The mathematical model, say, unsteady convective flow of magnetized second grade fluid, is based on nonfractional approach consisting of ramped conditions. In order to investigate the heat transfer and velocity field profile, we invoked Sumudu and Laplace transforms for finding the hidden aspects of unsteady convective flow of magnetized second grade fluid. For the sake of the comparative analysis, the graphical illustration is depicted that reflects effective results for the first time in the open literature. In short, the obtained profiles of temperature and velocity fields with Laplace and Sumudu transforms are in good agreement on the basis of numerical simulations.

We consider a class of uncertain fractional difference equation of the Liouville-Caputo type (UFLCDE). An equivalent uncertain fractional sum equation is found to the UFLCDE by using the basic properties. The successive Picard iteration method for finding a solution to the UFLCDE is introduced. Using the theory of Banach contraction under the Lipschitz constant condition, we investigate the structure of algebras of existence and uniqueness of the UFLCDE. The article finally exhibits three examples to show the effectiveness of the proposed investigation.

In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.

In this article, we are interested in some well-known dynamic inequalities on time scales. For this reason, we will prove some new Hermite–Hadamard (H-H) and Opial dynamic inequalities on time scales. The main results here will be derived via the dynamic integration by parts and chain rule formulas on time scales. In addition, we will extend and unify the inequalities for the convex functions.

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (HH-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter “γ” and real valued function “Ω” selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications.