Nonlinear Vibration of a Continuum Rotor with Transverse Electromagnetic and Bearing Excitations by Haiyang Luo, Yuefang Wang

“…The stability of steady response is analyzed and the amplitude-frequency curve of primary resonance is illustrated. The Runge-Kutta method is adopted to numerically solve transient response of the rotor-bearing system. …”
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A comprehensive numerical study exploring the significance of thermally reactive bioconvection in Falkner-Skan flow of Williamson nanomaterials influenced by activation energy and... by M. Israr Ur Rehman, Haibo Chen, Aamir Hamid, Wu Qian, Refka Ghodhbani, Mohamed Hussien

“…Similarity transformations are used to convert the system of partial differential equations into a system of ordinary differential equations, which are then numerically solved using the Runge–Kutta–Fehlberg (RKF-45) method. After the system has been altered, the physical parameters that result are examined and shown graphically. …”
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Impact of Awareness to Control Malaria Disease: A Mathematical Modeling Approach by Malik Muhammad Ibrahim, Muhammad Ahmad Kamran, Malik Muhammad Naeem Mannan, Sangil Kim, Il Hyo Jung

“…The solution to the proposed model is derived through an iterative numerical technique, the Runge–Kutta method. The proposed model is simulated for different numeric values of the population of humans and anopheles in each class. …”
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Vortex-Induced Vibration of a Cable-Stayed Bridge by M. T. Song, D. Q. Cao, W. D. Zhu

“…The dynamic response of the cable-stayed bridge is calculated using Runge-Kutta-Felhberg method in MATLAB for two cases with and without geometric nonlinear terms. …”
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Vibration Characteristic Analysis of a Coupling System for Compound Biaxial and Inter-Shaft Bearings by Kun Li, Hongchang Ding, Guohua Cao, Han Hou

“…Considering the moment of inertia, gyro effect, and structural damping, a one-dimensional beam model is used to perform the MATLAB numerical calculation of the coupling system to obtain the modal characteristics of the system, and based on the ANSYS simulation analysis of the three-dimensional model, a modal test is used to obtain the system modal shape and test frequency response function to verify the rationality of the coupling system model. Based on the Runge-Kutta iterative method and Hamilton’s principle, the nonlinear amplitude-frequency response of the system is calculated while taking into account the influence of the compound biaxial eccentricity, as well as the stiffness and radial clearance of the inter-shaft bearing. …”
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Photokinetics of Photothermal Reactions by Mounir Maafi

“…The validity of the model equation was established against simulated data obtained by a fourth-order Runge–Kutta method. It was then used to describe and quantify several situations of photothermal reactions, such as the effects of initial concentration, spectator molecules, and incident radiation intensity, and the impact of the latter on the photonic yield. …”
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MAC Layer Energy Consumption and Routing Protocol Optimization Algorithm for Mobile Ad Hoc Networks by Yaohua Chen, Waixi Liu

“…In order to solve the problem that the analytic solution of differential equations cannot be obtained in the model, a numerical solution of the optimization algorithm is proposed based on the implicit Runge–Kutta method. The MAC layer energy consumption and routing protocol optimization algorithm proposed in this paper can be applied well in various fields.…”
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A Mathematical Model of Tumor-Immune and Host Cells Interactions with Chemotherapy and Optimal Control by Tarekegn Dinku, Boka Kumsa, Jyotirmoy Rana, Aiyappan Srinivasan

“…The optimal system is solved using the forward-backward sweep method with fourth-order Runge–Kutta approximation. Reduction in tumor cell growth has been observed due to the increase in recruitment of immune cells activated by tumor cell antigenicity and the rate of conversion of resting immune cells into active immune cells. …”
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Study on the Dynamics Characteristic of Gear Cutting Tooth Fault Under Variable Load Excitation by Shi Peiming, Zhao Na, Dang Hui, Tian Guangjun

“…On the basis of that,the variable step size four- order Runge- Kutta numerical method is used to solve to the dynamics model and the dynamics mechanism of gear cutting tooth is studied. …”
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Effect of thermophoresis on particle deposition on an inclined plate in variable viscosity fluid flow within a porous medium by J.S. Huang, H.H. Hsu

