Search Results (23)

The main aim of our paper is concerning the damped oscillations of 3D dynamical systems, depending on a single physical parameter. This system does not admit Hamilton–Poisson structure but can be explicitly integrated, and the exact parametric solutions are built via a smooth function. The influence of the physical parameter is semi-analytically analyzed using the Optimal Auxiliary Functions Method (OAFM). One of the advantages of the applied method is the small number of iterations due to the appropriate choice of auxiliary convergence control functions. The OAFM solutions are effectively in good agreement with corresponding numerical ones, represented qualitatively by figures and quantitatively by tables. The statistical tests of residuals highlighted the accuracy of our results. The proposed method can be considered an analytical tool for nonlinear vibration analysis of numerous applications from electrical engineering or mechanical structures based on damped rotatory oscillators to the field of image encryption. Full article

The dynamic model and nonlinear forced vibration of a laminated beam with magnetostrictive layers, embedded on a nonlinear elastic Winkler–Pasternak foundation, in the presence of an electromagnetic actuator, mechanical impact, dry friction, a longitudinal magnetic field, and van der Waals force is investigated in the present work. The dynamic equations of this complex system are established based on von Karman theory and Hamilton’s principle. Then, by means of the Galerkin–Bubnov procedure, the partial differential equations are transformed into ordinary differential equations. The Optimal Auxiliary Functions Method (OAFM) is applied to solve the nonlinear differential equation. The results obtained are validated by comparisons with numerical results given by the Runge–Kutta procedure. Local stability in the neighborhood of the primary resonance is examined by means of the homotopy perturbation method, the Jacobian matrix, and the Routh–Hurwitz criteria. Global stability is studied by introducing the control law input function and using the approximate solution obtained by the OAFM in the construction of the Lyapunov function. La Salle’s invariance principle and Potryagin’s principle complete our study. The effects of some parameters are graphically presented. Our paper reveals the immense potential of the OAFM in the study of complex nonlinear dynamical systems. Full article

The present paper treats the problem of steady laminar MHD flow of an incompressible viscous fluid for mixed convection stagnation-point flow over a vertical stretching sheet in the presence of an externally magnetic field. By means of the Optimal Auxiliary Functions Method (OAFM), the resulting nonlinear ODEs are semi-analytically solved. The impact of various physical parameters, such as the velocity ratio parameter A, the Prandtl number $Pr$, and the Hartmann number $Ha$, on the behavior of velocity and temperature profiles is analyzed. Both assisting ($\lambda >0$) and opposing ($\lambda <0$) flows are considered. The influence of these parameters is tabulated and graphically presented. The originality of this work lies in the development of effective semi-analytical solutions and in the excellent agreement between these solutions and the corresponding numerical solutions. This highlights the accuracy of the proposed method applied to steady laminar MHD flow. A comparative analysis underlines the advantages of the OAFM compared to the iterative method. The obtained results confirm that the OAFM represents a competitive mathematical tool to explore a large class of nonlinear problems with applications in engineering. Full article

The Shimizu–Morioka dynamical system is analytically investigated in this paper by means of the Optimal Auxiliary Functions Method (OAFM). This system has a chaotic dynamical behavior, specified for more physical applications as chaos synchronization, an attractive phenomenon involving various real-life processes. Semi-analytical solutions for the Shimizu–Morioka system are provided. A comparative analysis between the obtained results via the OAFM method and the corresponding numerical solution highlights the accuracy and efficiency of the involved method. The choice of the OAFM method is justified by the performance in comparison with the iterative method with 7–10 iterations. The physical parameters’ influence is investigated on damped oscillations and periodical behaviors of the obtained solutions. Full article

This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. By comparing the solutions derived from these methods to exact solutions, it is clear that they provide accurate and efficient outcomes. These findings highlight the straightforward yet effective use of these methodologies in modeling engineering systems. Navier–Stokes equations have numerous practical uses, including analyzing fluid flow in pipelines and channels, predicting weather patterns, and constructing aircraft and vehicles. Full article

The behavior of the Rucklidge-type dynamical system was investigated, providing some semi-analytical solutions, in this paper. This system was analytically investigated by means of the Optimal Auxiliary Functions Method (OAFM) for two cases. An exact parametric solution was obtained. The effect of the physical parameters was investigated on the asymptotic behaviors and damped oscillations of the solutions. Damped oscillations are essential for analyzing and designing various mechanical, biological, and electrical systems. Many of the applications involving these systems represent the main reason of this work. A comparison between the obtained results via the OAFM, the analytical solution obtained with the iterative method, and the corresponding numerical solution was performed. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. Full article

The present paper focuses on some classes of dynamical systems involving Hamilton–Poisson structures, while neglecting their chaotic behaviors. Based on this, the closed-form solutions are obtained. These solutions are derived using the Optimal Auxiliary Functions Method (OAFM). The impact of the physical parameters of the system is also investigated. Periodic orbits around the equilibrium points are performed. There are homoclinic or heteroclinic orbits and they are obtained in exact form. The dynamical system is reduced to a second-order nonlinear differential equation, which is analytically solved through the OAFM procedure. The influence of initial conditions on the system is explored, specifically regarding the presence of symmetries. A good agreement between the analytical and corresponding numerical results is demonstrated, reflecting the accuracy of the proposed method. A comparative analysis underlines the advantages of the OAFM compared with the iterative method. The results of this work encourage the study of dynamical systems with bi-Hamiltonian structure and similar properties as physical and biological problems. Full article

