Linear Equations Research Articles

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Zero distribution of solutions of some linear differential equations

Assuming that A ( z ) = B 1 ( z ) e P ( z ) + B 2 ( z ) e ϱp ( z ) + Q ( z ) , where B 1 ( z ) , B 2 ( z ) , p ( z ) are nonzero polynomials, ϱ ∈ ( 0 , 1 ) , and Q ( z ) is an entire function of order less than deg ⁡ p , we study the oscillation of solutions of the linear differential equation f ( k ) + A ( z ) f = 0 . Some conditions on A ( z ) are given to guarantee that any non-trivial solution f satisfies λ ( f ) = ∞ or λ ( f ) ≥ σ ( A ) , where σ ( f ) and λ ( f ) denote respectively the order of f and the exponent of convergence of zeros of f. We also find some types of linear differential equations with zero-free solutions.

The Route to ROSC: Evaluating the Impact of Route and Timing of Epinephrine Administration in Out-of-Hospital Cardiac Arrest Outcomes

Objectives Previous investigations comparing intraosseous (IO) and intravenous (IV) epinephrine delivery in out-of-hospital cardiac arrest (OHCA) suggest that epinephrine is oftentimes more expeditiously administered via the IO route, but this temporal benefit doesn’t always translate to clinical benefit. However, very few studies adequately controlled for indication and resuscitation time biases, making the influence of first epinephrine route on OHCA outcomes unclear. To determine the association between first epinephrine route and return of spontaneous circulation (ROSC) while controlling for resuscitation time bias and other potential confounders. Methods We conducted a retrospective analysis using the 2020 ESO Data Collaborative dataset. Adult patients with a witnessed, non-traumatic OHCA prior to EMS arrival were included. Logistic regression was used to determine the association between medication route and ROSC. Linear regression was then used to calculate the probability of ROSC for each route across all call receipt-to-drug delivery intervals. Using these linear equations, the call receipt-to-drug delivery intervals were calculated that would yield equivalent probabilities of ROSC between the IV and IO routes. Results Data were available for 10,350 patients, of which 27.4% presented with a shockable rhythm, 29.7% received bystander CPR, and 39.6% experienced ROSC. After controlling for confounders, IO epinephrine was associated with decreased likelihood of ROSC (OR = 0.77, p < 0.001). The linear regression models provided differing slope coefficients for ROSC between each route, with the IV route associated with a higher likelihood of ROSC for any given call receipt-to-drug-delivery interval. From these equations, the additional time allowed to establish an IV and administer epinephrine intravenously beyond the time required for IO delivery, yet with an equivalent predicted probability of ROSC via the IO route, was calculated. This additional time interval for intravenous administration declined linearly from 9 min at a call receipt-to-intraosseous epinephrine interval of 4 min to no additional time at a call receipt-to-intraosseous epinephrine interval of 29 min. Conclusions This retrospective analysis of a national EMS database revealed that IO epinephrine was negatively associated with ROSC. Additionally, there appears to be a finite time window during which intravenous epinephrine remains superior to the intraosseous route even if there are brief initial delays in IV drug delivery.

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.

Meshless method and shifted Chebyshev polynomials for solving a class of VIEs of the third kind

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach.

An Interactive Fluid–Solid Approach for Numerical Modeling of Composite Metal Foam Behavior under Compression

Composite metal foams (CMF) are renowned for their high strength‐to‐density ratio, high stiffness, and energy absorption capabilities. Homogenized finite element models of CMF have been numerically solved to understand the mechanical behavior of the material under a variety of external loading conditions. This work aims to pioneer a comprehensive finite element model for steel–steel composite metal foam by incorporating the interactions between embedded stainless‐steel hollow spheres with entrapped air inside stainless‐steel matrix. The material behavior of hollow spheres, surrounding matrix, and air are modeled using Johnson–Cook (JC) plasticity, Deshpande–Fleck (D–F) foam, and linear polynomial equation of state in LS DYNA. Further, the finite element model utilizes a combination of Lagrangian solid elements and meshfree smooth particles hydrodynamics with appropriate contacts to effectively model the interaction of entrapped air within stainless‐steel hollow spheres with surrounding metallic spheres and matrix. The strain rate and the crosshead velocity of 65 s−1 and 2.4765 m s−1 are used for quasistatic compression analysis. Finally, the results obtained from the computational model are compared and validated using previously reported experimental quasistatic compression data. The numerical model corroborates stress–strain response of CMF with 5.6% error for plateau stresses average within 25% and 30% strain.

