Efficient Numerical Technique Research Articles

Capturing Long-Range Memory Structures with Tree-Geometry Process Tensors

We introduce a class of quantum non-Markovian processes—dubbed process trees—that exhibit polynomially decaying temporal correlations and memory distributed across timescales. This class of processes is described by a tensor network with treelike geometry whose component tensors are (1) causality-preserving maps (superprocesses) and (2) locality-preserving temporal change-of-scale transformations. We show that the long-range correlations in this class of processes tends to originate almost entirely from memory effects and can accommodate genuinely quantum power-law correlations in time. Importantly, this class allows efficient computation of multitime correlation functions. To showcase the potential utility of this model-agnostic class for numerical simulation of physical models, we show how it can efficiently approximate the strong memory dynamics of the paradigmatic spin-boson model, in terms of arbitrary multitime features. In contrast to an equivalently costly matrix-product-operator representation, the ansatz produces a fiducial characterization of the relevant physics. Finally, leveraging 2D tensor-network renormalization-group methods, we detail an algorithm for deriving a process tree from an underlying Hamiltonian via the Feynmann-Vernon influence functional. Our work lays the foundation for the development of more efficient numerical techniques in the field of strongly interacting open quantum systems, as well as the theoretical development of a temporal renormalization-group scheme. Published by the American Physical Society 2024

Flow of a biomagnetic fluid embedded by magnetic dipole over a continuously moving sheet

AbstractThis paper is concerned with the investigation of steady, two‐dimensional, and laminar boundary layer flow of a biomagnetic fluid over a continuously moving sheet in the presence of a magnetic dipole. The magnetic field resulting from the dipole is contemplated to be strong enough to saturate the biofluid. The magnetization of the fluid is regarded to be a linear function of temperature. The solution procedure involves the reduction of a nonlinear system of coupled PDEs into ODEs that comprise five parameters. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on the finite difference algorithm. The velocity, as well as temperature profiles within the boundary layer, are illustrated at specified values of free stream velocity (), wall velocity () and ferrohydrodynamic interaction parameter (). The demonstration of the Nusselt number and friction factor are achieved for various governing parameters. Considering a specified Prandtl number () and normalized velocity difference , higher values of the friction factor are obtained for increasing and than that of . Whereas for the Nusselt number, we attain higher values for decreasing and . Moreover, an increase in the velocity ratio results in a decrease in both heat transfer rate as well as friction factor. Furthermore, as increases, the heat transfer rate decreases yet we get higher values for higher and . In the case of friction factor, it increases with increasing and gives higher values for lower and higher . We have also depicted the streamlines for the 2‐D boundary layer flow of the biomagnetic fluid for different . The graphical results manifest that the flow field is greatly impacted by the ferrohydrodynamic field, which could be of interest in medical as well as bioengineering implementations, like, magnetic drug delivery in blood cells, separating RBCs as well as controlling the flow of blood during surgical procedures.

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.

Implicit EXP-RBF techniques for modeling unsaturated flow through soils with water uptake by plant roots

Modeling unsaturated flow through soils with water uptake by plan root has many applications in agriculture and water resources management. In this study, our aim is to develop efficient numerical techniques for solving the Richards equation with a sink term due to plant root water uptake. The Feddes model is used for water absorption by plant roots, and the van-Genuchten model is employed for capillary pressure. We introduce a numerical approach that combines the localized exponential radial basis function (EXP-RBF) method for space and the second-order backward differentiation formula (BDF2) for temporal discretization. The localized RBF methods eliminate the need for mesh generation and avoid ill-conditioning problems. This approach yields a sparse matrix for the global system, optimizing memory usage and computational time. The proposed implicit EXP-RBF techniques have advantages in terms of accuracy and computational efficiency thanks to the use of BDF2 and the localized RBF method. Modified Picards iteration method for the mixed form of the Richards equation is employed to linearize the system. Various numerical experiments are conducted to validate the proposed numerical model of infiltration with plant root water absorption. The obtained results conclusively demonstrate the effectiveness of the proposed numerical model in accurately predicting soil moisture dynamics under water uptake by plant roots. The proposed numerical techniques can be incorporated in the numerical models where unsaturated flows and water uptake by plant roots are involved such as in hydrology, agriculture, and water management.

