Algebraic Equations Research Articles

A Robust numerical technique based on the chromatic polynomials for the European options regulated by the time-fractional Black–Scholes equation

AbstractRisk mitigation and control are critical for investors in the finance sector. Purchasing significant instruments that eliminate the risk of price fluctuation helps investors manage these risks. In theory and practice, option pricing is a substantial issue among many financial derivatives. In this scenario, most investors adopt the Black–Scholes model to describe the behavior of the underlying asset in option pricing. The exceptional memory effect prevalent in fractional derivatives makes it easy to understand and explain the approximation of financial options in terms of their inherited characteristics prompted by the given reason. Finding numerical solutions that are both successful and suitably precise is crucial when working with financial fractional differential equations. Hence, this paper proposes an innovative method, designated the Chromatic polynomial collocation method (CPM), for the theoretical study of the Time fractional Black–Scholes equation (TFBSE) that regulates European call options. The newly developed numerical algorithm CPM is on a functional basis of the Chromatic polynomials of Complete graphs (Kn) and operational matrices of the basis polynomials. The CPM transforms the TFBSE into a framework of nonlinear algebraic equations with the help of operational matrices and equispaced collocation points. The fractional orders in the PDE are concerned in the Caputo sense. The CPM findings further corroborate the results of the most recent numerical schemes to show the effectiveness of the suggested numerical algorithm.

Flow dynamics over cylinder in two‐dimensional benchmark problem: A finite element approximation

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second‐order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady‐state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the and errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.

Effects of Virtual Algebra Tiles on the Performance of Year 8 Students in Solving Algebraic Linear Equations

This study aims to examine the effects of Virtual Algebra Tiles (VAT) on the performance of 41 Year 8 students in solving algebraic equations. The VAT used in this research is accessible and free through a website called Didax. Quantitative and qualitative data were collected from Class 8X (21 students) and Class 8Y (20 students) through convenient sampling in a public secondary school. The Didax VAT was used as an intervention tool to aid students in solving algebraic equations. Quantitative data, in the form of pre-test, post-test, and questionnaire, was administered to the students and analysed to determine the statistical significance of the findings. Qualitative data were used in focus group interviews to explore students’ perceptions of Didax VAT further. The conclusions of this study indicated that the result of the paired sample t-test was not significant, the intervention had a small positive effect on student learning. The qualitative data revealed most students positively perceived using Didax VAT in their learning.

Exact analytical solution of the equations for a quasiequilibrium two-phase domain: permeability and interdendritic spacing

This study is concerned with the theoretical description of a quasi-stationary process of directional crystallization of binary melts and solutions in the presence of a quasi-equilibrium two-phase region. The quasi-equilibrium process is ensured by the fact that the system supercooling is almost completely compensated by heat released during the phase transformation. Quasi-stationarity of the process determining constancy of the crystallization rate is ensured by given temperature gradients in the solid and liquid phases. The system of heat and mass transfer equations and boundary conditions to them under these assumptions is dependent on a single spatial variable in the reference frame moving with the crystallization rate relative to a laboratory coordinate system. Exact analytical solutions to the formulated problem in parametric form are obtained. The parameter of the solution is the solid phase fraction in a two-phase region. The distributions of temperature and impurity concentration in the solid, liquid and two-phase regions of the crystallizing system, the rate of solidification, and the spatial coordinate in the two-phase region depending on the solid phase fraction in it are found. An algebraic equation for the solid phase fraction at the interface between the solid material and the two-phase region is derived. Exact analytical solutions show that the impurity concentration in the two-phase layer increases as the solid phase fraction increases. Moreover, the solid phase fraction at the interface solid phase — two phase region and its thickness increase as the temperature gradient in the solid phase and the solidification rate increase. The developed theory allows us to determine analytically the permeability of the two-phase region and a characteristic interdendritic spacing in it. Analytical solutions show that the relative permeability in the two-phase region increases from a certain value at the interface with the solid phase to unity at the interface with the liquid phase. The selection theory of stable dendritic growth allows us to determine analytically a characteristic interdendritic distance in the two-phase layer that decreases as the temperature gradient in the solid phase increases. An increase of impurity in the molten phase gives a decrease in the interdendritic spacing within a two-phase region.

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

A coated hypotrochoidal compressible liquid inclusion neutral to a hydrostatic stress field after relaxation by interface slip and diffusion

We study the steady-state response of a three-phase composite composed of an internal hypotrochoidal compressible liquid inclusion, an intermediate isotropic elastic coating and an outer isotropic elastic matrix with simultaneous interface slip and diffusion occurring on the solid–solid interface when the matrix is subjected to a uniform hydrostatic stress field. We design a neutral coated hypotrochoidal liquid inclusion that does not disturb the prescribed uniform hydrostatic stress field in the surrounding matrix. The neutrality is achieved when the plane-strain bulk modulus or the compressibility of the elastic matrix is determined by solving a system of simultaneous linear algebraic equations for given geometric and material parameters of the coated liquid inclusion.

