Optimal Homotopy Asymptotic Analysis of the Dynamics of Eyring-Powell Fluid due to Convection Subject to Thermal Stratification and Heat Generation Effect by Solomon Bati Kejela, Emiru Rufo Adula

“…The governing equations of the flow are transformed from partial differential equations into a couple of nonlinear ordinary differential equations via similarity variables. …”
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A Higher-Order Thermomechanical Vibration Analysis of Temperature-Dependent FGM Beams with Porosities by Farzad Ebrahimi, Ali Jafari

“…The governing differential equations of motion are derived based on higher order shear deformation beam theory. …”
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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

“…The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. …”
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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

“…The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. …”
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Applications of the Kosambi–Cartan–Chern Theory to Hamiltonian Systems on a Cotangent Bundle: Linking Geometric Quantities to the Self-Similar Motions of Three Point Vortices by Yuma Hirakui, Takahiro Yajima

“…This study presents a differential geometric framework for Hamiltonian systems expressed in terms of first-order differential equations. For systems governed by second-order ordinary differential equations on tangent bundles, such as Euler–Lagrange systems, the stability of trajectories under perturbations is analyzed based on the eigenvalue of the deviation curvature tensor. …”
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Lamb Waves in a Functionally Graded Composite Plate with Nonintegral Power Function Volume Fractions by Xiaoshan Cao, Zhen Qu, Junping Shi, Yan Ru

“…Based on the theory of elastodynamics, differential equations with variable coefficients are established. …”
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Application of Laplace Transform For Solving Population Growth and Decay Models. by Natukunda, Pamellah

“…Laplace can be used to convert complex differential equations to a simpler form having polynomials, convert derivatives into multiple domain variables, and then convert the polynomials back to the differential equation using the Inverse Laplace transform, telecommunication field to send signals to both sides of the medium. …”
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A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building by Yuxing Wang, Chunyu Wei

“…The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. …”
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Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to 1:1:1 Simultaneous Resonance by Ali Kandil, Y. S. Hamed, Jan Awrejcewicz, Nasser A. Saeed

“…These deflections are governed by a system of two ordinary differential equations in order to describe their Cartesian directions. …”
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Supercritical Nonlinear Vibration of a Fluid-Conveying Pipe Subjected to a Strong External Excitation by Yan-Lei Zhang, Hui-Rong Feng, Li-Qun Chen

“…The motion measured from each of curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation with variable coefficients. The Galerkin method is employed to discretize the governing equation into a gyroscopic system consisting of a set of coupled nonlinear ordinary differential equations. …”
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A Numerical Assessment of Time-Dependent Magneto-Convective Thermal-Material Transfer over a Vertical Permeable Plate by Mohammed Jahir Uddin, R. Nasrin

“…The nondimensional partial differential equations of continuity, momentum, energy, and concentration are discussed using appropriate transformations. …”
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Modeling and Dynamical Behavior of Rotating Composite Shafts with SMA Wires by Yongsheng Ren, Qiyi Dai, Ruijun An, Youfeng Zhu

“…The equations of motion are derived based on the variational-asymptotical method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin method. …”
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Flow and Heat Transfer of Cu-Water Nanofluid between a Stretching Sheet and a Porous Surface in a Rotating System by M. Sheikholeslami, H. R. Ashorynejad, G. Domairry, I. Hashim

“…The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions using similarity transformation, which is then solved analytically using the homotopy analysis method (HAM). …”
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Nonlinear Dynamic Analysis of Macrofiber Composites Laminated Shells by Xiangying Guo, Dameng Liu, Wei Zhang, Lin Sun, Shuping Chen

“…The vibration mode-shape functions are obtained according to the boundary conditions, and then the Galerkin method is employed to transform partial differential equations into two nonlinear ordinary differential equations. …”
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Computational Study on Three-Dimensional Convective Casson Nanofluid Flow past a Stretching Sheet with Arrhenius Activation Energy and Exponential Heat Source Effects by P. Ragupathi, S. Saranya, H.V.R. Mittal, Qasem M. Al-Mdallal

“…The developed model of nonlinear partial differential equations (PDEs) has been transformed into ordinary differential equations (ODEs) using similarity transformations. …”
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A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics by Rafael J. Villanueva, Abraham J. Arenas, Gilberto González-Parra

“…This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. …”
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Dynamic Characteristics of Blade with Viscoelastic Damping Block Based on Complex Eigenvalue Method by Jiao Wang, Yue-hao Zhang, Tao Yu, Qing-kai Han

“…Then the differential equations of motion with 3-DOF are numerically solved by using complex eigenvalue method. …”
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Chemical Reaction Effects on the Flow of Titanium Dioxide–Water in a Mixed Convective Nanofluid along an Inclined Plate in a Porous Medium by Md. Nasir Uddin, Kakali Chowdhury, Jannatul Ferdows Mim

“…With the aid of a system of governing partial differential equations, a set of nonlinear ordinary differential equations for taking into consideration nanofluid flow have been derived using appropriate matching transformations. …”
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A Modified NM-PSO Method for Parameter Estimation Problems of Models by An Liu, Erwie Zahara, Ming-Ta Yang

“…Ordinary differential equations usefully describe the behavior of a wide range of dynamic physical systems. …”
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Thermal and Flow Dynamics of Magnetohydrodynamic Burgers' Fluid Induced by a Stretching Cylinder with Internal Heat Generation and Absorption by Fateh Mebarek-Oudina, G. Dharmaiah, J.L. Rama Prasad, H. Vaidya, Manda Aruna Kumari

“…By applying boundary layer theory, we transform the governing partial differential equations into a standard system of ordinary differential equations through similarity transformations. …”
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