RETRACTED ARTICLE: Quadratic multiple regression model and spectral relaxation approach for carreau nanofluid inclined magnetized dipole along stagnation point geometry by Sayed M. El Din, Adil Darvesh, Assad Ayub, Tanveer Sajid, Wasim Jamshed, Mohamed R. Eid, Syed M. Hussain, Manuel Sánchez-Chero, Sheda Méndez Ancca, Johana Milagritos Ramírez Cerna, Carmen Luisa Aquije Dapozzo

“…Physical model generates collection of partial differential equations (PDEs) and these PDEs are moved into ordinary differential equations by a similarity transformations scheme. …”
Get full text

Impact of nanoparticles shapes and fuzzy volume fraction on Al2O3 – H2O nanofluid flow past an unsteady expandable surface by Adnan Saeed Butt, Naveed Yaqoob, Sameea Akbar, Gul M. Shaikh, Adeeba Farhat, Fouzia Amir, Mohamed R. Ali

“…The goal is to simplify the nonlinear partial differential equations (PDEs) through appropriate transformations, converting them into ordinary differential equations (ODEs). …”
Get full text

Magnetohydrodynamic Peristaltic Propulsion of Casson Nanofluids With Slip Effects Over Heterogeneous Rough Channel by Hanumesh Vaidya, Fateh Mebarek‐Oudina, Rakesh Kumar, C. Rajashekhar, Kerehalli Vinayaka Prasad, Sangeeta Kalal, Kottakkaran Sooppy Nisar

“…A novel rough non‐uniform model is effectively governed by a set of nonlinear coupled governing partial differential equations, which are simplified under long wavelength and creeping flow approximations. …”
Get full text

Extrapolation of cavitation and hydrodynamic pressure in lubricated contacts: a physics-informed neural network approach by Faras Brumand-Poor, Freddy Kokou Azanledji, Nils Plückhahn, Florian Barlog, Lukas Boden, Katharina Schmitz

“…In particular, physics-informed neural networks (PINNs) are as effective hybrid solvers that combine data-driven and physics-based methods to solve the partial differential equations that drive EHL simulations. By integrating physical laws into the parameter optimization of the neural network (NN), PINNs provide accurate and fast solutions. …”
Get full text

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. …”
Get full text

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. …”
Get full text

Chemical reactions with the Casson nanofluid flow by the bioconvective behavior of microorganisms over a spinning disc by Prabhakar Sagadevan, Umadevi Raju, Meganathan Murugesan, Unai Fernandez-Gamiz, Samad Noeiaghdam

“…Using the right variables to change the nonlinear partial differential equations (PDEs) that govern the problem into a system of ordinary differential equations (ODEs) is a key step toward finding numerical solutions. …”
Get full text

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. …”
Get full text

A holistic physics-informed neural network solution for precise destruction of breast tumors using focused ultrasound on a realistic breast model by Salman Lari, Hossein Rajabzadeh, Mohammad Kohandel, Hyock Ju Kwon

“…Additionally, employing PINN for estimating partial differential equations (PDE) solutions can notably decrease the enormous number of discretized elements needed. …”
Get full text

Numerical computation of PDE formed characterizing thermal rheology of Cu/Al2O3/TiO2 ternary nanoparticles flow between coaxial cylinders by Noreen Sher Akbar, M.Fiaz Hussain, Dennis Ling Chuan Ching, Muhammad Farooq, Ilyas Khan

“…The governing partial differential equations associated with the single-phase simulation of ternary nanoparticles, including morphological effects, are analyzed. …”
Get full text

Numerical Analysis on the Storage of Nuclear Waste in Gas-Saturated Deep Coal Seam by Teng Teng, Yuming Wang, Xiaoyan Zhu, Xiangyang Zhang, Sihai Yi, Guowei Fan, Bin Liu

“…As the second step, a finite element numerical model and numerical simulation are developed to analyze the storage of nuclear waste in a gas-saturated deep coal seam based on the partial differential equations (PDE) solver of COMSOL Multiphysics with MATLAB. …”
Get full text

