Evaluating the Stability of Numerical Schemes for Fluid Solvers in Game Technology by Craig R. Stark, Declan A. Diver

“…However, determining the stability of a numerical algorithm is a complex and involved task when a coupled set of nonlinear partial differential equations need to be solved. …”
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Entropy generation of Engine Oil-based two phase tri-hybrid nanofluid due to Riga plate: Modified Magnetic field and Cattaneo-Christov heat Perceptions by A. Divya, Muhammad Jawad, Walid Abdelfattah, Sarfaraz Kamangar

“…Similarity framework is used to renovate the partial differential equations of dynamical system into a set of nonlinear ordinary differential equations. …”
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Shape factor analysis of water and aluminium oxide nanoparticles in a porous medium with slip effects by Prasun Choudhary, K. Loganathan, Kavita Jat, Kalpna Sharma, S. Eswaramoorthi

“…The impacts of first-order velocity slip and a porous medium are also examined. The governing flow equations are nonlinear partial differential equations that are reduced into ordinary differential equations by similarity transformations, and these reduced equations are subsequently solved numerically with the bvp4c MATLAB solver. …”
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Time Evolution of the Densities of Three Species Interacting in the Same Ecosystem and the Stability Analysis of Steady States for a Reaction–Diffusion PDE Model by Emmanuel Niyigaba, Wellars Banzi, Hamidou Touré

“…A system of three simultaneous nonlinear partial differential equations is used to represent this situation. …”
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Global asymptotic stability of inhomogeneous iterates by Yong-Zhuo Chen

“…Then we apply the results to investigate the stability of equilibrium of T when it satisfies certain type of sublinear conditions with respect to the partial order defined by a closed convex cone. The examples of application to nonlinear difference equations are also given.…”
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Advanced numerical simulation of hybrid nanofluid radiative flow with Cattaneo-Christov heat flux model over a rotating disk: Innovative iterative techniques by Muhammad Jebran Khan, Samina Zuhra, Zareen A. Khan, Mohsin Ali, Li Chen, Abdul Haq, Firas Zawaideh, Radwan M. Batyha, Diaa Salama AbdElminaam

“…The rotational motion of the disks generates the fluid flow, and the governing partial differential equations are transformed into dimensionless forms using similarity variables. …”
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Simulation of Bingham–Papanastasiou model within trapezoidal cavity with mixed convection effects by Arooj Tanveer, Muhammad Bilal Ashraf

“…The governing system of nonlinear dimensionless partial differential equations is analysed by using PDE solver of finite element method in COMSOL. …”
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Radiation Effects in Flow through Porous Medium over a Rotating Disk with Variable Fluid Properties by Shalini Jain, Shweta Bohra

“…Particular case considering these fluid properties’ constant is also discussed. The governing partial differential equations are converted into nonlinear normal differential equation using similarity alterations. …”
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Effects of heat generation and absorption on thermal radiative MHD flow of chemically reacting Casson nanofluids over a wedge by S. Venkateswarlu, B. Hari Babu, M. Veera Krishna

“…Silicon Dioxide (SiO₂) nanofluids are considered. The basic governing partial differential equations are converting into nonlinear ordinary differential equations with employing suitable similarity transformations. …”
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Thermal and velocity slip impacts on MHD tetra-hybrid nanofluids flow over a porous stretching surface by Prasun Choudhary, K. Loganathan, Kavita Jat, Krishna Prakash Arunachalam, S. Eswaramoorthi

“…Initially formulated flow governing equations are highly nonlinear partial differential equations that are changed into a set of linear ordinary differential equations adopting relevant similarity transformations. …”
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Computational investigation of magnetohydrodynamics Casson Micropolar fluid flowing past a permeable linearly stretchy wall with heat source by S. Alao, R. A. Oderinu, B. A. Sanusi, T. A. Oyeyinka, F. J. Ayanbukola

