Efficient Numerical Technique Research Articles

Biomagnetic flow with magnetic particles over a continuously moving sheet affected by a magnetic dipole

This research concentrates on the 2-D, steady, laminar, viscous, incompressible boundary layer flow of a biomagnetic fluid containing two different magnetic particles (CoFe2O4andFe3O4) over a continuously moving horizontal plate in the presence of a magnetic field generated by a magnetic dipole. For the mathematical formulation the comprehensive concept of Biomagnetic Fluid Dynamics (BFD) is adopted incorporating the principles of FerroHydroDynamics (FHD) and MagnetoHydroDynamics (MHD). The physical problem which is constituted by a coupled system of Partial Differential Equations (PDEs) along with corresponding boundary conditions, is transformed into a coupled system of nonlinear Ordinary Differential Equations (ODEs) subject to analogous boundary conditions by establishing newly simplified similarity transformations. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on a finite difference algorithm. Verification of this work has been also done by comparing the obtained results with previously published results and found in quite good agreement. The significant effects caused by the variation of the governing parameters such as the skin friction, heat transfer rate and wall pressure are presented more intricately. It has been contemplated that including magnetic particles with pure blood enhances the impact of the magnetic field on the flow, temperature and pressure profiles which could be of interest engineering implementations, like, magnetic drug delivering in blood cells, separating RBCs (Red Blood Cells), controlling the flow of blood during surgeries, treating cancer by producing magnetic hyperthermia etc.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

Borel control and efficient numerical techniques to solve the Allen–Cahn equation governed by temporal multiplicative noise

In this manuscript, we investigate a stochastic version of the Allen–Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations.

Liquid induced vibrations of partially liquid-filled elastic cylindrical-conical shells

The main objective of this study to develop efficient numerical techniques based on coupled finite and boundary element methods to estimate natural vibration frequencies in compound reservoirs with liquid. Elastic structures composed of cylindrical and conical shells connected by rings have been examined. The space between the shells is filled with an ideal, incompressible fluid. The advantage of the proposed approach lies in the ability to investigate both free and forced vibrations of empty and liquid-filled shells structures within the framework of the single computer technology. The benchmark test calculations demonstrated the high accuracy and efficiency of the proposed approach. The novelty and practical value of the obtained results lie in the ability to investigate fuel tanks of complex shapes under various conditions.

Explicit Representations of Solutions and Parameter Estimations Problems of Time Delay Tumor Cells Model

We consider ordinary differentials of cancerous tumor cells model with time delay that illustrates and represents the interactions of the tumor cells with effector cells include immune response of natural, dendritic and cytotoxic cells. The time delay is incorporated here into the model to justify the time required to stimulate the effector cells. This also help to study the behavior of the system without using all kinds of treatments. We study the stability of the model that confirms a free tumor steady state and displays two equilibrium points. The numerical simulations show that growing of the tumor cells reduces and the effector cells increase after few days for different values of time delay up to , even without considering the treatment strategies in the model. We present an efficient numerical technique that shows how we can first represent the periodic solution of the tumor nonlinear equation and then effectively estimate the unknown parameters of the model.

Efficient hybridized numerical scheme for singularly perturbed parabolic reaction–diffusion equations with Robin boundary conditions

An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov–Shishkin and Vulanović–Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

Inference of geoacoustic model parameters from acoustic field data: Perspectives on Geoacoustic Inversion

Estimation of parameters of geoacoustic models from acoustic field data has been a central research theme in acoustical oceanography and ocean acoustics. During the past several decades, highly efficient numerical inversion techniques have been developed that provide model parameter estimates and their uncertainties based on statistical inference methods. However, the methods are model-based and the inversions are prone to errors related to model mismatch. In any event, the inversions can generate only effective models of the true structure of the ocean bottom, which is generally highly variable over relatively small spatial scales in range and depth. There are also questions about the theory for modelling sound propagation in porous sediment media that raise doubt about the validity of inversion results. In most inversions, a visco-elastic theory is used, but is this the most appropriate propagation model? Another question is about the impact of neglecting shear waves in geoacoustic models. Most inversions assume a fluid model of the ocean bottom. This paper revisits issues that have raised questions about limitations of geoacoustic inversion methods, and discusses the impact of various mitigation measures that have been applied. The paper concludes with musings about new inversion techniques based on machine learning.

