Calculating the Gerber–Shiu function for Sparre Andersen process via collocation method

Electromagnetic radiative wavy flow analysis of shear thinning materials affected by heat and mass gradients: a computational study

A scale-3 wavelet collocation method for 2D time-fractional integro-differential equations with a weakly singular kernel

Fractional collocation method for linear neutral weakly singular Volterra integro-differential equations

Simulation of soil collapse based on the local radial basis function collocation method

• Waverider optimization via lower-surface polynomials and leading edge perturbations • GPU-accelerated CFD workflow for waverider optimization and trajectory analysis • Waverider optimization using analytical models and high-fidelity CFD • Key differences revealed between analytical and CFD-based optimization • Waveriders geometries optimized using different aero-performance models, Euler CFD flow fields at Mach 6.85, 25 km altitude. This work presents a methodology for designing waverider geometries that incorporate leading edge rounding and parameterized surface perturbations, and evaluates their performance using two distinct aerodynamic prediction approaches: a reduced-order analytical/empirical model and a GPU-based Euler CFD solver, both applied with an empirical approach to estimate the viscous contributions. Geometry optimization was performed using a global pattern search algorithm, and optimal trajectories were computed using a direct collocation method to assess the impact of aerodynamic modeling fidelity on mission performance. Cross-comparisons between these aerodynamic performance prediction methods revealed that CFD-based optimization produces geometries that capture off-design flow phenomena that are absent from analytical-based optimizations. As a result, CFD-optimized waveriders demonstrated higher average lift-to-drag ratios than those derived from the analytical model. The analytical trajectories had lower angle-of-attack control and degraded in accuracy for vehicles with high pressure losses. These findings highlight the limitations of traditional analytical methods and underscore the feasibility of using modern GPU-accelerated CFD for resource-intensive tasks such as vehicle optimization and trajectory analysis.

A multi-resolution hybrid Haar wavelet collocation method for solving nonlinear partial differential equations

Instability of MHD Poiseuille flow of nanoparticles FeO in water cylinder flow

A higher order collocation method for a singularly perturbed system having boundary turning points

In this article, a hybrid cubic B-spline collocation method has been proposed to find the numerical solutions of a well-known nonlinear Sine-Gordon equation. The finite difference scheme has been employed to discretize the time derivative, whereas cubic B-spline functions are used for spatial discretization. The efficiency of the applied method is checked through some test problems. Numerical outcomes are compared with the exact solutions available in the literature which illustrate the effectiveness of the proposed method. It can easily be concluded that the results obtained are reliable and consistent with those found in earlier research.

In this paper, meshfree collocation with fast-moving least-squares reproducing kernel was implemented to write down and execute numerical solutions for some applications in elastostatics and elastodynamics. The spatial discretization using meshfree collocation method was carried out on the equilibrium differential equations of elastostatics and elastodynamics and the corresponding boundary conditions. The resulting discrete forms were solved for benchmark problems in the one- and two-dimensional cases. In each case, a convergence study was conducted to ascertain the utility and efficacy of the developed solutions. For elastodynamics, the time domain, however, was discretized using the Newmark beta time-integration scheme. The latter combination was implemented to solve suitable benchmark problems in the one-dimensional and two-dimensional cases. In each case, a stability study was conducted to demonstrate, again, the method’s efficacy in handling elastodynamic problems.

The quasi-steady low-Reynolds-number flow induced by a linear chain of multiple slip spheres translating along their common axis in a Newtonian fluid is investigated. The particles are allowed to differ in radius, Navier slip coefficient, migration velocity, and interparticle spacing. A semi-analytical solution of the governing Stokes equation is obtained using a boundary collocation method. Hydrodynamic interactions among the particles are shown to be significant under appropriate geometric and surface conditions. For the two-sphere configuration, the computed hydrodynamic forces agree closely with previously published asymptotic solutions derived via the twin multipole expansion method. In the three-sphere case, the presence of a third particle substantially modifies the forces acting on the other two, demonstrating non-negligible many-body interaction effects. The interaction strength is found to be more pronounced for smaller particles or those with lower slip coefficients. Calculations for longer particle chains further reveal a clear hydrodynamic shielding effect within the assembly.

