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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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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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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Adjustment of Optimal Parameters to Control the Temperature in Different Conductors by Amirhossein Oudi, Samaneh Faramarzi, Maryam Hosseini, Yegane Davoodbeygi

“…In this method, partial differential equations were converted into ordinary differential equations with the help of finite difference, then a suitable controller was designed for temperature control. …”
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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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Numerical Solution of Binary Liquid-Phase Adsorption onto Porocel Clay Using Linear, Freundlich and Langmuir Isotherms by Olaosebikan A. Olafadehan, Alfred A. Susu

“…The scheme is based on orthogonal collocation for spatial discretization of the resulting set of coupled hyperbolic and parabolic partial differential equations for the macro-and micro-system, respectively. …”
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Computational analysis of EMHD nanofluid flow over a thermally conductive surface: Enhancing heat transfer for industrial applications by Faiza Zahid, Muhammad Bilal Riaz

“…The problem under scrutiny yields a non-similar solution necessitating the transformation of governing partial differential equations into ordinary differential equations by the use of similarity variables. …”
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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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$\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space by Duprez, Michel, Lleras, Vanessa, Lozinski, Alexei, Vuillemot, Killian

“…Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. …”
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Lie symmetry analysis and solitary wave solution of biofilm model Allen-Cahn by Muhammad Shakeel, Naseem Abbas, Muhammad Junaid U. Rehman, Fehaid Salem Alshammari, Abdullah Al-Yaari

“…Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. …”
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Analytical Investigation of Laminar Viscoelastic Fluid Flow over a Wedge in the Presence of Buoyancy Force Effects by B. Rostami, M. M. Rashidi, P. Rostami, E. Momoniat, N. Freidoonimehr

“…The two-dimensional boundary-layer governing partial differential equations (PDEs) are derived by the consideration of Boussinesq approximation. …”
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