$\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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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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Investigation of Two-Dimensional Viscoelastic Fluid with Nonuniform Heat Generation over Permeable Stretching Sheet with Slip Condition by Haroon Ur Rasheed, Zeeshan Khan, Saeed Islam, Ilyas Khan, Juan L. G. Guirao, Waris Khan

“…Here, in this research article, we have investigated an incompressible viscoelastic fluid flow over a uniform stretching surface sheet along with slip boundary conditions in the presence of porous media. The partial differential equations which govern the fluid flow are changed into ordinary differential equations through suitable similarity transformation variables. …”
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Evolution-Operator-Based Single-Step Method for Image Processing by Yuhui Sun, Peiru Wu, G. W. Wei, Ge Wang

“…The key component of the proposed method is a local spectral evolution kernel (LSEK) that analytically integrates a class of evolution partial differential equations (PDEs). From the point of view PDEs, the LSEK provides the analytical solution in a single time step, and is of spectral accuracy, free of instability constraint. …”
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Study of Heat Transfer under the Impact of Thermal Radiation, Ramped Velocity, and Ramped Temperature on the MHD Oldroyd-B Fluid Subject to Noninteger Differentiable Operators by Syed Tauseef Saeed, Ilyas Khan, Muhammad Bilal Riaz, Syed Muhammad Husnine

“…The mathematical analysis of fractional governing partial differential equations has been established using systematic and powerful techniques of Laplace transform with its numerical inversion algorithms. …”
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A novel computational analysis of boundary-driven two-dimensional heat flow with internal heat generation by Muhammad Abid, Madiha Bibi, Nasir Yasin, Muhammad Shahid

“…Accurate numerical solution of parabolic and elliptic partial differential equations governing two-dimensional heat transfer is critical for engineering simulations but computationally challenging.This work employs key numerical techniques finite differences, conjugate gradients, and Crank-Nicolson time stepping to solve the heat diffusion equation and analyze method performance.The Poisson equation is discretized using second-order central finite differences and solved with the conjugate gradient approach to determine the steady state solution. …”
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Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands by Yuan Feng, Jongwan Eun, Seunghee Kim, Yong-Rak Kim

“…Recently, physics-informed neural networks (PINNs), which incorporate partial differential equations (PDEs) to solve forward and inverse problems, have attracted increasing attention in machine learning research. …”
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Nonlinear Vibration Analysis for Stiffened Cylindrical Shells Subjected to Electromagnetic Environment by Xue-Qin Li, Guang-Chen Bai, Lu-Kai Song, Wei Zhang

“…By employing the Galerkin discretization procedure, the partial differential equations are diverted to a set of coupled nonlinear ordinary differential equations of motion. …”
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Nonlinear Dynamic Behavior of Winding Hoisting Rope under Head Sheave Axial Wobbles by Guoying Wang, Xingming Xiao, Chi Ma, Gang Cheng, Xiao Di

“…The governing equations are nonlinear infinite-dimensional partial differential equations, which are discretized into the finite-dimensional ordinary differential equations through the Galerkin method. …”
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Study on construction and identification of dynamic model of precooler in aviation bleed air system by SHI Zhouzheng, DONG Minghao, LIU Qi, LUO Ziyan, WANG Hongbo, QIN Xiansheng, WANG Zhanxi

“…The finite element method was used to establish the partial differential equation of the heat exchange mechanism, and the heat exchange mechanism of the precooler was analyzed. …”
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Combined Effects of Thermal Radiation and Nanoparticles on Free Convection Flow and Heat Transfer of Casson Fluid over a Vertical Plate by M. G. Sobamowo

“…The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are converted to systems of nonlinear ordinary differential equations through similarity transformations. …”
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Irreversibility and Sensitivity analysis of MHD squeezing various shaped nanofluid flow between parallel permeable disks by Abdul Awal, Md. Sarwar Alam

“…The system of partial differential equations governing the flow is converted to a coupled of non-dimension ordinary differential equations. …”
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Analysis of the Three-Dimensional Dynamic Problems by Using a New Numerical Method by Yao Rong, Yang Sun, LiQing Zhu, Xiao Xiao

“…In this paper, the scaled boundary element method (SBFEM) is used to analyze the displacement and pore pressure response of saturated soil due to consolidation under dynamic load. The partial differential equations of linear problems are transformed into ordinary differential equations and solved along the radial direction. …”
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On the possibility of controlling the temperature of gases and liquids in a transportation system by local cooling or heating with account salt deposition by Evgeny L. Pankratov

“…The model is based on an analytical solution of partial differential equations with nonlinearity and variation of their coefficients in space and time. …”
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Radionuclide Transport in Fractured Rock: Numerical Assessment for High Level Waste Repository by Claudia Siqueira da Silveira, Antonio Carlos Marques Alvim, Jose de Jesus Rivero Oliva

“…Transport in the fracture is assumed to obey an advection-diffusion equation, while molecular diffusion is considered the dominant mechanism of transport in porous matrix. The partial differential equations describing the movement of radionuclides were discretized by finite difference methods, namely, fully explicit, fully implicit, and Crank-Nicolson schemes. …”
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Application of fluid dynamics in modeling the spatial spread of infectious diseases with low mortality rate: A study using MUSCL scheme by Nnaji Daniel Ugochukwu, Kiogora Phineas Roy, Onah Ifeanyi Sunday, Mung’atu Joseph, Aguegboh Nnaemeka Stanley

“…By treating susceptible, infected, and treated population densities as fluids governed by a system of partial differential equations, the study simulates the epidemic’s spatial dynamics. …”
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Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches by Jiangen Lv, Zhicheng Yang, Xuebin Chen, Quanke Wu, Xiaoxia Zeng

“…Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. …”
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Theoretical Analysis and Experimental Validation on Galloping of Iced Transmission Lines in a Moderating Airflow by Bing Huo, Xuliang Li, Fujiang Cui, Shuo Yang

“…Galloping of an iced transmission line subjected to a moderating airflow has been analysed in this study, and a new form of galloping is discovered both theoretically and experimentally. The partial differential equations of the iced transmission line are established based on the Hamilton theory. …”
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Fast Attitude Maneuver of a Flexible Spacecraft with Passive Vibration Control Using Shunted Piezoelectric Transducers by Bassam A. Albassam

“…The control design process starts with deriving the nonlinear partial differential equations of motion for the spacecraft using Hamilton’s principle which accounts for the electromechanical coupling and the presence of resistive or resistive-inductive circuits. …”
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Analysis of non scalar control problems for parabolic systems by the block moment method by Boyer, Franck, Morancey, Morgan

“…We also deduce estimates on the cost of controllability when the final time goes to the minimal null control time.We illustrate how the method works on a few examples of such abstract controlled systems and then we deal with actual coupled systems of one dimensional parabolic partial differential equations. Our strategy enables us to tackle controllability issues that seem out of reach by existing techniques.…”
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