Fractal Fractional Derivative Models for Simulating Chemical Degradation in a Bioreactor

A three-differential-equation mathematical model is presented for the degradation of phenol and p-cresol combination in a bioreactor that is continually agitated. The stability analysis of the model’s equilibrium points, as established by the study, is covered. Additionally, we used three alternative kernels to analyze the model with the fractal–fractional derivatives, and we looked into the effects of the fractal size and fractional order. We have developed highly efficient numerical techniques for the concentration of biomass, phenol, and p-cresol. Lastly, numerical simulations are used to illustrate the accuracy of the suggested method.

A swarm-intelligence based formulation for solving nonlinear ODEs: γβII-(2+3)P method

A reverse engineering approach is taken to develop efficient numerical techniques for solving nonlinear ODEs through the numerical solution of a particular problem in engineering. The problem is the dynamic analysis of a mechanical system under a severely irregular seismic signal or earthquake load. It is a challenging problem which requires strong formulation to be tackled. Its solution is a rich source of information that could be exploited through artificial intelligence (AI) and statistical computation. This work devises a new strategy to conduct such a task and construct a series of 3∼5 points high-precision time integrators. The integrators belong to single-step two-derivative class of methods. The interconnectivity relationships between the weights of the integrators are formulated by linear regression and heuristics. These relationships make it possible to generate a wide spectrum of ODE-solving techniques in a continuous weighting space. In an innovative attempt, strong Hermite interpolators are constructed to couple with the integrators and improve the accuracy of the presented algorithms. They precisely evaluate the instep points of the integrator. All these issues are appropriately collected in the presented methodology. Four classes of weighting rules are presented for the integrators: 1) Symmetry weighting rule (SWR) 2) Consistency weighting rule (CWR), 3) Fundamental weighting rule (FWR), and 4) Auxiliary weighting rule (AWR). The first two rules are adopted from outward form of the well-configured quadratures available in literature. The third and the most important one, the so-called FWR, is disclosed here for the given integrator. It formulates the correlation between the weights of an optimal integrator. Finding the min-error solution of the vibration equation guides us to the FWR. Evolutionary grey wolf optimizer (GWO), accompanied with a statistical multilinear regression, extracts the FWR. This research pioneers the application of swarm-intelligence in formulation of the FWRs for the given two-derivative integrator. Finally, all the weighting rules and Hermite interpolators are combined into some unified algorithms so-called γβII−q+rP algorithms to efficiently cope with initial value ODEs. Great performance of the presented methods is demonstrated through the numerical examples.

The Impact of Discrete Element Method Parameters on Realistic Representation of Spherical Particles in a Packed Bed

Packed bed reactors play a crucial role in various industrial applications. This paper utilizes the Discrete Element Method (DEM), an efficient numerical technique for simulating the behavior of packed beds of particles as discrete phases. The focus is on generating densely packed particle beds. To ensure the model accuracy, specific DEM parameters were studied, including sub-step and rolling resistance. The analysis of the packed bed model extended to a detailed exploration of void fraction distribution along radial and vertical directions, considering the impact of wall interactions. Three different samples, spanning particle sizes from 0.3 mm to 6 mm, were used. Results indicated that the number of sub-steps significantly influences void fraction precision, a key criterion for comparing simulations with experimental results. Additionally, the study found that both loosely and densely packed beds of particles could be accurately represented by incorporating appropriate values for rolling friction. This value serves as an indicator of both inter-particle friction and friction between particles and the walls. An optimal rolling friction coefficient has been thereby suggested for the precise representation for the densely packed bed of spherical char particles.