The present study dealt with a comprehensive mathematical model to explore the nonlinear forced vibration of a magnetostrictive laminated beam. This system was subjected to mechanical impact, a nonlinear Winkler–Pasternak foundation, and an electromagnetic actuator considering the thickness effect. The expressions of the nonlinear differential equations were obtained for the pinned–pinned boundary conditions with the help of the Galerkin–Bubnov procedure and Hamiltonian approach. The nonlinear differential equations were studied using an original, explicit, and very efficient technique, namely the optimal auxiliary functions method (OAFM). It should be emphasized that our procedure assures a rapid convergence of the approximate analytical solutions after only one iteration, without the presence of a small parameter in the governing equations or boundary conditions. Detailed results are presented on the effects of some parameters, among them being analyzed were the damping, frequency, electromagnetic, and nonlinear elastic foundation coefficients. The local stability of the equilibrium points was performed by introducing two variable expansion method, the homotopy perturbation method, and then applying the Routh–Hurwitz criteria and eigenvalues of the Jacobian matrix. Full article

The aim of present paper is to obtain approximate semi-analytical solutions for the Qi-type dynamical system, while neglecting its chaotic behaviors. These solutions are derived using the Optimal Auxiliary Functions Method (OAFM). The impact of the system’s physical parameters is also investigated. A special case, involving a constant of motion, is considered for which closed-form solutions are obtained. The dynamical system is reduced to a second-order nonlinear differential equation, which is analytically solved through the OAFM procedure. The influence of initial conditions on the system is explored, specifically regarding the presence or absence of symmetries. An exact parametric solution is obtained for a particular case. A good agreement between the analytical and corresponding numerical results is demonstrated, highlighting the accuracy of the proposed method. A comparative analysis underlines the advantages of the OAFM compared to other analytical methods. These findings have numerous technological applications, such as in nonlinear circuits with three channels that involve adapted physical parameters to ensure effective functioning of electronic circuits, as well as in information storage, encryption, and communication systems. Full article

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function. Full article

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions. Full article

This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from fractional calculus. The OAFM provides a systematic approach to obtain approximate analytical solutions by constructing an auxiliary function. By optimizing the parameters of the auxiliary function, an approximate solution is derived that closely matches the behavior of the original system. The effectiveness and accuracy of the OAFM are demonstrated through numerical simulations and comparisons with existing methods. Fractional calculus enhances the understanding and modeling of the nonlinear dynamics in the Belousov–Zhabotinsky system. This study contributes to fractional calculus and nonlinear dynamics, offering a powerful tool for analyzing and solving complex systems such as the Belousov–Zhabotinsky equation. Full article

This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal–transverse vibration of a simple supported Euler–Bernoulli beam, which accounted for von Kármán geometric nonlinearity, including the first-order strain–displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler–Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton’s principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal–transverse governing equations were discretized using the Galerkin–Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh–Hurwitz criteria, and global stability was studied using the Lyapunov function. Full article

This paper presents the optimal auxiliary function method (OAFM) implementation to solve a nonlinear fractional system of the Jaulent–Miodek Equation with the Caputo operator. The OAFM is a vital method for solving different kinds of nonlinear equations. In this paper, the OAFM is applied to the fractional nonlinear system of the Jaulent–Miodek Equation, which describes the behavior of a physical system via a set of coupled nonlinear equations. The Caputo operator represents the fractional derivative in the equations, improving the system’s accuracy and applicability to the real world. This study demonstrates the effectiveness and efficiency of the OAFM in solving the fractional nonlinear system of the Jaulent–Miedek equation with the Caputo operator. This study’s findings provide important insights into the behavior of complex physical systems and may have practical applications in fields such as engineering, physics, and mathematics. Full article

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a constant current generating a static force and by an actuation generating a fast alternative force. The deformations between the solids in contact are supposed to be elastic and the contact is maintained. In this study, a single degree of freedom nonlinear damped oscillator under a static normal load is considered. An analytical approximate solution of this problem is obtained using the Optimal Auxiliary Functions Method (OAFM). By means of some auxiliary functions and introducing so-called convergence-control parameters, a very accurate approximate solution of the governing equation can be obtained. We need only the first iteration for this technique, applying a rigorous mathematical procedure in finding the optimal values of the convergence-control parameters. Local stability by means of the Routh-Hurwitz criteria and global stability using the Lyapunov function are also studied. It should be emphasized that the amplitude of AC excitation voltage is not considered much lower than bias voltage (in contrast to other studies). Also, the Hertzian contact coupled with EA is analytically studied for the first time in the present work. The approximate analytical solution is determined with a high accuracy on two domains. Local stability is established in five cases with some cases depending on the trace of the Jacobian matrix and of the discriminant of the characteristic equation. In the study of global stability, the estimate parameters which are components of the Lyapunov function are given in a closed form and a graphical form and therefore the Lyapunov function is well-determined. Full article