Development of an Impact Energy Absorption Structure by an Arc Shape Stroke Origami Type Hydraulic Damper by Jingchao Guan, Yuan Yao, Wei Zhao, Ichiro Hagiwara, Xilu Zhao

“…A nonlinear dynamics governing equation was established using the derived formula for the damping force of the origami hydraulic damper, and a numerical analysis using the Runge–Kutta method was established. An impact test device with an arc-shaped stroke was developed, and the error between the numerical analysis value of the impact displacement and the measured experimental value was confirmed to be sufficiently small. …”
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Analysis of Heat Transfer in Berman Flow of Nanofluids with Navier Slip, Viscous Dissipation, and Convective Cooling by O. D. Makinde, S. Khamis, M. S. Tshehla, O. Franks

“…These equations are solved analytically by regular perturbation methods with series improvement technique and numerically using an efficient Runge-Kutta Fehlberg integration technique coupled with shooting scheme. …”
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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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MHD Heat and Mass Transfer of Chemical Reaction Fluid Flow over a Moving Vertical Plate in Presence of Heat Source with Convective Surface Boundary Condition by B. R. Rout, S. K. Parida, S. Panda

“…The basic equations governing the flow, heat transfer, and concentration are reduced to a set of ordinary differential equations by using appropriate transformation for variables and solved numerically by Runge-Kutta fourth-order integration scheme in association with shooting method. …”
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Numerical Investigation of Contact Burning in an Air-Breathing Continuous Rotating Detonation Engine by Dapeng Xiong, Mingbo Sun, Haoyang Peng, Yanan Wang, Yixin Yang, Hongbo Wang, Jiangfei Yu, Zhenguo Wang

“…The second-order upwind advection upstream splitting method and second-order Runge-Kutta method are used to discretize space and time terms, and detailed 9-species 19-step hydrogen-oxygen reactions are applied in this study. …”
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The Impact of Media Coverage and Curfew on the Outbreak of Coronavirus Disease 2019 Model: Stability and Bifurcation by Afrah K. S. Al-Tameemi, Raid K. Naji

“…Numerical simulation is carried out in applying the model to the sample of the Iraqi population through solving the model using the Runge–Kutta fourth-order method with the help of Matlab. …”
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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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Bifurcation Study of Thin Plate with an All-Over Breathing Crack by Lihua Chen, Jian Xue, Zhijie Zhang, Wei Zhang

“…Lastly, the waveforms, bifurcation diagrams, and phase portraits of the model are gained by the Runge-Kutta method. It is found that complex nonlinear dynamic behaviors, such as quasi-periodic motion, bifurcation, and chaotic motion, appear in the breathing crack plate.…”
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Optimal control of infectious diseases using vaccination and treatment by Maryam Nikbakht

“…The experimental results are compared with those obtained from the fourth-order runge-kutta scheme. It should be noted that the proposed model is a public model suggested for contagious diseases and can be used as a method to prevent the spread of diseases such as influenza, coronavirus, and other infectious diseases.Findings: Numerical simulation, considered in a 90-day period, shows that using appropriate optimal control of vaccination and treatment will limit disease transmission and reduce the number of infected and infected people. …”
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Dynamic Model of Spur Gear Pair with Modulation Internal Excitation by Zhong Wang, Lei Zhang, Yuan-Qing Luo, Chang-Zheng Chen

“…The internal excitation was shown to exhibit obvious modulation sideband because of the modulation time-varying mesh stiffness and modulation transmission error. The Runge-Kutta method was used to solve the equations for analyzing the dynamic characteristics with the effect of modulation internal excitation. …”
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Chaotic Dynamics Analysis and FPGA Implementation Based on Gauss Legendre Integral by Li Wen, Li Cui, Hairong Lin, Fei Yu

“…We apply the Gauss–Legendre integral and the Runge–Kutta algorithm to the solution of dissipative chaotic systems for the first time and analyze and compare the differences between the two algorithms. …”
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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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Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane by Mingyue Shao, Jimei Wu, Yan Wang, Qiumin Wu

“…However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. …”
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Performance Analysis and Comparison for High Maneuver Target Track Based on Different Jerk Models by Qinghai Meng, Bowen Hou, Dong Li, Zhangming He, Jiongqi Wang

“…The simplified dynamic model is used to gain the standard trajectory with Runge-Kutta numerical integration method. The mathematical simulations show that Jerk model with self-adaptive noise variance has the best robustness while other models may diverge when the initial error is much larger. …”
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Nonlinear Vibration Analysis of Rotor considering Cogging and Harmonic Effects by Shanle Li, Feng Liu, Hongyan Wang, Haijun Song, Kuilong Yu

“…Motion equations of rotor system were established, and Runge–Kutta method was used to solve the equations. …”
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Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method by Indranil Ghosh, M. S. H. Chowdhury, Suazlan Mt Aznam, M. M. Rashid

“…Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.…”
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Dropping Shock Characteristics of the Suspension Cushioning System with Critical Components by Hui Li, Anjun Chen, Ningning Duan

“…The numerical results of shock response were gained using Runge-Kutta method. To evaluate the dropping shock characteristics of critical components, the dropping damage boundary curve was established, where the system parameter and the dimensionless shock velocity were selected as two coordinate parameters. …”
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A Time-Domain Boundary Element Method for Wave Diffraction in a Two-Layer Fluid by Ying Gou, Xin-jia Chen, Teng Bin

“…For the internal-wave mode, through the definition of a new function respected to velocity potentials of upper and lower fluid on the interface by using matching condition, a single set of linear equations is set up to compute the time histories of wave forces and wave profiles by using a fourth-order Runge-Kutta method. An artificial damping layer is adopted both on the free surface and interface to avoid the wave reflection. …”
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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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Introduction by Raymond Renard

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