Two efficient numerical techniques for solutions of fractional shallow water equation

In this paper, the nonlinear shallow water equation appears in the mathematical modeling of tsunami wave propagation along the coastline of an ocean is solved numerically. The model investigates fractional derivatives and is presented through a system of nonlinear partial differential equations. The variational iteration method (VIM) and the residual power series method (RPSM) have been applied to the fractional system of equations for different parameters. In addition, tsunami wave velocity and run-up height with respect to coast slope and sea depth are analyzed for different time and order in detail. A detailed error analysis is presented for tsunami velocity and tsunami wave propagation. The results obtained are compared to other existing results to show the efficiency and reliability of the proposed methods.

An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method

<abstract><p>In this article, the generalized Rosenau-Korteweg-de Vries-regularized long wave (GR–KDV–RLW) equation was numerically studied by employing the Fourier spectral collection method to discretize the space variable, while the central finite difference method was utilized for the time dependency. The efficiency, accuracy, and simplicity of the employed methodology were tested by solving eight different cases involving four examples of the single solitary wave with different parameter values, interactions between two solitary waves, interactions between three solitary waves, and evolution of solitons with Gaussian and undular bore initial conditions. The error norms were evaluated for the motion of the single solitary wave. The conservation properties of the GR-Rosenau–KDV–RLW equation were studied by computing the momentum and energy. Additionally, the numerical results were compared with the previous studies in the literature.</p></abstract>

Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation

<abstract><p>This study focuses on developing efficient numerical techniques for solving the fractional Keller-Segel (KS) model, which is critical in explaining chemotaxis events. Within the Caputo operator framework, the study applied two unique methodologies: The Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). These approaches were used to find precise solutions to the fractional KS equation, resulting in a better understanding of chemotactic behavior in biological systems. The comparative examination of the ARPSM and ATIM revealed their distinct strengths and applications in solving complicated fractional models. The work advances numerical approaches for fractional differential equations and improves our understanding of chemotaxis dynamics using a precise modeling approach.</p></abstract>

An Efficient Numerical Technique for the Simulation of Charge Transport in Polymeric Dielectrics

An Efficient Numerical Technique for the Simulation of Charge Transport in Polymeric Dielectrics

Solution of local fractional generalized coupled Korteweg–de Vries (cKdV) equation using local fractional homotopy analysis method and Adomian decomposition method

ABSTRACT In this study, time-fractional coupled Korteweg–de Vries (cKdV) equations are solved using an efficient and reliable numerical technique. The classical cKdV system has been generalized into the time-fractional cKdV system. We employ the local fractional homotopy analysis method (LFHAM) and the Adomian decomposition method (ADM) to propose an approximate solution for fractional cKdV equations. Both approaches determined findings are compared together. The findings clearly demonstrate that the suggested methods are appropriate and efficient for handling both linear and nonlinear issues in engineering and sciences. To demonstrate the suggested approaches' competencies, examples are provided. Convergent series form has been used to make the solutions. The relevance of the techniques is illustrated through graphic representations of the solution.

Magneto-thermal convection and entropy production of hybrid nanofluid in an inclined chamber having a solid block

