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This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.

The parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.

The IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.

This study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.

The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.

These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.

The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

This study aims to investigate the impact of different heater geometries (flat, rectangular, semi-elliptical and triangular) on hybrid nanofluidic (Cu–Al₂O₃–H₂O) convection in novel umbrella-shaped porous thermal systems. The system is top-cooled, and the identical heater surfaces are provided centrally at the bottom to identify the most enhanced configuration.

The thermal-fluid flow analysis is performed using a finite volume-based indigenous code, solving the nonlinear coupled transport equations with the Darcy number (10^–5 ≤ Da ≤ 10^–1), modified Rayleigh number (10 ≤ Ra_m ≤ 10⁴) and Hartmann number (0 ≤ Ha ≤ 70) as the dimensionless operating parameters. The semi-implicit method for pressure linked equations algorithm is used to solve the discretized transport equations over staggered nonuniform meshes.

The study demonstrates that altering the heater surface geometry improves heat transfer by up to 224% compared with a flat surface configuration. The triangular-shaped heating surface is the most effective in enhancing both heat transfer and flow strength. In general, flow strength and heat transfer increase with rising Ra_m and decrease with increasing Da and Ha. The study also proposes a mathematical correlation to predict thermal characteristics by integrating all geometric and flow control variables.

The present concept can be extended to further explore thermal performance with different curvature effects, orientations, boundary conditions, etc., numerically or experimentally.

The present geometry configurations can be applied in various engineering applications such as heat exchangers, crystallization, micro-electronic devices, energy storage systems, mixing processes, food processing and different biomedical systems (blood flow control, cancer treatment, medical equipment, targeted drug delivery, etc.).

This investigation contributes by exploring the effect of various geometric shapes of the heated bottom on the hydromagnetic convection of Cu–Al₂O₃–H₂O hybrid nanofluid flow in a complex umbrella-shaped porous thermal system involving curved surfaces and multiphysical conditions.

Evaluating warfighter lethality is a critical aspect of military performance. Raw metrics such as marksmanship speed and accuracy can provide some insight, yet interpreting subtle differences can be challenging. For example, is a speed difference of 300 milliseconds more important than a 10% accuracy difference on the same drill? Marksmanship evaluations must have objective methods to differentiate between critical factors while maintaining a holistic view of human performance.

Monte Carlo simulations are one method to circumvent speed/accuracy trade-offs within marksmanship evaluations. They can accommodate both speed and accuracy implications simultaneously without needing to hold one constant for the sake of the other. Moreover, Monte Carlo simulations can incorporate variability as a key element of performance. This approach thus allows analysts to determine consistency of performance expectations when projecting future outcomes.

The review divides outcomes into both theoretical overview and practical implication sections. Each aspect of the Monte Carlo simulation can be addressed separately, reviewed and then incorporated as a potential component of small arms combat modeling. This application allows for new human performance practitioners to more quickly adopt the method for different applications.

Performance implications are often presented as inferential statistics. By using the Monte Carlo simulations, practitioners can present outcomes in terms of lethality. This method should help convey the impact of any marksmanship evaluation to senior leadership better than current inferential statistics, such as effect size measures.

This study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method.

A preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case.

The solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented.

A combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.

The current analysis produces the fractional sample of non-Newtonian Casson and Williamson boundary layer flow considering the heat flux and the slip velocity. An extended sheet with a nonuniform thickness causes the steady boundary layer flow’s temperature and velocity fields. Our purpose in this research is to use Akbari Ganji method (AGM) to solve equations and compare the accuracy of this method with the spectral collocation method.

The trial polynomials that will be utilized to carry out the AGM are then used to solve the nonlinear governing system of the PDEs, which has been transformed into a nonlinear collection of linked ODEs.

The profile of temperature and dimensionless velocity for different parameters were displayed graphically. Also, the effect of two different parameters simultaneously on the temperature is displayed in three dimensions. The results demonstrate that the skin-friction coefficient rises with growing magnetic numbers, whereas the Casson and the local Williamson parameters show reverse manners.

Moreover, the usefulness and precision of the presented approach are pleasing, as can be seen by comparing the results with previous research. Also, the calculated solutions utilizing the provided procedure were physically sufficient and precise.

This paper aims to study numerically the steady natural convective heat transfer of a hybrid nanosuspension (Ag-MgO/H2O) within a partially heated/cooled trapezoidal region with linear temperature profiles at inclined walls under an effect of uniform Lorentz force. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates.

The governing equations formulated using the Oberbeck–Boussinesq approach and single-phase nanoliquid model are transformed to a non-dimensional form by using non-dimensional variables. The obtained equations with appropriate boundary conditions are resolved by the finite difference technique. The developed code has been validated comprehensively. Analysis has been performed for a wide range of governing parameters, including Rayleigh number (Ra = 105), Prandtl number (Pr = 6.82), Hartmann number (Ha = 0–100), magnetic field inclination angle (φ = 0–?/2) and nanoparticles volume fraction (φ_hnf = 0 and 2%).

