The significance of heat transfer through natural convection in stagnation point flow of prandtl fluid by Salman Zeb, Zakir Ullah, A.B. Albidah, Ilyas Khan, Waqar A. Khan

“…The analysis integrates velocity, concentration slips, and temperature jumps within a nonlinear partial differential equation framework. These equations are converted into nonlinear partial differential equations using appropriate dimensionless variables. …”
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Mathematical Modeling of Wax Deposition in Field-Scale Crude Oil Pipeline Systems by Francis Oketch Ochieng, Mathew Ngugi Kinyanjui, Jeconia Okelo Abonyo, Phineas Roy Kiogora

“…The coupled nonlinear partial differential equations governing the flow are discretized in time by a second-order semi-implicit time discretization scheme based on the Adams-Bashforth and Crank-Nicolson methods, which completely decouples the computation of the governing equations. …”
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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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Chemical reaction and Soret impacts on MHD heat and mass transfer Casson hybrid nanofluid (MoS2+ZnO) flow based on engine oil across a stretching sheet with radiation by M. Radhika, Y. Dharmendar Reddy

“…The relevant similarity variables convert the governing nonlinear partial differential equations to ordinary differential equations (ODEs). …”
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On the transitory performance of mixed convective heat and mass transfer around spherical region in the presence of fluctuating streams by Hossam A. Nabwey, Sahar Anwar, Muhammad Ashraf, A.M. Rashad

“…For this purpose, the coupled partial differential equations are first non-dimensionalized and their steady and oscillatory components are separated. …”
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Modeling Highly Buoyant Flows in the Castel Giorgio: Torre Alfina Deep Geothermal Reservoir by Giorgio Volpi, Fabien Magri, Francesca Colucci, Thomas Fisher, Mattia De Caro, Giovanni B. Crosta

“…The open source finite-element code OpenGeoSys is applied to solve the coupled systems of partial differential equations. The commercial software FEFLOW® is also used as additional numerical constraint. …”
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Accelerated pseudo-transient method for elastic, viscoelastic, and coupled hydromechanical problems with applications by Y. Alkhimenkov, Y. Y. Podladchikov

“…<p>The accelerated pseudo-transient (APT) method is a matrix-free approach used to solve partial differential equations (PDEs), characterized by its reliance on local operations, which makes it highly suitable for parallelization. …”
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Method for Calculating Transients in a Gas Pipeline by A. S. Fikov

“…An analytical solution of a system of partial differential equations describing the unsteady isothermal flow of real gases in gas pipelines is considered. …”
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A ternary hybrid model for analysis of enhanced heat transport in non-Newtonian nanofluid over cylindrical geometry with effect of infinite shear rate viscosity by Hakim AL Garalleh

“…Based on assumptions of current study, the governing partial differential equations (PDEs) are formed and then transformed into ODEs for numerical solution. …”
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A Novel Numerical Treatment to One Dimensional Heat and Wave Equations with Second Order Accuracy by Muhammad Abid, Muhammad Shahid

“…Numerical solutions to partial differential equations (PDEs) play a vital role in modeling complex physical phenomena across scientific computing and engineering disciplines. …”
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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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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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Qualitative financial modelling in fractal dimensions by Rami Ahmad El-Nabulsi, Waranont Anukool

“…Abstract The Black–Scholes equation is one of the most important partial differential equations governing the value of financial derivatives in financial markets. …”
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Thermal radiation and thermo-diffusion in Casson-ferrofluid over a magnetized porous surface: RSM analysis by S. Manjunatha, J. Santhosh Kumar, Khalil Ur Rehman, Wasfi Shatanawi, S.V.K. Varma

“…Similarity transformations convert the flow model's governing nonlinear coupled partial differential equations into ordinary coupled differential equations. …”
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Displacement of hydrocarbon liquids by water using the models of zonally heterogeneous deformable formations by Marat Ya. Khabibullin, Rustem I. Suleymanov, Razifa R. Stepanova, Alina Az. Gizzatullina, Arsen M. Khabibullin

“…It is known that the motion of two-phase hydrocarbon systems in deformable reservoirs is represented by complex nonlinear partial differential equations. An analytical solution of such equations is possible only with the use of special approaches. …”
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Computational analysis for efficient thermal transportation of ternary hybrid nanofluid flow across a stretching sheet with Cattaneo-Christov heat flux model by Wei Li, Shan Ali Khan, Muhammad Shafqat, Qamar Abbas, Taseer Muhammad, Muhammad Imran

“…The phenomenon were designed as partial differential equations. These are condensed to system of ODEs via similarity transformations. …”
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Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method by Yasir A. Madani, Khidir Shaib Mohamed, Sadia Yasin, Sehrish Ramzan, Khaled Aldwoah, Mohammed Hassan

“…The $$\phi ^{6}$$ ϕ 6 model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. …”
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Thermodynamics and solar radiative analysis in Jeffery-Hamel flow through non-parallel channel by novel improved residual power series method by Mohamed Bouzidi, Mohsin Ul Haq, Ikram Ullah, Saira Shukat, Hijaz Ahmad, Mohammad Mahtab Alam, Hamid Khan, Marouan Kouki

“…The governing system of partial differential equations can be transformed into ordinary differential equations (ODEs) using suitable transformations. …”
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Randomized radial basis function neural network for solving multiscale elliptic equations by Yuhang Wu, Ziyuan Liu, Wenjun Sun, Xu Qian

“…Ordinary deep neural network (DNN)-based methods frequently encounter difficulties when tackling multiscale and high-frequency partial differential equations. To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. …”
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Mixed convection hybrid nanofluid flow past a non-isothermal cone and wedge with radiation and convective boundary condition: Heat transfer optimization by Rusya Iryanti Yahaya, Norihan Md Arifin, Mohd Shafie Mustafa, Ioan Pop, Fadzilah Md Ali, Siti Suzilliana Putri Mohamed Isa

“…Non-linear ordinary differential equations, derived through similarity transformation of the governing partial differential equations and boundary conditions, are solved numerically using the bvp4c solver. …”
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