PurposeThis paper aims to examine the thermal transmission and entropy generation of hybrid nanofluid filled containers with solid body inside. The solid body is seen as being both isothermal and capable of producing heat. A time-dependent non-linear partial differential equation is used to represent the transfer of heat through a solid body. The current study’s objective is to investigate the key properties of nanoparticles, external forces and particular attention paid to the impact of hybrid nanoparticles on entropy formation. 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. Hybrid nanofluid has been proven to have useful qualities, making it an attractive coolant for an electrical device. The findings would help scientists and engineers better understand how to analyse convective heat transmission and how to forecast better heat transfer rates in cutting-edge technological systems used in industries such as heat transportation, power generation, chemical production and passive cooling systems for electronic devices.Design/methodology/approachThermal transmission and entropy generation of hybrid nanofluid are analysed within the enclosure. The domain of interest is a square chamber of size L, including a square solid block. The solid body is considered to be isothermal and generating heat. The flow driven by temperature gradient in the cavity is two-dimensional. The governing equations, formulated in dimensionless primitive variables with corresponding initial and boundary conditions, are worked out by using the finite volume technique with the SIMPLE algorithm on a uniformly staggered mesh. QUICK and central difference schemes were used to handle convective and diffusive elements. In-house code is developed using FORTRAN programming to visualize the isotherms, streamlines, heatlines and entropy contours, which are handled by Tecplot software. The influence of nanoparticles volume fraction, heat generation factor, external magnetic forces and an irreversibility ratio on energy transport and flow patterns is examined.FindingsThe results show that the hybrid nanoparticles concentration augments the thermal transmission and the entropy production increases also while the augmentation of temperature difference results in a diminution of entropy production. Finally, magnetic force has the significant impact on heat transfer, isotherms, streamlines and entropy. It has been observed that the external magnetic force plays a good role in thermal regulations.Research limitations/implicationsHybrid nanofluid is a desirable coolant for an electrical device. Various nanoparticles and their combinations can be analysed. Ferro-copper hybrid nanofluid considered with the help of prevailing literature review. The research would benefit scientists and engineers by improving their comprehension of how to analyses convective heat transmission and forecast more accurate heat transfer rates in various fields.Practical implicationsDue to its helpful characteristics, ferrous-copper hybrid nanofluid is a desirable coolant for an electrical device. The research would benefit scientists and engineers by improving their comprehension of how to analyse convective heat transmission and forecast more accurate heat transfer rates in cutting-edge technological systems used in sectors like thermal transportation, cooling systems for electronic devices, etc.Social implicationsEntropy generation is used for an evaluation of the system’s performance, which is an indicator of optimal design. Hence, in recent times, it does a good engineering sense to draw attention to irreversibility under magnetic force, and it has an indispensable impact on investigation of electronic devices.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyse convective energy transport and entropy generation in a chamber with internal block, which is capable of maintaining heat and producing heat. Effects of irreversibility ratio are scrutinized for the first time. Analysis of convective heat transfer and entropy production in an enclosure with internal isothermal/heat generating blocks gives the way to predict enhanced heat transfer rate and avoid the failure of advanced technical systems in industrial sectors.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

Numerical Investigation with Convergence and Stability Analyses of Integro-Differential Equations of Second Kind

In this paper, integro-differential equations are solved by using an efficient numerical technique, namely, Multistage Optimal Homotopy Asymptotic method. The existence and uniqueness of solutions are established by the application of Lipschitz condition. Convergence of approximate solutions along with stability are also carried out. Some examples are solved to highlight the vital characteristics of the applied numerical scheme. Error estimation and comparison of derived results with existing exact solutions and those results which already available in the literature through graphical illustrations and tables reveal that Multistage Optimal Homotopy Asymptotic algorithm is more efficient and fruitful.

Solving boundary value problems in heterogeneous catalysis with orthogonal collocation and arc‐length continuation

AbstractAn important part of the education of chemical engineers involves the design and assessment of heterogeneous catalytic reactions. They pose fundamental phenomena and constitute invaluable industrial processes. Progressing in this field, especially in graduate‐level courses, entails solving nonlinear differential equations systems, which can only be achieved by using efficient and reliable numerical techniques. In this work, we solve nonisothermal one‐dimensional reaction‐diffusion BVP considering convection in the boundary conditions, by applying orthogonal collocation and arc‐length continuation. Particularly, we predict the effectiveness factor as a function of the Thiele modulus for the main three particle geometries: plane slab, cylinder, and sphere. For chemical engineering students, such problems are at the zenith of the discipline, and the results of such calculations can elucidate basic and advanced concepts of mass and heat transfer as well as chemical reaction engineering. Both Chebyshev and shifted Legendre polynomials were used to transform the boundary value problem into a set of algebraic nonlinear equations. In addition, we also compare our findings against the available analytic solutions whenever possible. Codes in Scilab, Matlab®, and Mathematica© were developed for the solution of such problems and are available in the Supporting Information. There are two possible approaches to compute effectiveness factors: (a) integration as per definition versus (b) differentiation and nonlinear system. It is demonstrable that software performance is dependent on the approach, and that the differentiation approach yields better results.