Analysis of Model Parameters for a Polymer Filtration Simulator by N. Brackett-Rozinsky, S. Mondal, K. R. Fowler, E. W. Jenkins

“…The simulator is a three-dimensional, time-dependent discretization of a coupled system of nonlinear partial differential equations used to model fluid flow and debris transport, along with statistical relationships that define debris distributions and retention probabilities. …”
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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

“…The dynamical equation of the system is established and the Galerkin method is used to discretize the partial differential equations to a 3-DOF system so as to compute the dynamic natural frequencies and responses of the VE-blade. …”
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A Direct Eulerian GRP Scheme for the Prediction of Gas-Liquid Two-Phase Flow in HTHP Transient Wells by Jiuping Xu, Min Luo, Jiancheng Hu, Shize Wang, Bin Qi, Zhiguo Qiao

“…A coupled system model of partial differential equations is presented in this paper, which concerns the variation of the pressure and temperature, velocity, and density at different times and depths in high temperature-high pressure (HTHP) gas-liquid two-phase flow wells. …”
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A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods by Fanghua Pei, Fujun Cao, Yongbin Ge

“…Deep learning methods using neural networks for solving partial differential equations (PDEs) have emerged as a new paradigm. …”
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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 Reduced Order Model Based on ANN-POD Algorithm for Steady-State Neutronics and Thermal-Hydraulics Coupling Problem by Hanxing Liu, Han Zhang

“…Due to the high-dimensional partial differential equations (PDEs) derived from the N/TH system, it is usually time consuming to solve such a large-scale nonlinear equation by the traditional numerical solution method of PDEs. …”
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Effects of Thermodiffusion and Chemical Reaction on Magnetohydrodynamic-Radiated Unsteady Flow Past an Exponentially Accelerated Inclined Permeable Plate Embedded in a Porous Mediu... by B. Prabhakar Reddy, M. H. Simba, Alfred Hugo

“…The governing nondimensional set of coupled nonlinear partial differential equations with related initial and boundary conditions are solved numerically by using the accurate and efficient DuFort–Frankel’s explicit finite difference method. …”
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Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform by Nehad Ali Shah, Essam R. El-Zahar, Ali Akgül, Adnan Khan, Jeevan Kafle

“…In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.…”
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Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and... by Nikolaos Kazantzis, Vasiliki Kazantzi

“…The problem of interest is addressed within the context of singular partial differential equations (PDE) theory, and in particular, through a system of singular quasi-linear invariance PDEs for which a general set of conditions for solvability is provided. …”
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Conciliating accuracy and efficiency to empower engineering based on performance: a short journey by Chinesta, Francisco, Cueto, Elias

“…The art of modeling for describing the behavior of complex systems from the solution of partial differential equations that are expected to govern their responses. …”
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Vortex-Induced Vibration of a Cable-Stayed Bridge by M. T. Song, D. Q. Cao, W. D. Zhu

“…Orthogonality conditions of exact mode shapes of the linearized undamped cable-stayed bridge model are employed to convert coupled governing partial differential equations of the original cable-stayed bridge model with damping to a set of ordinary differential equations by using Galerkin method. …”
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Numerical Study on Resistance Change Characteristics of Phase Change Materials by Yifang Su, Yihang Zhang, Kaifeng Lin, Guanjun Zhao, Qinzheng Yu, Zuoqi Hu, Xiongfei Tao

“…To study the characteristics of the phase change materials, a numerical simulation model of the resistive change unit based on the finite element method and the classic nucleation/growth theory is established, while the partial differential equations of electricity and heat conduction and the discrete formula of the finite element are also derived. …”
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Insight on the complete process of photoacoustic generation and propagation of cerebrovascular in brain via multi-physics coupling method by Shangkun Hou, Anzhi Lv, Jiabin Lin, Bokang You, Shuaixin Huang, Xianlin Song

“…The diffusion equation is approximated by the partial differential equations in the form of coefficients in the mathematical module of COMSOL to describe laser propagation in brain tissue. …”
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Modeling bacterial attachment to surfaces as an early stage of biofilm development by Fadoua El Moustaid, Amina Eladdadi, Lafras Uys

“…While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation.In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. …”
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The Effects of Activation Energy and Thermophoretic Diffusion of Nanoparticles on Steady Micropolar Fluid along with Brownian Motion by Zulqurnain Sabir, Assad Ayub, Juan L. G. Guirao, Saira Bhatti, Syed Zahir Hussain Shah

“…The nonlinear partial differential equations are converted into nonlinear ordinary differential equations, and a famous shooting scheme is used to present the numerical solutions. …”
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Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott by Sami Aljhani, Mohd Salmi Md Noorani, Khaled M. Saad, A. K. Alomari

“…The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.…”
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Applying Fourier Neural Operator to insect wingbeat sound classification: Introducing CF-ResNet-1D by Béla J. Szekeres, Máté Natabara Gyöngyössy, János Botzheim

“…Despite recent advancements in Deep Learning, Fourier Neural Operators (FNO), efficient for solving Partial Differential Equations due to their global spectral representations, have yet to be thoroughly explored for real-world time series classification or regression tasks. …”
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Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions by Yufeng Zhang, Linlin Gui

“…Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. …”
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Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method by Wafaa B. Rabie, Hamdy M. Ahmed, Taher A. Nofal, E. M. Mohamed

“…Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. …”
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Circular thin plates buckling analysis with HRPIM method by Baid Sokayna, Hilali Youssef, Mesmoudi Saïd, Bourihane Oussama

“…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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