Soliton solutions, bifurcations, and sensitivity analysis to the higher-order nonlinear fractional Schrödinger equation in optical fibers by Md. Al Amin, M. Ali Akbar, M. Ashrafuzzaman Khan, Md. Sagib

“…A detailed sensitivity analysis was also performed on the dynamical system utilizing the Runge-Kutta method. These exact solitons play a vital role in understanding wave propagation and are essential for validating both numerical simulations and experimental findings in areas such as quantum mechanics, nonlinear optics and engineering.…”
Get full text

Revisiting the classical target cell limited dynamical within-host HIV model - Basic mathematical properties and stability analysis by Benjamin Wacker

“…As our second main contribution, we underline our theoretical findings through some numerical experiments with standard Runge–Kutta time stepping schemes. We conclude this work with a summary of our main results and a suggestion of an extension for more complex dynamical systems with regard to HIV-infection.…”
Get full text

Analysis of Metallic Nanoparticles (Cu, Al2O3, and SWCNTs) on Magnetohydrodynamics Water-Based Nanofluid through a Porous Medium by P. K. Pattnaik, S. K. Parida, S. R. Mishra, M. Ali Abbas, M. M. Bhatti

“…To solve the formulated nonlinear differential equations, the Runge–Kutta fourth-order numerical scheme is used in conjunction with the shooting technique. …”
Get full text

Bernstein Collocation Method for Solving MHD Jeffery–Hamel Blood Flow Problem with Error Estimations by Ahmad Sami Bataineh, Osman Rasit Isik, Ishak Hashim

“…As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. …”
Get full text

Effect of Gear Backlash Function on the Dynamics Characteristic of Helical Gear by Wang Xin, Kang Boqiang

“…The results of the nonlinear dynamic response under two different gear backlash functions are obtained by employing the established dynamics equation of helical cylindrical gear system with variable step- size Runge- kutta method numerical solution. The influences of frequency on the bifurcation and chaotic characteristics under two different gear backlash functions are analyzed by using bifurcation diagrams,top Lyapunov exponent（ TLE） diagrams,phase portraits,Poincaré maps and FFT spectrums. …”
Get full text

Mesh Stiffness Modelling and Dynamic Response Analysis of Spur Gear Pair with Assembly Error and Tooth Surface Friction by Nie Li, Xu Zhiliang, Xu Huachao

“…Numerical solutions characterizing the meshing behavior and dynamic response of the system were obtained using the Runge-Kutta method. The analysis reveals that, the mesh stiffness amplitude of the spur gear pair decreases with the increase of the tooth surface friction coefficient. …”
Get full text

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. …”
Get full text

Modeling and Control of the Public Opinion: An Agree-Disagree Opinion Model by Sara Bidah, Omar Zakary, Mostafa Rachik

“…We prove the existence of solutions to the control problem, we employ Pontryagin’s maximum principle to find the necessary conditions for the existence of the optimal controls, and Runge–Kutta forward-backward sweep numerical approximation method is used to solve the optimal control system, and we perform numerical simulations using various initial conditions and parameters to investigate several scenarios. …”
Get full text

Symmetry analysis, dynamical behavior, and conservation laws of the dual-mode nonlinear fluid model by Adil Jhangeer, Beenish, Lubomír Říha

“…The model is transformed into a planar dynamical system via Galilean transformation, with phase portraits generated using bifurcation parameters. Runge–Kutta method is utilized to compute both super nonlinear and nonlinear wave solutions, with all solutions illustrated in the phase plane. …”
Get full text

Effect of Activation Energy on Magnetized Couple Stress Fluid over an Inclined Stretching Permeable Cylinder by Suman Sharma, Shalini Jain

“…Subsequently, these transformed equations are solved using fourth order Runge-Kutta mechanism along with the shooting technique. …”
Get full text

Investigation of Nonlinear Vibrational Analysis of Circular Sector Oscillator by Using Cascade Learning by Naveed Ahmad Khan, Muhammad Sulaiman, Jamel Seidu, Fahad Sameer Alshammari

“…A data set for the supervised learning of the CNN-BLM algorithm for different angles α and radius R are generated by Runge–Kutta (RK-4) method. The BLM algorithm further interprets the dataset with log-sigmoid as an activation function for the solutions’ validation, testing, and training. …”
Get full text

Analysis of Spur Cylindrical Gear Pair Vibration Characteristic Considering Error and Randomness of Tooth Surface Friction by Zaida Gao, Shengbo Li, Shengping Fu, Polyakov Roman

“…The fourth-order Runge-Kutta method is used for numerical solution， the vibration response of the gear transmission is obtained. …”
Get full text

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. …”
Get full text

Dynamic Load Sharing Characteristic of Planetary Transmission Considering Number of Planetary Wheel and Clearance by Dong Hao, Liu Zhiyu, Liu Bo

“…The model is solved by Runge-Kutta numerical method， and the relationship between the number of planetary wheel， the clearance and the load sharing characteristics is established. …”
Get full text

Mixed Convection Squeezing Flow of Nanofluids in a Rotating Channel with Thermal Radiation by Wankui Bu, Hui Xu, Ilyas Khan, Sheikh Irfan Ullah Khan, Anwar Zeb

“…Then, the explanation was obtained numerically by the Runge–Kutta–Fehlberg (RKF) method. The emerging parameters such as quotient of the electric magnetic field to viscous forces (M), Prandtl number (Pr), and Reynolds number (Re), along with physical parameters such as the Nusselt number and skin friction coefficient, will be integrated graphically. …”
Get full text

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.…”
Get full text

Vibration Characteristic Analysis of Spur Gear System with Coupled Damage by Li Xike Wang Jian Chen Tao Zhang Jun

“…Then，the steady dynamics responses of the transmission system are solved by Runge-Kutta method，and the vibration characteristics of spur gear system with crack-wear coupled damage mode are analyzed. …”
Get full text

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. …”
Get full text

Reliability of Dynamic Kinematic Accuracy for Gear Transmission by Peng Xiao Tiyi Liu Tao Zhang Junfu Peng

“…Then， the four order Runge-Kutta method is used to solve the dynamic response of gear transmission， BP neural network is applied to build the input-output relationship model of dynamic system and motion accuracy reliability model of gear transmission system is then created. …”
Get full text

On Behavioral Response of 3D Squeezing Flow of Nanofluids in a Rotating Channel by Mubashir Qayyum, Omar Khan, Thabet Abdeljawad, Naveed Imran, Muhammad Sohail, Wael Al-Kouz

“…For validation purpose, system of equations is also solved through the Runge–Kutta–Fehlberg (RK45) scheme and results are compared with homotopy solutions, and excellent agreement has been found between analytical and numerical results. …”
Get full text