Search Results (18)

This study develops an analytical framework for investigating in-plane nonlinear subharmonic resonance in fixed–fixed circular arches under a vertical base-excitation, a phenomenon not adequately addressed in previous research. Based on Hamilton’s principle, the governing partial differential equation for in-plane nonlinear motion is first derived. The tangential displacement is then expressed as a modal superposition, and the system is reduced to a set of second-order ordinary differential equations via the Galerkin method. Using the method of multiple scales, the nonlinear 1/2-subharmonic resonance is solved, yielding closed-form, steady-state amplitude–phase relations and corresponding stability conditions. Validation against finite element simulations and Runge–Kutta analyses confirms the accuracy of the proposed approach. Dimensionless fundamental frequencies match finite element results exactly, with discrepancies in critical base-excitation below 2.5%. A close agreement is observed in both the amplitude–frequency and force–response curves with numerical predictions and Bolotin’s method, accurately capturing the characteristic hardening nonlinearity and three distinct dynamic regions spanning negligible vibration, stable resonance, and instability. Parametric studies further reveal key trends. Larger included angles intensify the vibration amplitude and promote saddle-node bifurcation, while narrowing stable operating regions. Higher slenderness ratios enhance structural flexibility and nonlinearity, shifting resonant peaks toward higher frequencies. Increased damping suppresses the response amplitude and raises the thresholds for vibration initiation and bifurcation. Full article

In a variety of applications, including signal processing, clock referencing, sensing, and others, microelectromechanical systems (MEMS) have been shown to be effective and broadly used. This study explores the dynamical response of a nonlinear MEMS resonator when subjected to a sudden mechanical shock under electrical excitation in the presence of quintic nonlinearity. The method of multiple scales (MMS) is utilized to construct the analytical formulas for analyzing the amplitude and phase response during primary resonance conditions. The analytical results are verified and compared with numerical simulations performed using the fourth-order Runge–Kutta method. Additionally, a parametric analysis is performed to examine the effect of different shock values on the resonator’s response and stability utilizing the Jacobian matrix. The agreement between analytical and numerical approaches proves MMS’s effectiveness in analyzing the shock impact on the MEMS resonator. The results provide valuable knowledge about the response and stability of MEMS resonators under mechanical shock, which is crucial for robust design in challenging conditions. Full article

To analyze the dynamic response of tension leg platform (TLP) tendons under irregular ocean wave action, the governing equations of coupled vibration between the platform and tendon under irregular wave action are established based on Hamilton’s principle and the Kirchhoff hypothesis. Using the spectrum representation–random function method, the power spectral density function of the irregular wave load is derived, and the lateral wave forces at different tendon locations are calculated. The coupled lateral and axial responses of the tendon system are obtained through the fourth-order Runge–Kutta method. Considering the parametric vibrations of both the platform and tendon, the extreme lateral deflection of the tendon is employed as the control index to derive the probability density curves of the tendon deflection under irregular wave load. The results show that the amplitude of the wave load increases gradually along the height of the tendon, with a faster growth rate at locations closer to the water surface. The tendon’s lateral deflection response changes more drastically due to coupled parametric vibration of the platform. Based on 628 complete samples of irregular wave loads, the probability density curve and cumulative distribution curve of the extreme lateral deflection of the tendon under irregular wave loads are obtained. Under typical sea state conditions generated from the P-M wave spectrum, the reliability of the tendon under irregular wave load increases with the initial tension force. Full article

We propose a new strategy for constructing Runge–Kutta (RK) methods using evolutionary computation techniques, with the goal of directly minimising global error rather than relying on traditional local properties. This approach is general and applicable to a wide range of differential equations. To highlight its effectiveness, we apply it to two benchmark problems with oscillatory behaviour: the (2+1)-dimensional nonlinear Schrödinger equation and the N-Body problem (the latter over a long interval), which are central in quantum physics and astronomy, respectively. The method optimises four free coefficients of a sixth-order, eight-stage parametric RK scheme using a novel objective function that compares global error against a benchmark method over a range of step lengths. It overcomes challenges such as local minima in the free coefficient search space and the absence of derivative information of the objective function. Notably, the optimisation relaxes standard RK node bounds ($c_i\in [0,1]$), leading to improved local stability, lower truncation error, and superior global accuracy. The results also reveal structural patterns in coefficient values when targeting high eccentricity and non-sinusoidal problems, offering insight for future RK method design. Full article