Comparative Analysis of the Equations Used in Modeling Individual Lactation Curves in Anatolian Water Buffalo (Bubalus bubalis)

In this study, the equations used in modeling the individual lactation curves of Anatolian Water Buffalo were comparatively examined and the most suitable model or models were tried to be determined. For this purpose, individuals with 8, 9 and 10 lactation records were selected among 8057 Anatolian Water Buffalo and a total of 1591 individual lactation curves were modeled. Guo-Salve, Grossman, Cappio-Borlino, Parabolic Exponential, Cobby and Le Du, Logarithmic Quadratic, Wilmink, Logarithmic Linear, Quadratic, Inverse Polynomial and Wood equations were used in the study. When comparing the models, coefficient of determination, mean squared error, Durbin-Watson autocorrelation test, the Akaike Information Criterion were considered. Among the models used in the study, the Cobby and Le Du, Wood and Logarithmic Quadratic models gave the best results, as opposed to the Guo-Salve model.

New and Modified Homotopy Perturbation Methods for Addressing Burger’s Non-linear Equation

Objectives: The aim of this study is to find a novel solution procedure for solving a fluid dynamical problem, especially to solve two-dimensional coupled Burger’s non-linear equation. Methods: New and Modified homotopy perturbation techniques are used to solve the two-dimensional coupled Burger’s equation. The methods intend to make homotopy perturbation method a more robust and trustworthy tool for fluid dynamics researchers by addressing convergence concerns, improving solution accuracy and allowing it to handle a broader range of problems. Findings: The solution for two-dimensional coupled Burger’s non-linear equation is obtained by constructing homotopies. Numerical results thus obtained are compared with the exact solution. The obtained results and the exact solutions to the stated problem are closely related. Novelty: The New and Modified Homotopy Perturbation Method proposed in this paper are effective mathematical tools for getting an exact solution to the coupled Burgers' equations. It is also a potential approach for solving various linear and nonlinear partial differential equations. Keywords: Perturbation, Homotopy, New homotopy perturbation, Modified homotopy perturbation, Two­dimension, Linear, Coupled burgers' equation

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

Leveraging External Aggregated Information for the Marginal Accelerated Failure Time Model.

It is becoming increasingly common for researchers to consider leveraging information from external sources to enhance the analysis of small-scale studies. While much attention has focused on univariate survival data, correlated survival data are prevalent in epidemiological investigations. In this article, we propose a unified framework to improve the estimation of the marginal accelerated failure time model with correlated survival data by integrating additional information given in the form of covariate effects evaluated in a reduced accelerated failure time model. Such auxiliary information can be summarized by using valid estimating equations and hence can then be combined with the internal linear rank-estimating equations via the generalized method of moments. We investigate the asymptotic properties of the proposed estimator and show that it is more efficient than the conventional estimator using internal data only. When population heterogeneity exists, we revise the proposed estimation procedure and present a shrinkage estimator to protect against bias and loss of efficiency. Moreover, the proposed estimation procedure can be further refined to accommodate the non-negligible uncertainty in the auxiliary information, leading to more trustable inference conclusions. Simulation results demonstrate the finite sample performance of the proposed methods, and empirical application on the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial substantiates its practical relevance.

Three-dimensional airborne electromagnetic forward modeling based on multiscale hexahedral finite-element method

Slow forward modeling is the main factor that restricts the practical use of three-dimensional (3D) inversion and interpretation of airborne electromagnetic (AEM) data. To improve the modeling efficiency in 3D AEM, we propose a new multiscale finite-element (MsFE) method based on unstructured hexahedral meshes. Compared with the traditional 3D AEM forward modeling, the main advantage of our newly developed method is that it can simulate complex underground structures in the earth quickly. Since we can fit the earth's topography or the anomalous bodies underground using a small number of hexahedral grids, we can quickly model them using MsFE. The main idea of the MsFE forward modeling method is to construct an interpolation operator between a coarse and a dense mesh and use the interpolation operator to map the conventional FE coefficient matrix to the MsFE coefficient matrix and thus reduce the number of unknowns in the modeling process. This will vastly reduce the scale of the linear equations system. We validate our method by simulating a typical mountain peak model and demonstrate its effectiveness by simulating numerous synthetic models and a model from Voisey Bay's Ovoid sulfide deposit, Canada.