Complex, powerful, and representative fractional SDDMs are used to model fractional-order systems of stochasticity and time delays. This study aims to construct an approximate spectral collocation scheme, numerically solving certain types of these models. Specifically, a suitable model of conformable fractional operator sense where the stochastic term is of the standard one-dimensional SBM type, and the time delay is discrete and constant, is established and solved. The proposed scheme is based on the spectral collocation method, with SLP basis functions and SLGL collocation points. To obtain the desired approximate solutions utilizing the present method, the domain under consideration is discretized into steps, and at each step, the solution is approximated using the spectral collocation technique. Simply, complicated problems are evolved into a system of algebraic equations where the unknowns are the Legendre coefficients. These equations are solved utilizing an appropriate numerical method executed by MATHEMATICA. For convenience, the convergence analysis under Lipschitz properties is presented. To validate the reality of the method, some applications of linear and nonlinear types of the suggested model are solved, and the errors are computed. Moreover, the Log-Log plots are sketched to confirm the efficiency of the presented method when more collocation points are used. The results formulated in tables, figures, and discussions illustrate the high accuracy and capability of the proposed methodology. Final remarks and future work are reviewed and debated, too.

Enhancement of heat transfer rate through fins is one of the prevalent options, and the highly adaptable design of fins enables their application in many systems, such as radiators and electronic cooling devices. Pyramidal spine fins (PSF) are designed to enhance heat transfer and provide better performance compared to other fin designs. Due to their effectiveness, these fins are implemented in heat sinks. Motivated by this practical application, the current study examines the thermal behavior of a wetted PSF, taking into account the effects of both convection and radiation mechanisms. The highly nonlinear energy equation of PSF is converted into dimensionless form using appropriate dimensionless terms. To get the solution for the resultant equation, the Schröder polynomial collocation method (SPCM) is applied. The impact of important parameters on the thermal profile, heat transfer rate, and efficiency of PSF is illustrated graphically to address the thermal performance. The study reveals that increasing the value of the radiation-conduction parameter from 0 to 0.2 and the conduction-convection parameter from 0 to 0.5 enhances the heat transfer rate by approximately 1.35% and 28.08%, respectively. Additionally, a larger rate of heat transmission of about 2.23% is seen in the wetted PSF compared to the dry PSF.

Robust numerical frameworks are essential for the simulation, design, monitoring, and control of membrane-based separation units, particularly under highly nonlinear and industrially relevant operating conditions. In this context, a comprehensive phenomenological and numerical framework is proposed for the simulation of hollow-fiber membrane modules, incorporating coupled mass, momentum (through pressure drop), and energy transport equations. The governing equations are discretized using a rigorous orthogonal collocation formulation, and the performances of two numerical solution strategies are systematically investigated for the first time to allow the in-line and real-time implementation of the model: a steady-state approach based on the Newton-Raphson method with careful treatment of initial estimates, and a pseudotransient formulation. Particularly, an original and consistent numerical treatment is introduced for the energy balance at boundaries where the permeate flow vanishes, enabling the stable incorporation of thermal effects and Joule-Thomson phenomena. The results clearly show that the steady-state Newton-Raphson approach provides the best overall performance in terms of computational efficiency, numerical robustness, and accuracy when physically consistent initial profiles are employed. In particular, the combination of a linear initial guess and a numerical mesh constituted of four collocation points yielded the most favorable balance between convergence speed, numerical robustness, and accuracy for the base-case sensitivity analysis. For monitoring-oriented applications, the numerical choice should be weighted primarily toward computational performance once physical consistency and convergence criteria are satisfied, rather than toward maximum mesh-refinement accuracy. In this context, small differences in internal-fiber profiles can be compensated through real-time permeance estimation and are negligible when compared with measurement uncertainty in real industrial processes. Under extreme operating conditions involving low concentrations, low flow rates, and highly permeable species, the pseudotransient formulation proved to be a reliable auxiliary strategy, enabling robust convergence when suitable initial guesses were not readily available. The proposed framework is validated against experimental data from the literature and subjected to extensive convergence and sensitivity analyses, providing a reliable basis for simulation and for assessing computational feasibility in in-line and real-time monitoring-oriented applications. A full demonstration of digital-twin integration, online parameter updating, reduced-order coupling, and closed-loop control is beyond the scope of the present study and will be addressed in future work.

