Role of ternary hybrid nanoparticle and EMHD on boundary layer flow with stretching-shrinking plate: Optimization analyses using RMS by Ibrahim Mahariq, Rania Saadeh, Farhan Lafta Rashid, Ahmad Qazza, Hamiden Abd El-Wahed Khalifa, Mohamed Kezzar, Haifa Alqahtani, Mohamed Rafik Sari

“…The partial differential equations (PDEs) are transformed into model is simplified nonlinear ordinary differential equations using similarity transformation. …”
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Heat transfer enhancement using Cu+Al2O3 ferro magnetic nanoparticle confession in an elliptic duct device with energy conservation by Noreen Sher Akbar, M. Fiaz Hussain, Javaria Akram, Fahid Riaz, Muhammad Farooq

“…The resulting nonlinear and coupled system of partial differential equations are tackled through Galerkin finite element method with the help of MATLAB. …”
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Circular thin plates buckling analysis with HRPIM method by Baid Sokayna, Hilali Youssef, Mesmoudi Saïd, Bourihane Oussama

“…The in-plane displacement is approximated with RPIM method, while the HRPIM approach is used to compute the transverse component and its derivatives. The governing partial differential equations are discretized using the Galerkin method and solved by combining a Taylor series expansion with a continuation procedure. …”
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Data-driven model discovery and model selection for noisy biological systems. by Xiaojun Wu, MeiLu McDermott, Adam L MacLean

“…Machine learning methods offer alternative means of model construction: differential equation models can be learnt from data via model discovery using sparse identification of nonlinear dynamics (SINDy). …”
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MHD radiative mixed convective flow of a sodium alginate-based hybrid nanofluid over a convectively heated extending sheet with Joule heating by Yasmin Humaira, Bossly Rawan, Alduais Fuad S., Al-Bossly Afrah, Saeed Anwar

“…The physical phenomena are modeled in a nonlinear partial differential equation, which is then converted into ordinary differential equations with the help of suitable similarity variables. …”
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Novel FTLRNN with Gamma Memory for Short-Term and Long-Term Predictions of Chaotic Time Series by Sanjay L. Badjate, Sanjay V. Dudul

“…For coupled partial differential equations generated chaotic time series such as Mackey Glass and Duffing, FTLRNN-based predictor performs consistently well for different depths of predictions ranging from short term to long term, with only slight deterioration after k is increased beyond 50. …”
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Impact of oscillation on hybrid nanofluid convective heat Transfer: Implications for climate change by M.M. Nour, Hossam A. Nabwey, Farhat Imtiaz, Muhammad Ashraf, A.M. Rashad, Mohamed aldawood

“…To explore fluid behaviour, a mathematical model involving nonlinear coupled partial differential equations is developed based on the flow configuration. …”
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Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation. by Moritz Augustin, Josef Ladenbauer, Fabian Baumann, Klaus Obermayer

“…Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. …”
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Heat and Mass Transfer of Boundary Layer Jeffrey Ternary Hybrid Nanofluid Flow Over Porous Wedge With Surface-Catalysed Chemical Reactions: An ANFIS Model by M. Shanmugapriya, A. S. Ashwinth Jeffrey, R. Sundareswaran

“…Initially, the mathematically modelled partial differential equations (PDEs) are solved numerically by the help of the shooting method. …”
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RETRACTED ARTICLE: Experimental and TDDFT materials simulation of thermal characteristics and entropy optimized of Williamson Cu-methanol and Al2O3-methanol nanofluid flowing throu... by Wasim Jamshed, Mohamed R. Eid, Ahmed F. Al-Hossainy, Zehba Raizah, El Sayed M. Tag El Din, Tanveer Sajid

“…Adequate similarity transformations were applied to convert the partial differential equation (PDEs) into nonlinear ordinary differential equations (ODEs) with the corresponding boundary constraints. …”
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Tsallis Entropy-Based Nonparametric Focusing Algorithm Under Dual-Stage Processing Framework for THz SAR Imaging by Siyu Chen, Yong Wang, Hongzhi Li