Side-Channel Linearization Attack on Unrolled Trivium Hardware

This paper presents a new side-channel attack (SCA) on unrolled implementations of stream ciphers, with a particular focus on Trivium. Most conventional SCAs predominantly concentrate on leakage of some first rounds prior to the sufficient diffusion of the secret key and initial vector (IV). However, recently, unrolled hardware implementation has become common and practical, which achieves higher throughput and energy efficiency compared to a round-based hardware. The applicability of conventional SCAs to such unrolled hardware is unclear because the leakage of the first rounds from unrolled hardware is hardly observed. In this paper, focusing on Trivium, we propose a novel SCA on unrolled stream cipher hardware, which can exploit leakage of rounds latter than 80, while existing SCAs exploited intermediate values earlier than 80 rounds. We first analyze the algebraic equations representing the intermediate values of these rounds and present the recursive restricted linear decomposition (RRLD) strategy. This approach uses correlation power analysis (CPA) to estimate the intermediate values of latter rounds. Furthermore, we present a chosen-IV strategy for a successful key recovery through linearization. We experimentally demonstrate that the proposed SCA achieves the key recovery of a 288-round unrolled Trivium hardware implementation using 360,000 traces. Finally, we evaluate the performance of unrolled Trivium hardware implementations to clarify the trade-off between performance and SCA (in)security. The proposed SCA requires 34.5 M traces for a key recovery of 384-round unrolled Trivium implementation and is not applicable to 576-round unrolled hardware.

12518 Development Of A Four-compartment, Dynamic (In Vivo) Model For Insulin Disposition In Healthy Humans: Use Of Fick’s Law To Model Plasma-interstitial Insulin Flux

Abstract Disclosure: R.I. Dorin: None. C.R. Qualls: None. Introduction: Metabolic elimination of insulin (I) in vivo includes pathways mediated by high-affinity and potentially saturable insulin receptor binding; endogenous insulin secreted into the portal vein is subject to first pass elimination. The rate of insulin flux between vascular (Vp) and interstitial (Vi) compartments is often modeled by first order rate constants; here we analyzed an alternative approach using Fick’s law. Methods: The four-compartment, insulin (I) model is expressed by four differential equations that represent I flow (mass/time) in 4 volumes (V) (or compartments) having 3 corresponding elimination rate functions (α): (i) vascular plasma (Vp), (ii) interstitial (Vi, αi), (iii) hepatic (Vh, αh), and (iv) kidney (Vk, αk). Fick’s law was used to model diffusion of insulin between Vp and Vi; Michaelis-Menten like functions were used for saturable elimination functions (αh and αi); rates of hepatic (HPF) and renal (KPF) plasma flow were from literature. Realistic bounds for solved parameters (insulin permeability constant [k, L/min] and αi [min-[1]]) were developed by test bed analysis of published plasma (Ip) and interstitial (Ii) insulin concentration data. The inverse problem (estimating parameters) used Ip data from insulin-modified, frequently sampled iv glucose tolerance (IM-FSIVGT) studies performed in 40 healthy, non-diabetic women (1). Parameter solutions for individual subjects were obtained by iterative least squares method minimizing differences between model-predicted and measured Ip. Results: For infusion of exogenous I (ZI) at a constant rate (e.g., insulin clamp), reported steady state Ii/Ip data suggest that interstitial insulin elimination (αi) is conditionally saturable. Together with non-steady state data for Ip and Ii, initial estimates of k and αi were obtained. Steady state solutions give rise to four algebraic equations: (1) Ii/Ip = a, where a < 1 is consistent with experimental data, (2) Ih is a reduced version of a linear combination of Ip and endogenous insulin secretion (Zβ), where the factor of reduction (b) is <1, (3) Ik/Ip = d, where d<1, and (4) Ip is a linear combination of (i) ZI and (ii) the reduced (by factor b) Zβ. For Ip, this linear combination of ZI and reduced Zβ is also normalized by factor c (weighted sum of k, HPF, and KPF). Conclusions: (i) The proposed insulin flow model using Fick’s law was advantageous in providing realistic parameters and solutions that are tractable to physiologic interpretation, (ii) a simplified version of this model is generalizable to C-peptide, and (iii) since measurement of Ii is impractical in the clinical setting, the proposed prediction model may provide useful modeling adjunct to personalize pharmacokinetic and pharmacodynamic strategies for insulin administered by subcutaneous, iv, and hybrid closed-loop insulin pump routes. 1. Gower et al. Nutrition & Metabolism (2016) 13:2 Presentation: 6/1/2024

Simplified approach to retention times of narrow binary pulses in the case of ideal chromatography model and Langmuir isotherm

Analytical solution to ideal chromatography model has been established for binary Langmuir isotherm and rectangular injections. However, retention time of the less adsorbed species, which is of great theoretical and practical significance, cannot be given in a closed form and is conventionally solved by numerical integration with a floating boundary. A simplified approach is provided in this article. A 4th order algebraic equation was derived and used to solve the maximum concentration that can be further used to explicitly calculate retention time. Under most practical conditions, reliable initial guess can be easily acquired, allowing for the application of Newton-Raphson method for rapid determination of the root of the 4th order equation. In addition, derivatives of retention time with respective to isotherm parameters can be given in analytical forms.

