Search Results (134)

This paper introduces a novel Runge–Kutta (RK) pair of orders $8\left(6\right)$ designed specifically for solving linear inhomogeneous initial value problems (IVPs) with constant coefficients. The proposed method requires only 11 stages per iteration, a significant improvement over conventional RK pairs of orders $8\left(7\right)$, which typically demand 13 stages. The reduction in stages is achieved by leveraging a smaller set of order conditions tailored to linear inhomogeneous problems, where traditional simplification techniques are not applicable. To address the complexity of deriving such methods, the authors employ the Differential Evolution algorithm, a global optimization technique, to solve the resulting system of equations. The new RK pair, named NEW8(6)Lin, is tested on several benchmark problems, including scalar, linear inhomogeneous, and larger systems, demonstrating a superior performance in terms of accuracy and computational efficiency. The method’s high phase-lag accuracy and efficiency make it particularly suitable for problems requiring high precision over extended intervals. The coefficients of the method are provided with high precision, enabling direct implementation in computational environments like Mathematica. The results highlight the method’s potential as a robust tool for solving linear inhomogeneous IVPs, offering a balance between computational cost and accuracy. This work contributes to the ongoing development of specialized numerical methods for differential equations, particularly in scenarios where traditional approaches struggle with efficiency or stability. Full article

Urban street trees often exhibit distinct architectural characteristics compared to their counterparts in natural forests. Allometric equations for the stem (M_S), branches (M_B), and total dry aboveground biomass of urban trees (M_T) were developed, based on 52 destructively sampled specimens, belonging to 10 different species, growing in the Municipality of Athens, Greece. Linear, log-linear, and nonlinear regression analyses were applied, and fit statistics were used to select the most appropriate model. The results indicated that diameter at breast height (D_1.3) and tree height (H) are needed for accurately predicting M_S, while M_B may be estimated based on D_1.3. To circumvent the caveat of the additivity property for estimating the biomass of different tree component, nonlinear seemingly unrelated regression (NSUR) was implemented. The 95% prediction intervals for M_S, M_B, and M_T efficiently captured the variability of the sampled trees. Finally, the predictions were compared with estimates from i-Tree, the most widely used model suite for urban and rural forestry analysis, and a mean deviation of 134% (ranging from 3% to 520%) was reported. Therefore, in the absence of urban-specific allometries, the obtained empirical models are proposed for estimating biomass in street trees, particularly in cities with Mediterranean-like climatic influences. Full article

The initial value estimation of uncertain differential equations refers to the process of estimating the initial state of a time-varying system using observed data when we cannot know exactly the initial state of the system in uncertain environments. In order to study the initial value estimation problem of uncertain differential equations, this paper constructs the confidence interval and point estimation of the initial value based on the residuals corresponding to the observed data. In order to further explain the initial value estimation based on residuals, this paper gives the confidence intervals and point estimations of the initial value for several specific uncertain differential equations, including linear, exponential, and mean-reversion uncertain differential equations. Finally, two numerical examples and an empirical study are also provided to illustrate the effectiveness of the proposed method. Full article

In engineering, physics, and other fields, implicit ordinary differential equations are essential to simulate complex systems. However, because of their intrinsic nonlinearity and difficulty separating higher-order derivatives, implicit ordinary differential equations pose substantial challenges. When applied to these types of equations, traditional numerical methods frequently have problems with convergence or require a significant amount of computing power. In this work, we present the multi-stage differential transform method, a novel semi-analytical approach for effectively solving first- and second-order implicit ordinary differential systems, in conjunction with Adomian polynomials. The main contribution of this method is that it simplifies the solution procedure and lowers processing costs by enabling the differential transform method to be applied directly to implicit systems without transforming them into explicit or quasi-linear forms. We obtain straightforward and effective algorithms that build solutions incrementally utilizing the characteristics of Adomian polynomials, providing benefits in theory and practice. By solving several implicit ODE systems that are difficult for traditional software programs such as Maple 2024, Mathematica 14, or Matlab 24.1, we validate our approach. The multi-stage differential transform method’s contribution includes expanded convergence intervals for numerical results, more accurate approximate solutions for wider domains, and the efficient calculation of exact solutions as a convergent power series. Because of its ease of implementation in educational computational tools and substantial advantages in terms of simplicity and efficiency, our method is suitable for researchers and practitioners working with complex implicit differential equations. Full article