“…In Stage Two, we investigate the dual-strategy focusing algorithm based on specific simple harmonic model and image quality measurement criterion to address different defocusing factors. Second, as a specific implementation of the above framework, we construct the THz SAR signal model by considering the multicomponent time-varying amplitude vibration and other nonideal factors, and further propose Tsallis entropy-based THz SAR image focusing method by equivalent nonlinear equations system conversion. …”
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Effects of Chemical Reaction and Joule Heating on MHD Generalized Couette Flow between Two Parallel Vertical Porous Plates with Induced Magnetic Field and Newtonian Heating/Cooling by Jumanne Mng’ang’a

“…The mathematical model used for the MHD generalized Couette flow takes into account the effect of viscous dissipation. The system of nonlinear partial differential equations governing the flow was solved numerically using the finite difference method. …”
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Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy by Alshehry Azzh Saad, Noor Saima, Abas Syed Arshad, Fiza Mehreen, Ullah Hakeem

“…The hybrid nanofluid model over two different geometries is governed by partial differential equations and converted to ordinary differential equations using similarity variables. …”
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Influence of induced magnetic field, electric load and viscous dissipation on electro-hydromagnetic Jeffrey fluid flowing in an insulated vertical channel: An application of electr... by Shabiha Naz, Tamizharasi R

“…The integration of the electric load parameter is particularly significant, as it generates electromagnetic forces that govern the flow patterns. The governing nonlinear partial differential equations are solved using a finite difference scheme, with numerical simulations performed via MATLAB's ODE15s solver. …”
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Development and Validation of Subchannel Analysis Program for LMR Based on FEM by ZHU Xinyang, CHEN Ronghua, WANG Jinshun, TIAN Jiahao, QIU Suizheng, SU Guanghui, TIAN Wenxi

“…To solve the coupled nonlinear partial differential equations, the Jacobians-Free Newton-Krylov (JFNK) algorithm was used. …”
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Inertial aftermaths on peristaltic drive of MHD micropolar fluid in a two-dimensional asymmetric inclined porous soaked channel independent of wavelength: Galerkin finite element s... by Bilal Ahmed, Fizza Anwar

“…This approach introduces nonlinearity into the modeled equations and allows for significant Reynolds numbers, thereby enhancing our understanding of the peristaltic phenomenon. …”
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An Application of Variational Minimization: Quasi-Harmonic Coons Patches by Daud Ahmad, Kiran Naz, Sadia Bashir, Abdul Bariq

“…For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation (pde), whose solution in general is the desired surface as the unknown function of surface parameters. …”
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Artificial neural networks analysis of entropy generation in magnetic-micropolar nanofluid flow equipped with porous media and Darcy-Forchheimer law by Zahoor Iqbal, Md Fayz- Al- Asad, Huiying Xu, Xinzhong Zhu, Muhammad Sajjad Hossain, Ridha Selmi, M.M. Alqarni, Sharifah E. Alhazmi, Fahima Hajjej, M.M.H. Imran

“…By using the appropriate magnitude analysis, the governing partial differential equations are transformed into a system of ordinary differential equations. …”
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An evaluation of flow-routing algorithms for calculating contributing area on regular grids by A. B. Prescott, J. D. Pelletier, S. Chataut, S. Ananthanarayan

“…We assess the performance of IDS by comparing its results to those of FLO-2D for a variety of real and idealized landforms and to an analytic solution of the shallow-water equations. We also demonstrate how IDS can be modified to solve other fluid-dynamical nonlinear partial differential equations arising in Earth surface processes, such as the Boussinesq equation for the height of the water table in an unconfined aquifer.…”
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Heat Transfer in a Porous Radial Fin: Analysis of Numerically Obtained Solutions by R. Jooma, C. Harley

“…A time dependent nonlinear partial differential equation modelling heat transfer in a porous radial fin is considered. …”
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