Localized space-time Trefftz method for diffusion equations in complex domains

This paper introduces an advanced localized space-time Trefftz method tackling boundary value predicaments within complex two-dimensional domains governed by diffusion equations. In contrast to the widespread space-time collocation Trefftz method, which typically produces dense and ill-conditioned matrices, the proposed strategy employs a localized collocation scheme to remove these constraints. In particular, this is beneficial in multi-connected configurations or when dealing with significant variations in field values. To the best of our knowledge, this is the first space-time collocation Trefftz method adaptation, which is referred to as the localized space-time Trefftz method in this paper. The latter combines the space-time collocation Trefftz method principles, which allows to eliminate the need for mesh and numerical quadrature in its application. The localized space-time Trefftz method represents each interior node expressed as a linear blend of its immediate neighbors, while the space-time collocation Trefftz method applies numerical techniques within distinct subdomains. A sparse system of linear algebraic equations with internal points satisfying the governing equation, and boundary points satisfying the boundary conditions, allows to obtain numerical solutions using matrix systems. The localized space-time Trefftz method retains the easy-to-use properties and mesh-free structure of the space-time collocation Trefftz method, and it mitigates its ill-conditioning characteristics. Due to the localization principle and the consideration of overlapping subdomains, the solutions in the proposed localized space-time Trefftz method are more simply and compactly expressed compared with those in the space-time collocation Trefftz method, especially when dealing with multiply-connected domains. Numerical examples for simply-connected and multiply-connected domains are then provided to demonstrate the high precision and simplicity of the proposed localized space-time Trefftz method. The obtained results show that the latter has very high accuracy in solving two-dimensional diffusion problems. Compared with the traditional space-time collocation Trefftz method, the proposed mesh-free strategy yields solutions with higher precision while significantly reducing the instability.

A general alternating-direction implicit Newton method for solving continuous-time algebraic Riccati equation

The complex continuous-time algebraic Riccati equation (CCARE) is quadratic, which is closely related to the analysis of the optimal control problem. In this paper, we apply Newton method as the outer iteration and an efficient general alternating-direction implicit (GADI) method as the inner iteration to solve CCARE. Meanwhile, we propose the inexact Newton-GADI method to further improve the efficiency of the algorithm. We give the convergence analysis of our proposed method and prove that its convergence rate is faster than the classical Newton-ADI method. Finally, some numerical examples are given to illustrate the effectiveness of our algorithms and the correctness of the theoretical analysis.

Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

Purpose This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems. Design/methodology/approach Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method. Findings The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method. Originality/value Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.

Buckling disappearance via merging/divergence in a nonlinear three-d.o.f. system with linear constitutive law

The phenomenon of buckling disappearance, occurring in a parameter-dependent family of systems admitting a nontrivial fundamental path, is studied. Two different forms of disappearance are detected, namely: (i) the divergence, in which the critical load continuously tends to infinity, and (ii) the merging, in which two critical loads approach each other, coalesce, and then disappear at a finite value of the critical load. It is shown that the two phenomena can be exhibited by the same mechanical system, when a suitable elasto-geometric parameter is varied. More importantly, it is proved that merging continuously changes into divergence when a second parameter is changed. A paradigmatic system is chosen to illustrate the two forms of buckling, i.e., a three degree-of-freedom spherical pendulum, elastically constrained at the ground, loaded by a transverse force and/or a conservative couple, made of two longitudinal potential forces. The springs are taken elastically linear, to stress the fact that divergence not necessarily calls for introducing a nonlinear constitutive law, as also mentioned in literature. Only a linear bifurcation analysis is carried out here, aimed to find the bifurcation points along the nonlinear fundamental path. However, due to the presence of non-negligible prestrains, such a bifurcation problem is governed by nonlinear algebraic equations, whose number of roots cannot be predicted in advance.