The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice. Full article

This paper investigates the influence of soil with finite depth on the vibrational behavior of a multi-span continuous beam resting on an elastic foundation. The simplified model of the Timoshenko beam supported on soil with finite depth is established, introducing the foundation displacement decay function. The numerical solution of the continuous beam’s vibration response on the elastic foundation is obtained by using the transfer matrix method (TMM) and fourth-order Runge-Kutta method (RK4). Taking a two-span continuous beam as an illustrative example, the validity of the calculation theory is validated by comparing it with the outcomes obtained from the finite element method (FEM). Utilizing numerical computation and parametric analysis, the vibration response of continuous beams is evaluated in terms of its influence by various factors such as soil thickness, viscous damping coefficient of the soil, subgrade reaction coefficient, and span ratio. The findings indicate that the inertial motion of the soil with a finite depth significantly reduces the continuous beam’s inherent frequency and enhances the structure’s resonance effect. The rise of the subgrade response coefficient increases the system’s resonant frequency while decreasing the displacement response amplitude. The ratio between the adjacent spans determines the effect of beam span vibration energy transfer to adjacent spans. In addition, compared with the span directly excited by a concentrated harmonic load, the impact of soil thickness, subgrade reaction coefficient, and viscous damping, the coefficient of the soil is more significant on the indirect influence span of a continuous beam. Full article

Computer simulation of continuous chaotic systems is usually performed using numerical methods. The discretization may introduce new properties into finite-difference models compared to their continuous prototypes and can therefore lead to new types of dynamical behavior exhibited by discrete chaotic systems. It is known that one can control the dynamics of a discrete system using a special class of integration methods. One of the applications of such a phenomenon is chaos-based communication systems, which have recently attracted attention due to their high covertness and broadband transmission capability. Proper modulation of chaotic carrier signals is one of the key problems in chaos-based communication system design. It is challenging to modulate and demodulate a chaotic signal in the same way as a conventional signal due to its noise-like shape and broadband characteristics. Therefore, the development of new modulation–demodulation techniques is of great interest in the field. One possible approach here is to use adaptive numerical integration, which allows control of the properties of the finite-difference chaotic model. In this study, we describe a novel modulation technique for chaos-based communication systems based on generalized explicit second-order Runge–Kutta methods. We use a specially designed test bench to evaluate the efficiency of the proposed modulation method and compare it with state-of-the-art solutions. Experimental results show that the proposed modulation technique outperforms the conventional parametric modulation method in both coverage and noise immunity. The obtained results can be efficiently applied to the design of advanced chaos-based communication systems as well as being used to improve existing architectures. Full article

We investigate the numerical solution of the nonlinear Schrödinger equation in two spatial dimensions and one temporal dimension. We develop a parametric Runge–Kutta method with four of their coefficients considered as free parameters, and we provide the full process of constructing the method and the explicit formulas of all other coefficients. Consequently, we produce an adaptable method with four degrees of freedom, which permit further optimisation. In fact, with this methodology, we produce a family of methods, each of which can be tailored to a specific problem. We then optimise the new parametric method to obtain an optimal Runge–Kutta method that performs efficiently for the nonlinear Schrödinger equation. We perform a stability analysis, and utilise an exact dark soliton solution to measure the global error and mass error of the new method with and without the use of finite difference schemes for the spatial semi-discretisation. We also compare the efficiency of the new method and other numerical integrators, in terms of accuracy versus computational cost, revealing the superiority of the new method. The proposed methodology is general and can be applied to a variety of problems, without being limited to linear problems or problems with oscillatory/periodic solutions. Full article

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems. Full article

Diabetes is sweeping the world as a silent epidemic, posing a growing threat to public health. Modeling diabetes is an effective method to monitor the increasing prevalence of diabetes and develop cost-effective strategies that control the incidence of diabetes and its complications. This paper focuses on a mathematical model known as the diabetes complication (DC) model. The DC model is analyzed using different numerical methods to monitor the diabetic population over time. This is by analyzing the model using five different numerical methods. Furthermore, the effect of the time step size and the various parameters affecting the diabetic situation is examined. The DC model is dependent on some parameters whose values play a vital role in the convergence of the model. Thus, parametric analysis was implemented and later discussed in this paper. Essentially, the Runge–Kutta (RK) method provides the highest accuracy. Moreover, Adam–Moulton’s method also provides good results. Ultimately, a comprehensive understanding of the development of diabetes complications after diagnosis is provided in this paper. The results can be used to understand how to improve the overall public health of a country, as governments ought to develop effective strategic initiatives for the screening and treatment of diabetes. Full article

