A comprehensive numerical study exploring the significance of thermally reactive bioconvection in Falkner-Skan flow of Williamson nanomaterials influenced by activation energy and... by M. Israr Ur Rehman, Haibo Chen, Aamir Hamid, Wu Qian, Refka Ghodhbani, Mohamed Hussien

“…Similarity transformations are used to convert the system of partial differential equations into a system of ordinary differential equations, which are then numerically solved using the Runge–Kutta–Fehlberg (RKF-45) method. …”
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Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents by Raimund BÜrger, Gerardo Chowell, Elvis GavilÁn, Pep Mulet, Luis M. Villada

“…An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. …”
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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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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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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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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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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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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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Non-local piezoelasticity to incorporate the influence of small-scale factors on the resonance behavior of the Mindlin piezoelectric polymeric nanoplates by Narinderjit Singh Sawaran Singh, Waqed H. Hassan, Zainab Mоhammed Ameen Ahmed, Younis Mohamed Atiah Al-zahy, Soheil Salahshour, Mostafa Pirmoradian

“…The Galerkin method is utilized to solve the partial differential equations governing the dynamics of the piezoelectric polymeric nanoplate, marking a significant methodological contribution to the field. …”
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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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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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Thermo-Magneto-Solutal Squeezing Flow of Nanofluid between Two Parallel Disks Embedded in a Porous Medium: Effects of Nanoparticle Geometry, Slip and Temperature Jump Conditions by M. G. Sobamowo, A. T. Akinshilo, A. A. Yinusa

“…Similarity variables are used to transform the developed governing systems of nonlinear partial differential equations to systems of nonlinear ordinary differential equations. …”
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Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method by Mohammed Hamaidi, Ahmed Naji, Ahmed Taik

“…The advantages of such formulation are (i) time discretization as implicit, explicit, θ-method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. …”
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Melting ternary hybrid nanofluid stagnation point flow with velocity slip past a stretching/shrinking sheet: Numerical simulation and validation via P2SATRA by Nur Syahirah Wahid, Nur Ezlin Zamri, Siti Zulaikha Mohd Jamaludin, Nur Hazirah Adilla Norzawary, Mohd Shareduwan Mohd Kasihmuddin, Mohd. Asyraf Mansor, Norihan Md Arifin, Ioan Pop

“…The governing partial differential equations (PDEs) are initially formulated and subsequently reduced to ordinary differential equations (ODEs). …”
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MOVIMENTO BROWNIANO: APLICAÇÃO EM ESTRATÉGIAS DE BUSCA by Matheus Dias Carvalho, Ricardo de Carvalho Falcão, Antonio Marcos de Oliveira Siqueira

“…This paper structured partial differential equations governing this process for the immortal case of walker, and later found analytical solutions to these expressions. …”
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Analysis of a Two-Phase MHD Free Convection Generalized Water–Ethylene Glycol (50 : 50) Dusty Brinkman-Type Nanofluid Pass through Microchannel by Dolat Khan, Musawa Yahya Almusawa, Waleed Hamali, M. Ali Akbar

“…While the left plate moves at a consistent velocity and the right plate stays stationary, the fluid is also evenly dispersed with all dust particles that have a spherical form. Partial differential equations (PDE) are used to present the mathematical modeling. …”
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