Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation by Tahir Ayaz, Farhad Ali, Wali Khan Mashwani, Israr Ali Khan, Zabidin Salleh, Ikramullah

“…The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. …”
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Analytic Approximate Solution for Falkner-Skan Equation by Vasile Marinca, Remus-Daniel Ene, Bogdan Marinca

“…This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. …”
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The effects of heterogeneity on Darcy-Benard convection saturated with a ferrofluid by Moussa Rachida, Lagziri Hajar, El Khanoussi Fadoua, El Fakiri Hanae

“…A dimensionless formulation and linear stability analysis are employed to derive an ordinary differential equation, which is solved using the shooting method and a Runge-Kuttasolver. …”
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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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A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs by I. B. Aiguobasimwin, R. I. Okuonghae

“…In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). …”
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IMPROVED FINITE ELEMENT TRANSFER MATRIX METHOD OF PLANE BEAM ELEMENTS USING THE ABSOLUTE NODAL COORDINATE FORMULATION by QIAN ShuangLin, HE Bin, YAO LiKe, NI ChunHai, LV JianDong

“…Here ordinary differential equation numerical methods can be applied conveniently. …”
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Asymptotic and numerical solutions for diffusion models for compounded risk reserves with dividend payments by S. Shao, C. L. Chang

“…After defining the process of conditional probability in finite time, martingale theory turns the nonlinear stochastic differential equation to a special class of boundary value problems defined by a parabolic equation with a nonsmooth coefficient of the convection term. …”
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An Improved Cockroach Swarm Optimization by I. C. Obagbuwa, A. O. Adewumi

“…The original CSO was modelled with three components: chase-swarming, dispersion, and ruthless; additional hunger component which is modelled using partial differential equation (PDE) method is included in this paper. …”
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Analytical Solutions for Nonlinear Dispersive Physical Model by Wen-Xiu Ma, Mohamed R. Ali, R. Sadat

“…The time FBBM equation is reduced to a nonlinear fractional ordinary differential equation (NLFODE) using its Lie symmetries. …”
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New Exact Solutions of Burgers’ Equation Using Power Index Method by Khalil Ahmad, Khudija Bibi

“…In this method, we choose suitable indexes of independent variables and similarity transformation so that the partial differential equation may be converted into ODE. We have obtained analytic solution of the ODE by using symbolic package Maple. …”
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Enhancement of stability of metastable states in the presence of Lévy noise by Alexander A. Dubkov, Claudio Guarcello, Bernardo Spagnolo

“…Starting from the fractional Fokker-Planck equation, we derive an integro-differential equation along with the necessary conditions to calculate the mean residence time of a particle within a fixed interval. …”
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Dynamical Behavior of Stochastic Markov Switching Hepatitis B Epidemic Model with Saturated Incidence Rate by Chun Lu

“…Finally, we propose a definition of the solution to an impulsive stochastic functional differential equation with Markovian switching (ISFDM).…”
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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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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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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 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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Multi-State Dependent Impulsive Control for Holling I Predator-Prey Model by Huidong Cheng, Fang Wang, Tongqian Zhang

“…According to the different effects of biological and chemical control, we propose a model for Holling I functional response predator-prey system concerning pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove that the existence of order one periodic solution of such system and the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. …”
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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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Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions by Hui Li, Libo Wang, Minghe Pei

“…We investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form x(4)(t)=f(t,x(t),x′(t),x′′(t),x′′′(t)), t∈[0,1], x(0)=x′(1)=0, x′′(0)=∫01h(s,x(s),x′(s),x′′(s))ds, x′′′(1)=0, where f∈C([0,1]×ℝ4), h∈C([0,1]×ℝ3). …”
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