In this article, we study the novel features of morphological effects for hybrid nanofluid flow subject to expanding/contracting geometry. The nanoparticles are incorporated due to their extraordinary thermal conductivity and innovative work for hybrid nanofluids, which are assembled of aluminum oxides, Al₂O₃ metallic oxides, and metallic copper Cu. Cu nanoparticles demonstrate very strong catalytic activity, while Al₂O₃ nanoparticles perform well as an electrical insulator. The governing partial differential equations of the elaborated model are transformed into a system of nonlinear ordinary differential equations with the use of similarity variables, and these equations are numerically solved through a shooting technique based on the Runge–Kutta method. We develop a hybrid correlation for thermophysical properties based on a single-phase approach. A favorable comparison between shape and size factors for metallic and metallic-oxide nanoparticles is discussed via tables and figures. Moreover, the effect of embedding flow factors on concentration, velocity, and temperature is shaped in line with parametric studies, such as the permeable Reynolds number, nanoparticle volume fractions, and expansion/contraction parameters. The fluid velocity, temperature, and concentration are demonstrated in the presence of hybrid nanoparticles and are discussed in detail, while physical parameters such as the shear stress, flow of heat, and mass transfer at the lower and upper disks are demonstrated in a table. The hybrid nanoparticles show significant results as compared to the nanofluids. If we increase the nanoparticle volume fraction, this increases the thermal performance for an injection/suction case as well. The above collaborative research provides a strong foundation in the field of biomedical equipment and for the development of nanotechnology-oriented computers. Full article

Evolutionary approaches have a critical role in different disciplines such as real-world problems, computer programming, machine learning, biological sciences, and many more. The design of the stochastic model is based on transition probabilities and non-parametric techniques. Positivity, boundedness, and equilibria are investigated in deterministic and stochastic senses. An essential tool, Euler–Maruyama, is studied for the solution of said model. Standard and nonstandard evolutionary approaches are presented for the stochastic model in terms of efficiency and low-cost approximations. The standard evolutionary procedures like stochastic Euler–Maruyama and stochastic Runge–Kutta fail to restore the essential features of biological problems. On the other hand, the proposed method is efficient, of meager cost, and adopts all the desired feasible properties. At the end of this paper the comparison section is presented to support efficient analysis. Full article

Squeezing flow is a flow where the material is squeezed out or disfigured within two parallel plates. Such flow is beneficial in various fields, for instance, in welding engineering and rheometry. The current study investigates the squeezing flow of a hybrid nanofluid (propylene glycol–water mixture combined with paraffin wax–sand) between two parallel plates with activation energy and entropy generation. The governing equations are converted into ordinary differential equations using appropriate similarity transformations. The shooting strategy (combined with Runge–Kutta fourth order method) is applied to solve these transformed equations. The results of the conducted parametric study are explained and revealed in graphs. This study uses a statistical tool (correlation coefficient) to illustrate the impact of the relevant parameters on the engineering parameters of interest, such as the surface friction factor at both plates. This study concludes that the squeezing number intensifies the velocity profiles, and the rotating parameter decreases the fluid velocity. In addition, the magnetic field, rotation parameter, and nanoparticle volumetric parameter have a strong negative relationship with the friction factor at the lower plate. Furthermore, heat source has a strong negative relationship with heat transfer rate near the lower plate, and a strong positive correlation with the same phenomena near the upper plate. In conclusion, the current study reveals that the entropy generation is increased with the Brinkman number and reduced with the squeezing parameter. Moreover, the results of the current study verify and show a decent agreement with the data from earlier published research outcomes. Full article

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, $R_0$, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when $R_0<1$. A similar result is obtained for the endemic equilibrium when $R_0>1$. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note. Full article

A numerical analysis of unsteady fluid and heat transport of compressible Helium–Xenon binary gas through a rectangular porous channel subjected to a transverse magnetic field is herein presented. The binary gas mixture consists of Helium (He) and Xenon (Xe). In addition, the compressible gas properties are temperature-dependent. The set of governing equations are nondimensionalized via appropriate dimensionless parameters. The dimensionless equations involve a number of dimensionless groups employed for detailed parametric study. Consequently, the set of equations is discretized using a compact finite difference scheme and solved by using the 3rd-order Runge–Kutta method. The model’s computed results are compared with data from past literature, and very favorable agreement is achieved. The results show that the magnetic field, compressibility and variable fluid properties profoundly affect heat and fluid transport. Variations of density with temperature as well as pressure result in an asymmetric mass flow profile. Furthermore, the friction coefficient is greater for the upper wall than for the lower wall due to larger velocity gradients along the top wall. Full article