“…This study utilizes the similarity transformation technique, Runge-Kutta integration, and a shooting approach to address the governing equations for continuity, momentum, energy, and concentration fields. …”
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Reappraising double pendulum dynamics across multiple computational platforms by Sandy Herho, Faiz Fajary, Katarina Herho, Iwan Anwar, Rusmawan Suwarman, Dasapta Irawan

“…These were solved numerically using the efficient Runge-Kutta-Fehlberg method, implemented in Python, R, GNU Octave, and Julia, while runtimes and memory usage were extensively benchmarked across these environments. …”
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Unsteady flow and heat transfer optimization of viscous fluid with bioconvection over a rotating stretchable disk and gyrotactic motile microorganisms by Mazhar Hussain, Samya Shahid, Noreen Sher Akbar, Mohammed Kbiri Alaoui

“…The boundary value problem is numerically solved using shooting method with Runga-Kutta technique by converting equations into initial value problem. …”
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Hybrid in-situ and ex-situ hydrolysis of catalytic epoxidation neem oil via a peracid mechanism by Ismail Md. Rasib, Mohd Jumain Jalil, Nabisab Mujawar Mubarak, Intan Suhada Azmi

“…Lastly, a mathematical model was developed using MATLAB, applying the fourth-order Runge–Kutta method and simulated annealing optimization to determine the best-fitting kinetic model. …”
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Analysis of Dynamic Characteristics of Grease-Lubricated Tapered Roller Bearings by Zheng-Hai Wu, Ying-Qiang Xu, Si-Er Deng

“…The impacts of load, grease rheological properties, and temperature on the roller tilt and skew and the bearing slip were simulated by using the fourth-order Runge–Kutta method. The results show that the roller tilt angle in the unloaded zone is obviously larger than that in the loaded zone, while the roller skew angle in the unloaded zone is smaller than that in the loaded zone. …”
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Breather and solitonic behavior of parametric Sine–Gordon equation with phase-shift and driven term by Taj Munir, Muhammad Zaman, Can Kang, Hussan Zeb, Alrazi Abdeljabbar, Mohammed Daher Albalwi

“…Numerically, the explicit finite difference scheme and the fourth-order Runge–Kutta finite difference method are implemented. The model equation is formulated for a 0−π−0 junction with appropriate initial and boundary conditions. …”
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A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation by Wenyuan Liao

“…Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. …”
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Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments by Koji Hashimoto, Koshiro Matsuo, Masaki Murata, Gakuto Ogiwara, Daichi Takeda

“…Our machine reads spatially and temporarily inhomogeneous linear response data of the condensed matter system, and incorporates a novel layer that implements the Runge–Kutta method to achieve better numerical control. …”
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Investigation of Dynamic Load Sharing Behavior for Herringbone Planetary Gears considering Multicoupling Manufacturing Errors by Fei Ren, Jinchen Ji, Guofu Luo, Shaofu Zhao, Liya Zhao, Guiqin Shi, Xiaoling Wu, Ning Wang

“…By employing the variable-step Runge–Kutta approach to calculate the system dynamic response, in conjunction with the presented calculation approach of the HPGT load sharing coefficient, the relationships among manufacturing errors, component floating, and load sharing are numerically obtained. …”
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Financial development, research in finance, and economic growth by Chadi Azmeh, Marwan Al-Raeei

“…Utilizing the two-step system GMS method and the Runge-Kutta framework, the research defines and addresses a differential equation, achieving remarkable precision in economic modeling by fine-tuning step sizes and adapting the algorithm to the specific economic context. …”
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Inclined Magnetic Field on Mixed Convection Darcy–Forchheimer Maxwell Nanofluid Flow Over a Permeable Stretching Sheet With Variable Thermal Conductivity: The Numerical Approach by Gossaye Aliy Adem, Adamu Gizachew Chanie

“…These ODEs are then solved using the shooting method and a fourth-order Runge-Kutta methodology. The study explores how fluid flow velocity is affected by magnetic field inclination and suction/injection effects. …”
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