Policy iteration based cooperative linear quadratic differential games with unknown dynamics

This article investigates the Pareto optimality of infinite horizon cooperative linear quadratic (LQ) differential games by policy iteration technique where the system dynamics are partially or completely unknown. Firstly, the policy iteration algorithm for the approximate solutions of the corresponding algebraic Riccati equation (ARE) without any prior knowledge of the matrix parameters of the dynamic system is derived by collecting the input and state information of each player. Secondly, when the presented specific rank condition is satisfied, the convergence of the proposed algorithm is rigorously demonstrated by recursion. Moreover, the weighting approach is employed to obtain the Pareto optimal strategy and the Pareto optimal solutions on the basis of the convex optimization theory. Finally, simulation results are reported to verify the feasibility and correctness of the proposed theoretical results.

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.

Numerical solution of fuzzy non-linear equations under generalized trapezoidal fuzziness by Newton–Raphson method

In the real world, we are faced with many problems that cannot be easily described by crisp values, as there is always some uncertainty involved in real-life situations and in those situations, we end up receiving fuzzy values rather than crisp ones in those situations. Some preliminary concepts of fuzzy sets and operations on trapezoidal fuzzy numbers incorporated with the membership function have been discussed here. This paper aims to demonstrate the advantages of the newly developed fuzzified newton-raphson method with the help of trapezoidal fuzzy numbers for solving algebraic nonlinear equations arising in fuzzy environments. The algorithm of newton-raphson techniques for solving the same has been described. This paper presents several examples of how the new fuzzified newton-raphson method works.

Physics-informed neural networks for dynamic process operations with limited physical knowledge and data

In chemical engineering, process data are expensive to acquire, and complex phenomena are difficult to fully model. We explore the use of physics-informed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semi-explicit differential–algebraic equation systems and scarce process data. In particular, we focus on estimating states for which neither direct observational data nor constitutive equations are available. We propose an easy-to-apply heuristic to assess whether estimation of such states may be possible. As numerical examples, we consider a continuously stirred tank reactor and a liquid-liquid separator. We find that PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. We thus show that PINNs are capable of modeling processes when relatively few experimental data and only partially known mechanistic descriptions are available, and conclude that they constitute a promising avenue that warrants further investigation.

Hyperslip velocity of melting ice sliding down inclined parallel ridges

A geometric and physical model for melting ice sliding over inclined superhydrophobic (SH) surfaces with parallel ridges is presented. By analyzing the micro-shear flows of molten liquid films between the ice layer and SH surfaces, the hyperslip velocities of melting ice sliding are investigated. The stick-slip boundary condition of the SH surface is used to establish the dual-series equations analytically, and the numerical solutions are implemented by truncating Fourier series and transforming the dual-series equations into linear algebraic equations to determine the hyperslip velocities of melting ice sliding. The numerical results indicate that the non-dimensional hyperslip velocities increase nonlinearly from near 0 to approximately 1.1 for longitudinal sliding and from near 0 to approximately 0.55 for transverse sliding with an increasing air groove ratio (a). The hyperslip velocities increase with increasing δ at the beginning initially (δ < 1), after which they tend toward asymptotic solutions as δ = 1. The hyperslip velocity ratio (Wh/Uh) shows that longitudinal ridges are at least twice as effective as transverse ridges in enhancing the ice hyperslip velocity, with the velocities accounting for more than 60% of the ice sliding velocities for arbitrary θ at a = 0.95 and δ = 0.1. The relative deviations between the numerical and asymptotic solutions are less than 5% at δ = 1, with the maximum relative deviation occurring at a = 0.65 for arbitrary θ.

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

Entropy optimization of MHD second-grade nanofluid thermal transmission along stretched sheet with variable density and thermal-concentration slip effects

The goal of present investigation is to explore the influence of exponential variable density and entropy optimization on second-grade nanofluid heating efficiency and mass-concentration transmission along extended surface using external magnetic-field and temperature-concentration slip effects. To enhance the motion of nanoparticles and thermal efficiency, the influence of exponential form of temperature-based density on magnetically charged second-grade nanomaterial is main novelty of this research. For higher temperature difference, the entropy optimization is used. The defined formulation of stream functions and similarities are used to convert leading second-grade nanofluid model into ordinary differential form. The efficient Keller box method and Newton Raphson technique are applied to compute numerical results. The final algebraic equations are solved through global matrix for unknown physical quantities. The consequence of all physical constraints on velocity/U profile, temperature/θ field, concentration/ϕ shapes, skin friction coefficient, Nusselt and Sherwood number are analyzed pictorially and numerically. The following range of parameters 0.1 ≤ ≤ 2.0, 0.0 ≤ ≤ 1.2, 0.1 ≤ ≤ 2.0, 0.07 ≤ ≤ 7.0, 0.01 ≤ ≤ 0.8, 0.01 ≤ ≤ 0.9 is used. It is found that velocity field increases with maximum amplitude as variable density, magnetic force and temperature-slip constraint. It is noted that the slip behavior in temperature field and concentration field are increased with convective boundary conditions. It is depicted that local Nusselt quantity and local Sherwood quantity increases as buoyancy force and Prandtl coefficient increases.