Stability Analysis of Hermite Collocation Method for Pulp Washing Models

Pulp washing is concerned with detaching cellulose fibres from black liquor with the use of a minimal amount of wash liquor. An efficient numerical technique of hermite collocation method is used for the solution of mathematical models related to pulp washing. The linear and non-linear models are solved using Quintic Hermite collocation method with Dirichlet’s and mixed Robin’s boundary conditions. Numerical solution of the models are derived using MATLAB ode15s. This study deals with the justification of accuracy of the method with stability analysis. The present method is more convenient, simple and elegant for solving the two point boundary value problems and the results found are very much stable from numerical point of view

On nonlinear dynamical analysis of a fractional-order two-strains Nipah virus model

The Nipah virus (NiV) is one of the most lethal viruses which can infect humans and lead to fatal encephalitis. The recent significant awareness of NiV is due to its elevated death rate and effective transmission capabilities among humans. With its recurrent outbreaks and exceptionally high mortality rate, the NiV infections have emerged as one of the most concerning hazards to public health. The exploration of NiV and its characteristics revealed that NiV has two distinct strains, namely, the Malaysia (NiVM) strain and the Bangladesh (NiVB) strain. In this paper, we propose a novel Caputo fractional order mathematical model to simulate the dynamics of the two strains NiV. The positivity and boundedness of the model’s solutions are investigated. The existence and asymptotic stability of the equilibrium points of the model are examined. The analysis of basic reproduction number is presented to determine whether the infection will die out or persist. The effects of key parameters of the model on its dynamical behaviors are also explored. Finally, efficient numerical technique is used to confirm the analytical results through detailed numerical simulations.

Non-polynomial spline method for computational study of reaction diffusion system

This work addresses an efficient and new numerical technique utilizing non-polynomial splines to solve system of reaction diffusion equations (RDS). These system of equations arise in pattern formation of some special biological and chemical reactions. Different types of RDS are in the form of spirals, hexagons, stripes, and dissipative solitons. Chemical concentrations can travel as waves in reaction-diffusion systems, where wave like behaviour can be seen. The purpose of this research is to develop a stable, highly accurate and convergent scheme for the solution of aforementioned model. The method proposed in this paper utilizes forward difference for time discretization whereas for spatial discretization cubic non-polynomial spline is used to get approximate solution of the system under consideration. Furthermore, stability of the scheme is discussed via Von-Neumann criteria. Different orders of convergence is achieved for the scheme during a theoretical convergence test. Suggested method is tested for performance on various well known models such as, Brusselator, Schnakenberg, isothermal as well as linear models. Accuracy and efficiency of the scheme is checked in terms of relative error (E R ) and L ∞ norms for different time and space step sizes. The newly obtained results are analyzed and compared with those available in literature.

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.

INVESTIGATING VIRUS SPREAD ANALYSIS IN COMPUTER NETWORKS WITH ATANGANA–BALEANU FRACTIONAL DERIVATIVE MODELS

This paper proposes a mathematical model to investigate the dynamic behaviors of a modified computer virus model using the Atangana–Baleanu fractional derivative in the Caputo sense, aiming to elucidate the connection between its parameters and network attributes. By introducing a relatively new numerical method, we address the memory-dependent and nonlocal features of the system. The existence and uniqueness of the model’s solution are confirmed. To explore solution trajectories and assess the impact of various input factors on computer virus dynamics, we employ an efficient numerical technique. Our simulations provide insights into the consequences of fractional order, anti-virus measures, asymptotic fraction, damage rate, and removal rate in the system. These findings illuminate the relationships between model parameters, facilitating the design of networks that minimize the risk of virus outbreaks and prevent future cyber threats. By identifying critical factors involved in the progression of viruses, these results enable the development of more effective defense mechanisms.

Dynamic behaviors of a modified computer virus model: Insights into parameters and network attributes

Securing computers is crucial to prevent data breaches, identity theft, and financial losses. Virus incursions disrupt operations, causing downtime and costly repairs. Protective measures, including anti-virus software and cybersecurity practices, maintain network integrity and reduce the spread of malware. Combining robust cybersecurity with green computing strategies ensures efficient energy usage and sustainable network environments, safeguarding against viruses while contributing to both security and environmental goals In this study, we explore the dynamic behaviors of a modified version of the computer virus model and elucidate the connection between its parameters and network attributes. We employ Banach’s and Schaefer’s fixed-point theorems to assess the existence and uniqueness of solutions of the suggested model. Furthermore, we establish sufficient conditions for Ulam–Hyers stability within the envisioned computer virus model. To analyze solution trajectories and the impact of various input factors on computer virus dynamics, we utilize an efficient numerical technique, providing insight into the relationships between model parameters and enabling the design of networks that minimize the risk of virus outbreaks under various bifurcation conditions.

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.

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.