During switching in electrical systems, transient electromagnetic processes occur. The resulting dangerous current surges are best studied by computer simulation. However, the time required for computer simulation of such processes is significant for complex electromagnetic devices, which is undesirable. The use of spectral methods can significantly speed up the calculation of transient processes and ensure high accuracy. At present, we are not aware of publications showing the use of spectral methods for calculating transient processes in electromagnetic devices containing ferromagnetic cores. The purpose of the work: The objective of this work is to develop a highly effective method for calculating electromagnetic transient processes in a coil with a ferromagnetic magnetic core connected to a voltage source. The method involves the use of nonlinear magnetoelectric substitution circuits for electromagnetic devices and a spectral method for representing solution functions using orthogonal polynomials. Additionally, a schematic model for applying the spectral method is developed. Obtained Results: A method for calculating transients in magnetoelectric circuits based on approximating solution functions with algebraic orthogonal polynomial series is proposed and studied. This helps to transform integro-differential state equations into linear algebraic equations for the representations of the solution functions. The developed schematic model simplifies the use of the calculation method. Representations of true electric and magnetic current functions are interpreted as direct currents in the proposed substitution circuit. Based on these methods, a computer program is created to simulate transient processes in a magnetoelectric circuit. Comparing the application of various polynomials enables the selection of the optimal polynomial type. The proposed method has advantages over other known methods. These advantages include reducing the simulation time for electromagnetic transient processes (in the examples considered, by more than 12 times than calculations using the implicit Euler method) while ensuring the same level of accuracy. The simulation of processes over a long time interval demonstrate error reduction and stabilization. This indicates the potential of the proposed method for simulating processes in more complex electromagnetic devices, (for example, transformers). Full article

Accurate prediction of amino acid requirements in fast-growing broilers is crucial for cost-effective diet formulation and reducing nitrogen excretion to mitigate environmental impact. This study developed a dynamic model to predict standardized ileal digestible amino acid requirements throughout broiler growth using a factorial approach and the comparative slaughter technique, considering maintenance, growth, and gender factors. The model was based on an experiment were designed using 480 15-day-old Arbor Acres chickens randomly assigned to 10 groups. A linear equation was derived using established growth and protein deposition curves to calculate maintenance and growth coefficients. Models for five essential amino acids under different amino-acid-to-protein ratios were created (R² > 0.70). The model effectively estimated daily amino acid needs and specific time intervals. Comparisons with NRC (1994), BTPS (2011), and Arbor Acres manual (2018) showed higher predicted requirements for lysine, methionine, valine, and threonine than Arbor Acres (2018) and BTPS (2011), significantly exceeding NRC (1994). Arginine predictions aligned with BTPS in early stages, but were slightly lower in later stages. This supports the further development of dynamic amino acid models. Full article

In the literature so far, for Caputo fractional boundary value problems of order $2q$ when $1<2q<2,$ the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order $2q.$ Further, even the initial conditions for a Caputo fractional differential equation of order $nq$ use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘$nq$’ will only be a $C^{nq}$ solution and not a $C^n$ solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order $kq$, $k=0,1,\dots (n-1)$ when the order of the fractional derivative involved in the differential equation is $nq$. The Caputo left derivative can be computed only when the function can be expressed as $f(x-a)$. Then the Caputo right derivative of the same function will be computed for the function $f(b-x).$ Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when $q=1.$ Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data. Full article

Background: Diabetes is a growing global health crisis that requires effective therapeutic strategies to optimize treatment outcomes. This study aims to address this challenge by developing and characterizing extended-release polymeric matrix tablets containing metformin hydrochloride (M-HCl), a first-line treatment for type 2 diabetes, and honokiol (HNK), a bioactive compound with potential therapeutic benefits. The objective is to enhance glycemic control and overall therapeutic outcomes through an innovative dual-drug delivery system. Methods: The tablets were formulated using hydrophilic polymers, such as Carbopol^® 71G NF and Noveon^® AA-1. The release kinetics of M-HCl and HNK were investigated through advanced mathematical models, including fractal and multifractal dynamics, to capture the non-linear and time-dependent release processes. Traditional kinetic models (zero-order, first-order, Higuchi equations) were also evaluated for comparison. In vitro dissolution studies were conducted to determine the release profiles of the active ingredients under varying polymer concentrations. Results: The study revealed distinct release profiles for the two active ingredients. M-HCl exhibited a rapid release phase, with 80% of the drug released within 4–7 h depending on polymer concentration. In contrast, HNK demonstrated a slower release profile, achieving 80% release after 9–10 h, indicating a greater sensitivity to polymer concentration. At shorter intervals, drug release followed classical kinetic models, while multifractal dynamics dominated at longer intervals. Higher polymer concentrations resulted in slower drug release rates due to the formation of a gel-like structure upon hydration, which hindered drug diffusion. The mechanical properties and stability of the matrix tablets confirmed their suitability for extended-release applications. Mathematical modeling validated the experimental findings and provided insights into the structural and time-dependent factors influencing drug release. Conclusions: This study successfully developed dual-drug extended-release matrix tablets containing metformin hydrochloride and honokiol, highlighting the potential of hydrophilic polymers to regulate drug release. The findings emphasize the utility of advanced mathematical models for predicting release kinetics and underscore the potential of these formulations to improve patient compliance and therapeutic outcomes in diabetes management. Full article