ANFIS-PSO analysis on axisymmetric tetra hybrid nanofluid flow of Cu-CNT-Graphene-Tio2 with WEG-Blood under linear thermal radiation and inclined magnetic field: A bio-medicine app... by Maddina Dinesh kumar, José Luis Díaz Palencia, G. Dharmaiah, A. Wakif, S. Noeiaghdam, U. Fernandez-Gamiz, S. Dinarvand

“…Method: Applying the ODE45 integration technique to the given numerical solutions yields non-linear, non-dimensionalized, and highly partial differential equations that control the momentum, energy, and concentration. …”
Get full text

Mathematical modeling in autoimmune diseases: from theory to clinical application by Yaroslav Ugolkov, Yaroslav Ugolkov, Yaroslav Ugolkov, Antonina Nikitich, Antonina Nikitich, Cristina Leon, Gabriel Helmlinger, Kirill Peskov, Kirill Peskov, Kirill Peskov, Kirill Peskov, Victor Sokolov, Victor Sokolov, Alina Volkova, Alina Volkova

“…. ≥70% of the models were developed as nonlinear systems of ordinary differential equations, others – as partial differential equations, integro-differential equations, Boolean networks, or probabilistic models. …”
Get full text

Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U-net backbone by Christophe Bonneville, Nathan Bieberdorf, Arun Hegde, Mark Asta, Habib N. Najm, Laurent Capolungo, Cosmin Safta

“…However, the LMD governing equations in these models often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. …”
Get full text

Heat transportation of 3D chemically reactive flow of Jeffrey nanofluid over a porous frame with variable thermal conductivity by Nahid Fatima, Aaqib Majeed, Taoufik Saidani, Nouman Ijaz, Kamal Barghout, Nidal Abu-Libdeh

“…To model these phenomena, we employ the boundary layer approximation to derive a system of partial differential equations (PDEs). These PDEs are subsequently simplified into more manageable ordinary differential equations (ODEs) using the similarity variables. …”
Get full text

Entropy generation and heat convection analysis of second-grade viscoelastic nanofluid flow in a tilted lid-driven square enclosure: A finite difference approach by Anil Ahlawat, Mukesh Kumar Sharma, K. Loganathan

“…The stream function approach removes the pressure gradient term from the linear momentum equation, and the resulting partial differential equations are discretized via finite differences and then resulting algebraic equations are solved with SOR and SUR methods employing self-developed MATLAB codes. …”
Get full text

Thermal enhancement using variable characteristics and tripartite diffusion features of solar aircraft wings in context of Reiner-Philippoff hybrid nanofluid passing through a para... by Esraa N. Thabet, A.M. Abd-Alla, S.M.M. El-Kabeir

“…With the utilization of the appropriate similarity transformations, partial differential equations that represent the mathematical model can be simplified to ordinary differential equations. …”
Get full text

Application of artificial intelligence brain structure-based paradigm to predict the slip condition impact on magnetized thermal Casson viscoplastic fluid model under combined temp... by Umar Farooq, Shan Ali Khan, Haihu Liu, Muhammad Imran, Lotfi Ben Said, Aleena Ramzan, Taseer Muhammad

“…With the help of a similarity transformation, the complex partial differential equations governing the flow and energy take on the form of nonlinear ordinary differential equations. …”
Get full text

Second order slip micropolar MHD hybrid nanofluid flow over a stretching surface with uniform heat source and activation energy: Numerical computational approach by Syed Arshad Abas, Hakeem Ullah, Mehreen Fiza, Ali Akgul, Aasim Ullah Jan, Magda Abd El-Rahman, Seham M. Al-Mekhlafi

“…Methodology: A mathematical model is formulated based on boundary-layer approximations, leading to a system of partial differential equations (PDEs) that describe the flow, thermal, and concentration fields. …”
Get full text

ANN-based two hidden layers computational procedure for analysis of heat transport dynamics in polymer-based trihybrid Carreau nanofluid flow over needle geometry by Adil Darvesh, Fethi Mohamed Maiz, Basma Souayeh, Luis Jaime Collantes Santisteban, Hakim AL. Garalleh, Afnan Al Agha, Lucerito Katherine Ortiz García, Nicole Anarella Sánchez-Miranda

“…Methodology: The physical model is originally formed with the help of partial differential equations (PDEs), that are formulated with pre-defined assumption of fluid flow mechanism. …”
Get full text