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.

Fluorescence Visualization Quantitative Detection of Tetracycline and Nitrofurantoin in Food and Natural Water by Zn2+@Eu-bpdc Composite.

Quantitative detection of tetracycline (TC) and nitrofurantoin (NFT) in food and water is of importance for food safety and environmental protection. Herein, Zn2+ was introduced into a europium metal-organic framework Eu-bpdc (H2bpdc = 2,2'-bipyridyl-5,5'-dicarboxylic acid) to prepare a composite of Zn2+@Eu-bpdc, which was developed as a fluorescence sensor for TC and NFT. The fluorescence mechanism concerns with bpdc2- ligand-to-Eu(III) charge transfer, and the detection mechanism is the inner filter effect. Zn2+@Eu-bpdc is a ratiometric fluorescence sensor for TC with the linear fitting equation of I520/I618 = 1.94 × 104 M-1CTC, whose limit of detection (LOD) is 0.148 μmol·L-1 (μM); it is also a fluorescence "turn-off" sensor for NFT with the fitting equation of (I0-I)/I = 3.62 × 104 M-1CNFT and LOD = 0.0792 μM. Zn2+@Eu-bpdc can detect TC or NFT in lake water, honey, and milk with high accuracy. The emission color changes of paper-based Zn2+@Eu-bpdc depending on CTC or CNFT reveal the visualization detections of TC and NFT. With the red and green values as input signals, smartphone-assisted on-site detection is utilized to recognize the antibiotic residuals of TC and NFT by a self-programmed APP. Zn2+@Eu-bpdc is promising in a smartphone-assisted intelligent platform for on-site detection of TC and NFT.

Estimation of added effects and their frequency dependence in various fluid–structure interaction problems

This paper focuses on a fluid–structure interaction topic—the determination of added effects caused by fluid forces acting on a body, considering the standard linear equation of motion. We present various problems that assume small-displacement oscillations of single and multiple bodies in inviscid irrotational (potential) flow or viscous incompressible flow in both closed domain and external flow. For inviscid flow, effects of geometric parameters on the added effects were studied. The presented results extend results known from the literature. For viscous flow, frequency dependence of the added effects was studied for a wide range of frequency. The added effects were computed from data from numerical simulations of fluid flow, where the body oscillations were modeled using the dynamic mesh approach. Effects of the phase shift caused by the dynamic mesh were addressed. The added mass was compared with the corresponding value determined for inviscid flow where applicable. The results show strong dependence of the added effects on many parameters, making their proper computation challenging even for simplified cases.

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.

Unpacking Needs Protection

Most of the previous attacks on Dilithium exploit side-channel information which is leaked during the computation of the polynomial multiplication cs1, where s1 is a small-norm secret and c is a verifier's challenge. In this paper, we present a new attack utilizing leakage during secret key unpacking in the signing algorithm. The unpacking is also used in other post-quantum cryptographic algorithms, including Kyber, because inputs and outputs of their API functions are byte arrays. Exploiting leakage during unpacking is more challenging than exploiting leakage during the computation of cs1 since c varies for each signing, while the unpacked secret key remains constant. Therefore, post-processing is required in the latter case to recover a full secret key. We present two variants of post-processing. In the first one, a half of the coefficients of the secret s1 and the error s2 is recovered by profiled deep learning-assisted power analysis and the rest is derived by solving linear equations based on t = As1 + s2, where A and t are parts of the public key. This case assumes knowledge of the least significant bits of t, t0. The second variant uses lattice reduction to derive s1 without the knowledge of t0. However, it needs a larger portion of s1 to be recovered by power analysis. We evaluate both variants on an ARM Cortex-M4 implementation of Dilithium-2. The experiments show that the attack assuming the knowledge of t0 can recover s1 from a single trace captured from a different from profiling device with a non-negligible probability.

