Similarity Solutions for Flow and Heat Transfer of Non-Newtonian Fluid over a Stretching Surface by Atta Sojoudi, Ali Mazloomi, Suvash C. Saha, Y. T. Gu

“…The nonlinear coupled partial differential equations (PDE) governing the flow and the boundary conditions are converted to a system of ordinary differential equations (ODE) using two-parameter groups. …”
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Double-Diffusive MHD Viscous Fluid Flow in a Porous Medium in the Presence of Cattaneo-Christov Theories by Bidemi Olumide Falodun, Adeola John Omowaye, Funmilayo Helen Oyelami, Homan Emadifar, Ahmed A. Hamoud, S. M. Atif

“…A set of partial differential equations governs the current design (PDEs). …”
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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 Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation by Wenyuan Liao

“…The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. …”
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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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On Modeling and Constrained Model Predictive Control of Open Irrigation Canals by Lihui Cen, Ziqiang Wu, Xiaofang Chen, Yanggui Zou, Shaohui Zhang

“…As a set of hyperbolic partial differential equations, they are not solved explicitly and difficult to design optimal control algorithms. …”
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The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses by Yohan Chandrasukmana, Helena Margaretha, Kie Van Ivanky Saputra

“…This paper presents the Hadamard-Physics-Informed Neural Network (H-PINN) for solving inverse problems in partial differential equations (PDEs), specifically the heat equation and the Korteweg–de Vries (KdV) equation. …”
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Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth by Ping Wang, Zunshui Cheng

“…The present formulas, different from the perturbation solutions, are highly accurate and uniformly valid without assuming that these nonlinear partial differential equations (PDEs) have small parameters necessarily. …”
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Stochastic control with state constraints via the Fokker–Planck equation. Application to renewable energy plants with batteries by Bermúdez, Alfredo, Padín, Iago

“…For this purpose, advantage is taken from the fact that optimal control problems for stochastic ordinary differential equations (SDE) can be equivalently formulated as optimal control problems for deterministic partial differential equations (PDE), namely, the corresponding Fokker–Planck equation.…”
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Hydrodynamic Boundary Layer Flow of Chemically Reactive Fluid over Exponentially Stretching Vertical Surface with Transverse Magnetic Field in Unsteady Porous Medium by M. Sulemana, I. Y. Seini, O. D. Makinde

“…The flow problem is modelled as time depended dimensional partial differential equations which are transformed to dimensionless equations and solved by means of approximate analytic method. …”
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Nonlinear iterative approximation of steady incompressible chemically reacting flows by Gazca-Orozco, Pablo Alexei, Heid, Pascal, Süli, Endre

“…We consider a system of nonlinear partial differential equations modelling steady flow of an incompressible chemically reacting non-Newtonian fluid, whose viscosity depends on both the shear-rate and the concentration; in particular, the viscosity is of power-law type, with a power-law index that depends on the concentration. …”
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An efficient technique to study of time fractional Whitham–Broer–Kaup equations by Nishant Bhatnagar, Kanak Modi, Lokesh Kumar Yadav, Ravi Shanker Dubey

“…The method’s novelty and straightforward implementation establish it as a reliable and efficient analytical technique for solving both linear and nonlinear fractional partial differential equations.…”
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Analytical Solutions of Fractional Walter’s B Fluid with Applications by Qasem Al-Mdallal, Kashif Ali Abro, Ilyas Khan

“…By employing the dimensional analysis, the dimensional governing partial differential equations have been transformed into dimensionless form. …”
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Bat population dynamics: multilevel model based on individuals' energetics by Paula Federico, Dobromir T. Dimitrov, Gary F. McCracken

“…A structured population model based on extended McKendrick-von Foerster partial differential equations integrates those individual dynamics and provides insight into possible regulatory mechanisms of population size as well as conditions of population survival and extinction. …”
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On the Fractional View Analysis of Keller–Segel Equations with Sensitivity Functions by Haobin Liu, Hassan Khan, Rasool Shah, A. A. Alderremy, Shaban Aly, Dumitru Baleanu

“…The suggested procedure is very attractive and straight forward and therefore can be modified to solve high nonlinear fractional partial differential equations and their systems.…”
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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 Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations by Philku Lee, George V. Popescu, Seongjai Kim

“…After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. …”
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