The Rayleigh quotient and dynamic programming by Richard Bellman

“…The purpose of this paper is to derive a nonlinear partial differential equation for which λ given by (1.3), is one value of the solution. …”
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High-Frequency Electromagnetic Field Coupling to Small Antennae in a Rectangular Resonator by Sergey Tkachenko, Jürgen Nitsch, Moawia Al-Hamid

“…The integral-differential equation for the current of an electrically small antenna, inside a resonator, which is induced by given sources, is approximately solved by the so-called “Method of Small Antenna,” both for dipole and loop antennas. …”
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Perihelion Precessions of Inner Planets in Einstein’s Theory and Predicted Values for the Cosmological Constant by Alvaro H. Salas S, Jairo E Castillo H, Lorenzo J. Martınez H

“…In this paper, we obtain the approximate value of 42.9815 arcsec/century for Mercury’s perihelion precession by solving both numerically and analytically the nonlinear ordinary differential equation derived from the geodesic equation in Einstein’s Theory of Relativity. …”
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Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems by Zhenhua Hu, Shuqing Zhou

“…We first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equation div(A(x,∇u))=div f(x,u), where A(x,∇u), f(x,u) are two n×N matrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.…”
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Application of PDE Constrained Optimization in Internal Combustion Engine Pollution Control by Hongliang Guan

“…Aiming at the problem of secondary pollution of waters due to the difficulty of controlling the dosage of purifiers in the treatment of internal combustion engine pollution, a partial differential equation (referred to as PDE) constrained optimization algorithm based on l1-norm is proposed. …”
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Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction by Weili Kong, Yuanfu Shao

“…Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. …”
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On a Fractional SPDE Driven by Fractional Noise and a Pure Jump Lévy Noise in ℝd by Xichao Sun, Zhi Wang, Jing Cui

“…We study a stochastic partial differential equation in the whole space x∈ℝd, with arbitrary dimension d≥1, driven by fractional noise and a pure jump Lévy space-time white noise. …”
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Reducing the sign problem with line integrals by Rasmus N. Larsen

“…We lay out how to write down an ordinary differential equation for the line integrals. As an example of its usage, we apply the results to a 1d quantum mechanical anharmonic oscillator with a x 4 potential in real time, finite temperature.…”
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The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems by Yanling Shi, Jia Li

“…We study the following two-order differential equation, (Φp(x'))'+f(x,t)Φp(x')+g(x,t)=0, where Φp(s)=|s|(p-2)s, p>0. f(x,t) and g(x,t) are real analytic functions in x and t, 2aπp- periodic in x, and quasi-periodic in t with frequencies (ω1,…,ωm). …”
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The Effect of Pulse Vaccination and Treatment on SIR Epidemic Model with Media Impact by Zhi-Long He, Lin-Fei Nie

“…By using the geometry theory of differential equation and method of successor function, the existence of positive order-1 periodic solution is studied. …”
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The traveling wave solution and dynamics analysis of the fractional order generalized Pochhammer–Chree equation by Chunyan Liu

“…First, the fractional order generalized Pochhammer–Chree equation is transformed into an ordinary differential equation. Second, the dynamic behavior is analyzed using the planar dynamical system, and some three-dimensional and two-dimensional phase portraits are drawn using Maple software to reflect its chaotic behaviors. …”
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Travelling Wave Solutions for Nonlinear Schrödinger Equation with a Higher-Order Dispersive Term by Rui Cao

“…A nonlinear Schrödinger equation with a higher-order dispersive term describing the propagation of ultrashort femtosecond pulses in optical fibres is considered and is transformed into a second-order nonlinear ordinary differential equation. We investigate the exact travelling wave solutions of the nonlinear Schrödinger equation using three methods, namely, the auxiliary equation method, the first integral method, and the direct integral method. …”
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Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation by Zhonglong Zhao, Yufeng Zhang, Tiecheng Xia

“…Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of (2+1)-dimensional nonlinear partial differential equation. The resulting theory is applied to the (2+1)-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. …”
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A Numerical Method for Computing the Principal Square Root of a Matrix by F. Soleymani, S. Shateyi, F. Khaksar Haghani

“…Finally, application of the presented approach is illustrated in solving a matrix differential equation.…”
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Comparison of Analytical Solution of DGLAP Equations for... by Neelakshi N. K. Borah, D. K. Choudhury, P. K. Sahariah

“…The DGLAP equation for the nonsinglet structure function F 2 N S ( x , t ) at LO is solved analytically at low x by converting it into a partial differential equation in two variables: Bjorken x and t    ( t = l n ( Q 2 / Λ 2 ) and then solved by two methods: Lagrange’s auxiliary method and the method of characteristics. …”
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Research on Spatial Location of a Spherical 3-RRR Parallel Mechanism by Wang Li

“…Aiming at the large-scale asymptotically stable dynamic quality of the system, the conditions that the angular velocity of the moving platform should satisfy are obtained, and the quaternion differential equation of motion of the moving platform in the positioning process is established. …”
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DYNAMIC CHARACTERISTICS ANALYSIS OF FLEXIBLE FOUNDATION VIBRATION ISOLATION SYSTEM by ZHANG DaoKun, HUO Rui, LI ShuYing, WANG ZhiDong

“…By transforming the system dynamic equation into an expression of initial value problem of differential equation,the calculation method of transmitted power flow is deduced,which is put forward to evaluation and analysis of isolation effectiveness. …”
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Computational Modelling of Thermal Stability in a Reactive Slab with Reactant Consumption by O. D. Makinde

“…The nonlinear partial differential equation governing the transient reaction-diffusion problem is solved numerically using a semidiscretization finite difference technique. …”
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Rolling Bearing Dynamics Response and Parameter Sensitivity Analysis based on the Response Surface Method by Shi Jialian, Ma Xiaogang, Li Chang, Xie Kuiqiu

“…The dynamics model of rolling bearing of automotive engine bearing-rotor system is established and the nonlinear dynamics differential equation is derived. The numerical solution is obtained by MATLAB after nondimensionalization,the change rule of dynamic response of rolling bearing is obtained. …”
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Mathematical Modeling of the Transmission Dynamics of Gumboro Disease by J. S. Musaili, I. Chepkwony, W. N. Mutuku

“…The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog<1 and an asymptotic stable EE whenever Rog>1. …”
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