8307 Molecularly Targeted Therapy in Oncogene-Driven Human Thyroid Cancer Cell Models: Assay Selection and Response Assessment

Abstract Disclosure: A.T. Yang: None. T.W. Laetsch: Consulting Fee; Self; Cellectis, Deciphera, GentiBio, Jazz Pharmaceuticals, Jumo Health, MassiveBio, Novartis Pharmaceuticals, Pyramid Biosciences, Y-mAbs Therapeutics, Advanced Microbubbles, AI Therapeutics, Bayer, Inc.. A.T. Franco: None. Background: The most common types of pediatric thyroid cancer (TC) demonstrate kinase-activating genomic alterations that are clinically inhibitor responsive – however, optimal integration of molecular therapy into practice remains uncertain. Human TC cell models can facilitate evaluation of combination therapies. However, prior studies vary from using traditional cell counts to newer reaction-based assays; we assessed whether a linear relationship between cell count and alternative assays exist for human TC cell models. Methods: Standardized curves for Cell TiterGlo (CTG; measures total ATP using luciferase-based luminescence) and NucBlue (DAPI; binds to DNA with nuclear fluorescence) were generated using TC cells seeded at escalating concentrations (5k to 300k cells/well). A linear equation was fitted to the CTG data and used to estimate cell count. DAPI counts were obtained with EVOS automation. Human TC cell lines TPC1 (RET fusion) / CUTC5 (BRAF V600E) were seeded at low (1.5k/2k) and high (3k/4k) cells per well, exposed for 3 days to 0-256 nM selpercatinib (RET inhibitor) or 0-192 nM dabrafenib (BRAF inhibitor), respectively, and response was measured using CTG and DAPI. All were measured in triplicate. Results: Standardized curves of CTG and DAPI demonstrated a 9-23% difference between known and measured cell counts. CTG was accurate within 20k-40k cells/well; DAPI showed linearity within 10k-80k cells/well. TC cells exhibited different IC50 depending on the assay and seeding concentration. Prior studies for selpercatinib and dabrafenib reported in vitro IC50 of 0.4-67 nM and 0.5-6 nM, respectively. In this study, IC50 for CTG/DAPI were closer at higher cell counts: TPC1 was 6.1 / 8.2 nM respectively; CUTC5 was 10.6 / 5.1 nM. Wells seeded at lower counts demonstrated larger inter-assay difference: IC50 for TPC1 was 10.1 / 5.9 nM respectively; CUTC5 was 38.4 / 5.9 nM. Conclusion: IC50 values varied by cell seeding concentrations and assay used, with a two to six fold difference at lower seeding and loss of linearity between luminescence measurements and cell count. This variability complicates evaluating the concordance among data from different laboratories, and in vitro and clinical response. Alternative assays intended to assess viability of inhibitor-exposed human TC cell models may require additional evaluation to ensure measured values demonstrate the expected relationship to the intended measure (ie cell number) and ultimately clinical outcome. Presentation: 6/3/2024

Stability analysis of a linear system coupling wave and heat equations with different time scales

In this paper we consider a system coupling a wave equation with a heat equation through its boundary conditions. The existence of a small parameter in the heat equation, as a factor multiplying the time derivative, implies the existence of different time scales between the constituents of the system. This suggests the idea of applying a singular perturbation method to study stability properties. In fact, we prove that this method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which allows us to approximate the solution of the coupled system using some appropriate uncoupled subsystems.

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.

Energy storage configuration method for distribution networks based on moment difference analysis

The integration of a high proportion of distributed PV into distribution networks can cause power backfeeding and voltage limit violations. This paper introduces moment difference analysis theory for distribution networks, reframing PV consumption as the balancing moment difference equations. When maximum node voltage approaches the upper limit, the moment difference between PV and load stabilizes to an approximately constant, termed the standard moment difference. The standard moment difference represents the limit of the network's capacity to consume distributed PV. Essentially, the PV moment is the target for integration, while the load moment serves as the resource to consume the PV moment. Based on this theory, a method for energy storage configuration is proposed. Simplifying a complex multi-branch distribution network into single-branch lines and solving linear equations determines the optimal storage configuration. This method was tested on the Jibei Power Grid and the IEEE 33 and 69 bus systems, increasing computational efficiency by 1611.47–4973.82 times compared to the PSO algorithm, while maintaining an error margin of less than 2.6 %. Furthermore, the discharging strategy reduces the ESS capacity to between 12.39 % and 31.69 % of the original estimate, thereby enhancing both the economic and practical appeal of the approach.