This study discusses the performance comparison of three numerical approaches, namely the Shooting method, the Finite Difference Method (FDM), and the Collocation method, in solving Boundary Value Problems (BVP) for three categories of problems: linear non-stiff, linear stiff, and non-linear. Each type of problem is tested with two types of boundary conditions, namely Dirichlet and Neumann. The evaluation is based on two main criteria: accuracy, measured using error norms with respect to the exact solution, and computational efficiency, quantified in terms of CPU execution time. The results show that in the non-stiff case with Dirichlet boundary conditions, the Shooting methods based on LSODA and RK5 provide very high accuracy with good efficiency, while the Finite Difference Method excels in efficiency but is slightly inferior in accuracy. Under Neumann boundary conditions, the Finite Difference Method tends to be less accurate, whereas the Collocation method delivers very good accuracy but with relatively lower efficiency. For stiff problems, the Shooting method maintains high accuracy, while the Finite Difference and Collocation methods show varying performance depending on the type of boundary condition. In the non-linear case, the Shooting method becomes the most accurate option, although with slightly lower efficiency compared to the Finite Difference Method. These findings provide practical guidance in selecting appropriate numerical methods for BVPs based on problem characteristics and boundary conditions.

Abstract A new numerical method is presented for the solution of the Burgers' equation(BE). Quadratic B-splines are used for the time integration of the Burgers' equation, while the spatial integration is handled using extended cubic B-spline collocation method. Thus, a combination of both quadratic and cubic B-spline collocation methods is utilized to obtain a system of algebraic equations, which is then solved to find the solutions of the BE. The applicability and simplicity of the proposed algorithm are demonstrated through three test problems

Coupled heat and mass transfer in electroosmotic microfluidic systems is difficult to investigate since the phenomenon is nonlinear and exhibits memory effects that cannot be accurately captured by classical integer-order models. The existing literature is mostly based on deterministic numerical methods and is not very keen on the predictive performance of data-driven models, namely artificial neural networks (ANNs), of parameter-sensitive transport processes. This research gap leads to this research that examines a fractal-fractional electroosmotic fluid flow model in a microchannel under sinusoidal wall oscillation and time-varying thermal and concentration fields, to improve the physical realism and predictive performance. The momentum, energy and concentration equations are constructed with proper physical assumptions and generalized with fractal-fractional derivative of singular kernel to include memory effects and geometric heterogeneity. The resulting non-dimensional equations are solved numerically using local radial basis function (LRBF) collocation method that offers a stable, mesh-free and very accurate computational framework to complex transport problems. Furthermore, an ANN-based surrogate model using Levenberg–Marquardt backpropagation algorithm is developed on the data generated by LRBF. The hybrid LRBF-ANN model provides an efficient solution and detailed parametric sensitivity analysis of each parameter, along with effective prediction of the heat and mass transfer characteristics. The ANN model performance is strictly checked through regression analysis, curve fitting, and error histogram analysis, revealing strong agreement with the numerical solution and excellent predictive accuracy. The parametric analysis shows that the fractal-fractional orders have a strong influence on the velocity, temperature distribution and mass diffusion profiles and provide a more expressive and vivid picture of the electroosmotic transport than classical formulations. These findings highlight the usefulness of ANN-assisted modelling of complex heat and mass transfer phenomena and provide a powerful computational tool to optimize electroosmotic transport in microfluidic systems, biomedical engineering systems, lab-on-a-chip systems and thermal management systems.

Thermal and mass management is one of the major components of material processing in various productive industries. A significant predictive analytical way is presented in this workto predict the heat and mass exchange during many industrial processes. More specifically, this work is very relevant to exploring the physics of ternary hybrid nanofluid (THNF), induced magnetic field, first-order chemical reaction, Darcy Forchheimer effects, suction/injection, modified heat and mass fluxes effects. The convective surface constraints are imposed to address the surface behavior of the dynamic disk. The typical potential Homann-type flow equations and the thermal and mass balance aspects are mathematically expressed via a set of nonlinear flow, thermal, mass, and induced magnetic field equations. However, the material composition is based on the effective inclusion of nanoparticles, CoFe2O4, ZnO and Au in the base liquid ethylene glycols. The influence of such physical factors is described through explicit graphical and numerical tables, while using the modified collocation method. Moreover, the valid behaviors of each controlling parameter are presented through the THNF temperature, concentration, velocity, resistive forces, and induced magnetic field. The ratio of stress and strain caused a significant enhancement in the flow components of the material. The thermal expansion factor reduced the material temperature significantly away from the surface. The Darcy and non-Darcy Forchiemer factors both cause a reduction in the flow speed of the composite THNF materials. The ratio of strain rate and disk linear deformation enhanced the flow speed, and a reduction is noted for the higher Eckert and thermal relaxation factor. The Biot number appeared in the surface condition, which decreased the thermal trend of the material. The time relaxation for the mass fraction factor diminished the material concentration, and the same is enhanced via the first chemical reaction factor. An authentic and justified comparison is generated to show the validity of the numerical method.

Optimal convergence of IgA collocation methods