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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Mathematical model and stability of SWCNT- and MWCNT-based nanofluid flow with thermal and chemically reactive effects inside a porous vertical cone by Haihua Xu, Fuad A. Awwad, Emad A. A. Ismail, Waris Khan

“…Based on the flow assumptions, the fundamental flow equations are modeled as partial differential equations (PDEs). Using the appropriate transformation, the PDEs are converted to ordinary differential equations and then solved via RK4 in MATLAB. …”
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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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Two-phase Agrawal hybrid nanofluid flow for thermal and solutal transport fluxes induced by a permeable stretching/shrinking disk by Hatem Gasmi, Muhammad Waqas, Umair Khan, Aurang Zaib, Anuar Ishak, Imtiaz Khan, Ali Elrashidi, Mohammed Zakarya

“…Through the utilization of similarity ansatz, the governing partial differential equations are simplified into a class of ordinary differential (similarity) equations. …”
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Machine learning-driven analysis of heat transfer in chemically reactive fluid flow considering Soret-Dufour effects by Shazia Habib, Saleem Nasir, Zeeshan Khan, Abdallah S. Berrouk, Waseem, Saeed Islam

“…By transforming the governing system of nonlinear partial differential equations into ordinary differential equations via similarity transformation, the methodology attains computational efficiency and solution precision. …”
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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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Stochastic Diffusive Modeling of CO₂ Emissions with Population and Energy Dynamics by Muhammad Shoaib Arif, Kamaleldin Abodayeh, Hisham M. Al-Khawar, Yasir Nawaz

“…A novel numerical scheme, an extension of the Euler-Maruyama algorithm, is proposed to solve stochastic time-dependent partial differential equations governing the model. The scheme's consistency and stability are rigorously analyzed in the mean square sense. …”
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Spatially Distributed Morphogen Production and Morphogen Gradient Formation by Arthur D. Lander, Qing Nie, Frederic Y. M. Wan

“…Partial differential equations and auxiliaryconditions governing the activities of the morphogen Dpp in Drosophila wingimaginal discs were formulated and analyzed as Systems B, R, and C in[7][9][10]. …”
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Heat transport efficiency in rheology of radiated casson material due to porous shrinking cylinder by Muhammad Yasir, N. Ameer Ahammad, Aisha M. Alqahtani, Yahia Said

“…The physical characteristics of the problem are governed by partial differential equations, which are converted to ordinary differential equations using appropriate similarity variables. …”
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Entropy Analysis of Carreau Nanofluid Flow in the Presence of Joule Heating and Viscous Dissipation Past Unsteady Stretching Cylinder by Eshetu Haile Gorfie, Gizachew Bayou Zegeye, Gurju Awgichew Zergaw

“…The governing partial differential equations are transformed into first‐order initial value problems by similarity transformation and linearization. …”
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Investigation of natural convection heat transfer in MHD fluid within a hexagonal cavity with circular obstacles by Mohammad Dehghan Afifi, Ali Jahangiri, Mohammad Ameri

“…The finite element method is employed to solve the governing partial differential equations of the system. This research focuses on the impact of innovative geometry, variations in Rayleigh and Hartmann numbers on flow patterns, temperature distribution and concentration fields. …”
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