“…To achieve this, a similarity transformation is adopted, to convert the fundamental partial differential equations into nonlinear ordinary differential equations which are then solved numerically using collocating weighted residual scheme that adopts Hermite polynomials as the basis functions and Gauss-Lobatto points for spatial discretization. …”
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Entropy driven optimization of non-linear radiative chemically reactive sutterby nanofluid flow in presence of gyrotactic micro-organism with Hall effect and activation energy by Muhammad Jameel, Zahir Shah, Muhammad Rooman, Mansoor H. Alshehri, Narcisa Vrinceanu, Elisabeta Antonescu

“…This model reveals the dynamics of Sutterby nanofluid behavior by using partial differential equations (PDEs) and successively converted into an ordinary differential equation (ODE) system. …”
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Stability Analysis of Dual Solutions of Convective Flow of Casson Nanofluid past a Shrinking/Stretching Slippery Sheet with Thermophoresis and Brownian Motion in Porous Media by Kifle Adula Duguma, Oluwole Daniel Makinde, Lemi Guta Enyadene

“…The governing modeled partial differential equations with boundary conditions are transformed into nonlinear ordinary differential equations using similarity transformations and finally converted to the first-order initial value problem. …”
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Analysis of thermophoretic and Brownian diffusions in hydromagnetic curvilinear flow of Carreau nanofluid with activation energy and heat generation by M. Imran, M. Naveed

“…The investigation includes the impact of heat generation and activation energy, which are incorporated into the energy and concentration equations, respectively. The mathematical model of the boundary layer flow is formulated as a set of nonlinear partial differential equations using curvilinear coordinates. …”
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Enhanced heat transfer efficiency in Williamson Radiated hybrid nanofluid over a porous Surface: Role of activation energy, chemical reaction and motile Microbes by Muhammad Umar Farooq, Aaqib Majeed, Parvez Ali, Taoufik Saidani

“…The governing coupled nonlinear partial differential equations (PDEs) are reduced to a system of nonlinear ordinary differential equations (ODEs) through similarity transformations. …”
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Two-Phase Turbulent Fluid Flow in a Geothermal Pipe with Chemical Reaction by Nyariki M. Eric, Mathew N. Kinyanjui, Jeconia O. Abonyo

“…In this study, two-phase non-Newtonian turbulent fluid flow in an inclined geothermal pipe with chemical reaction was considered. The governing nonlinear partial differential equations derived were solved numerically using Finite Difference Method. …”
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Effect of the Velocity Second Slip Boundary Condition on the Peristaltic Flow of Nanofluids in an Asymmetric Channel: Exact Solution by Emad H. Aly, Abdelhalim Ebaid

“…The system governing the current flow was found as a set of nonlinear partial differential equations in the stream function, pressure gradient, nanoparticle concentration, and temperature distribution. …”
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Exploration of Casson hybrid nanofluid (Cu−Al2O3/EG) flow over an exponentially heated stretchy sheet with radiation absorption and viscous dissipation: A modified Buongiorno model... by S. Baskaran, R. Sowrirajan, S. Eswaramoorthi, K. Loganathan, Balachandra Pattanaik

“…Employing suitable transformations, the system of governing nonlinear partial differential equations are reformed into ordinary differential equations, which are numerically computed by adopting the bvp4c solver in MATLAB. …”
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Reduced Gravity and Magnetohydrodynamic Effects on Transient Mixed Convection Flow Past a Magnetized Heated Cone Embedded in Porous Medium by Hossam A. Nabwey, Zia Ullah, Asifa Ilyas, Muhammad Ashraf, Ahmed M. Rashad, Sumayyah I. Alshber, Miad Abu Hawsah

“…The mathematical form of the nonlinear partial differential equations subject to the boundary conditions for the proposed unsteady model is presented. …”
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Unsteady Hydromagnetic Flow of Radiating Fluid Past a Convectively Heated Vertical Plate with the Navier Slip by O. D. Makinde, M. S. Tshehla

“…This paper investigates the unsteady hydromagnetic-free convection of an incompressible electrical conducting Boussinesq’s radiating fluid past a moving vertical plate in an optically thin environment with the Navier slip, viscous dissipation, and Ohmic and Newtonian heating. The nonlinear partial differential equations governing the transient problem are obtained and tackled numerically using a semidiscretization finite difference method coupled with Runge-Kutta Fehlberg integration technique. …”
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