It has been shown that inclined magnetic field can be used to manage the energy transport performance. An inclusion of nanoparticles without Lorentz force influence allows forming more stable convective regime with descending heat plume in the central zone, while such a regime was performed for clear fluid only for moderate and high Hartmann numbers. Moreover, the average overall entropy generation can be decreased with a growth of the Hartmann number, while an addition of hybrid nanoparticles allows reducing this parameter for Ha = 30 and 50. The average Nusselt number can be increased with a growth of the nanoparticles concentration for low values of the magnetic field intensity.

Governing equations written using the conservation laws and dimensionless non-primitive variables have been resolved by the finite difference approach. The created numerical code has been verified by applying the grid independence test and computational outcomes of other researchers. The comprehensive analysis for various key parameters has been performed.

The purpose of this paper is to develop surrogate models, using deep learning (DL), that can facilitate the application of EM analysis software. In the current status quo, electrical systems can be found in an ever-increasing range of products that are part of everyone’s daily live. With the advances in technology, industries such as the automotive, communications and medical devices have been disrupted with new electrical and electronic systems. The innovation and development of such systems with increasing complexity over time has been supported by the increased use of electromagnetic (EM) analysis software. Such software enables engineers to virtually design, analyze and optimize EM systems without the need for building physical prototypes, thus helping to shorten the development cycles and consequently cut costs.

The industry standard for simulating EM problems is using either the finite difference method or the finite element method (FEM). Optimization of the design process using such methods requires significant computational resources and time. With the emergence of artificial intelligence, along with specialized tools for automatic differentiation, the use of DL has become computationally much more efficient and cheaper. These advances in machine learning have ushered in a new era in EM simulations where engineers can compute results much faster while maintaining a certain level of accuracy.

This paper proposed two different models that can compute the magnetic field distribution in EM systems. The first model is based on a recurrent neural network, which is trained through a data-driven supervised learning method. The second model is an extension to the first with the incorporation of additional physics-based information to the authors’ model. Such a DL model, which is constrained by the laws of physics, is known as a physics-informed neural network. The solutions when compared with the ground truth, computed using FEM, show promising accuracy for the authors’ DL models while reducing the computation time and resources required, as compared to previous implementations in the literature.

The paper proposes a neural network architecture and is trained with two different learning methodologies, namely, supervised and physics-based. The working of the network along with the different learning methodologies is validated over several EM problems with varying levels of complexity. Furthermore, a comparative study is performed regarding performance accuracy and computational cost to establish the efficacy of different architectures and learning methodologies.

The purpose of this study is to determine the impact of two-phase power law nanofluid on a curved arterial blood flow under the presence of ovelapped stenosis. Over the past couple of decades, the percentage of deaths associated with blood vessel diseases has risen sharply to nearly one third of all fatalities. For vascular disease to be stopped in its tracks, it is essential to understand the vascular geometry and blood flow within the artery. In recent scenarios, because of higher thermal properties and the ability to move across stenosis and tumor cells, nanoparticles are becoming a more common and effective approach in treating cardiovascular diseases and cancer cells.

The present mathematical study investigates the blood flow behavior in the overlapped stenosed curved artery with cylinder shape catheter. The induced magnetic field and entropy generation for blood flow in the presence of a heat source, magnetic field and nanoparticle (Fe₃O₄) have been analyzed numerically. Blood is considered in artery as two-phases: core and plasma region. Power-law fluid has been considered for core region fluid, whereas Newtonian fluid is considered in the plasma region. Strongly implicit Stone’s method has been considered to solve the system of nonlinear partial differential equations (PDE’s) with 10–6 tolerance error.

The influence of various parameters has been discussed graphically. This study concludes that arterial curvature increases the probability of atherosclerosis deposition, while using an external heating source flow temperature and entropy production. In addition, if the thermal treatment procedure is carried out inside a magnetic field, it will aid in controlling blood flow velocity.

The findings of this computational analysis hold great significance for clinical researchers and biologists, as they offer the ability to anticipate the occurrence of endothelial cell injury and plaque accumulation in curved arteries with specific wall shear stress patterns. Consequently, these insights may contribute to the potential alleviation of the severity of these illnesses. Furthermore, the application of nanoparticles and external heat sources in the discipline of blood circulation has potential in the medically healing of illness conditions such as stenosis, cancer cells and muscular discomfort through the usage of beneficial effects.

The aim of this study is to offer a contemporary approach for getting optical soliton and traveling wave solutions for the Date–Jimbo–Kashiwara–Miwa equation.

The approach is based on a recently constructed ansätze strategy. This method is an alternative to the Painleve test analysis, producing results similarly, but in a more practical, straightforward manner.

The approach proved the existence of both singular and optical soliton solutions. The method and its application show how much better and simpler this new strategy is than current ones. The most significant benefit is that it may be used to solve a wide range of partial differential equations that are encountered in practical applications.

The approach has been developed recently, and this is the first time that this method is applied successfully to extract soliton solutions to the Date–Jimbo–Kashiwara–Miwa equation.