In urban areas prone to extreme weather, it has become crucial to implement effective strategies to improve living conditions for residents reliant on medical and educational facilities. This research highlighted the importance of urban green spaces in cooling European cities and examined the planning and maintenance of these areas alongside economic losses due to water consumption during heatwaves. Key findings using an SEM (structural equation model) showed that hot summer days indirectly impacted water prices by increasing cumulative temperature days. The confidence interval (0.015, 0.038) confirmed this effect. Additionally, tropical nights indirectly impacted water prices, as shown by the cooling degree days, which indicated the need for air conditioning. The increased use of energy for cooling resulted in higher water prices due to the water required for power generation. This effect was statistically significant, with an estimated value of 0.029 (p < 0.001). A generalized linear model (GLM) indicated an inverse relationship between urban green space and impervious surfaces (slope: −0.69996 ± 0.025561, intercept: 53.675 ± 0.97709, p < 0.01), which was important for reducing impervious surfaces and improving water management, ultimately leading to cooler urban temperatures. Practical recommendations for decision-makers, urban planners, and residents are provided to adapt to changing extreme weather conditions. These include improving the soil environment in current locations and increasing access to green spaces, which can enhance well-being and address health issues. Full article

For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions. Full article

The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice. Full article

Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction–diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction–diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection–reaction–diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves. Full article

Background: China’s rapidly aging population presents challenges for cognitive health and mental well-being among the older adults. This study examines how the number of children affects cognitive function in middle-aged and older adults and whether depressive symptoms mediate this relationship. Methods: This study analyzed data from waves 1 to 5 (2011–2020) of the China Health and Retirement Longitudinal Study (CHARLS), involving 5932 participants aged 45 and older. Participants were grouped by the number of children: childless, only child and multiple children. We used Logarithmic Generalized Linear Models (LGLMs) to explore the relationships among the number of children, depressive symptoms, and cognitive function. Indirect effect coefficients and 95% bias-corrected and accelerated confidence intervals (BCaCI) were estimated using Simultaneous Equation Models (SEM) with three-stage least squares (3SLS) and the bootstrap method to assess the mediating effect of depressive symptoms. Results: In middle-aged and older adults, a negative association was observed between the number of children and overall cognitive functioning (all p < 0.01). This association remained significant even after adjusting for covariates in groups with three (β = −0.023, p < 0.05) and four or more children (β = −0.043, p < 0.001). Conversely, the positive association between the number of children and depression also persisted after adjusting for covariates, although it weakened as the number of children increased (all p < 0.01). Depressive symptoms consistently correlated negatively with overall cognitive function (p < 0.001) and partially mediated the relationship between the number of children and cognitive function (pMe = 20.36%, p < 0.05). The proportion of the mediating effect attributed to depression was more pronounced in middle-aged and older adults who had experienced the loss of children (pMe = 24.31%) or had two children (pMe = 25.39%), with stronger mediating effects observed in males (pMe = 48.84%) and urban residents (pMe = 64.58%). Conclusions: The findings indicate that depressive symptoms partially mediate the relationship between the number of children and cognitive function in middle-aged and older adults in China. These results highlight the significance of considering mental health factors when studying cognitive function in this demographic. Notably, in families without children and those with two children, depressive symptoms play a crucial role in explaining the decline in cognitive function. Full article

This paper presents a significant extension of perturbation theory techniques for estimating the roots of polynomials. Building upon foundational results and recent work by Pakdemirli and Yurtsever, as well as taking inspiration from the concept of probabilistic bounds introduced by Sheikh et al., we develop and prove several novel theorems that address a wide range of polynomial structures. These include polynomials with multiple large coefficients, coefficients of different orders, alternating coefficient orders, large linear and constant terms, and exponentially decreasing coefficients. Among the key contributions is a theorem that establishes an upper bound on the expected maximum modulus of the zeros of polynomials with uniformly distributed perturbations in their coefficients. The theorem considers the case where all but the leading coefficient receive a uniformly and independently distributed perturbation in the interval $[-1,1]$. Our approach provides a comprehensive framework for estimating the order of magnitude of polynomial roots based on the structure and magnitude of their coefficients without the need for explicit root-finding algorithms. The results offer valuable insights into the relationship between coefficient patterns and root behavior, extending the applicability of perturbation-based root estimation to a broader class of polynomials. This work has potential applications in various fields, including random polynomials, control systems design, signal processing, and numerical analysis, where quick and reliable estimation of polynomial roots is crucial. Our findings contribute to the theoretical understanding of polynomial properties and provide practical tools for engineers and scientists dealing with polynomial equations in diverse contexts. Full article

The present study deals with the transient interval analysis of a shallow beam having uncertainty in structural parameters viz. mass density and applied load. It is quite difficult to obtain information regarding the exact values of these parameters in several practical situations. Use of precise (deterministic) values of structural parameters in such a situation leads to erroneous results as the mathematical model built using deterministic structural parameters does not account for the uncertainty present in the system. In the present work, uncertainty present in the system is represented by interval parameters. In the research work carried out in the past quarter century, several methods were developed to model structural response of uncertain structural systems subjected to static loads under conditions of linear elasticity. The partial differential equations of motion of a Euler-Bernoulli beam are solved using Finite difference and finite element methods under conditions of linear elasticity. The resulting interval equations are solved using search and direct methods. Further, direct optimization approach is used to compute the bounds of displacement. The applicability and effectiveness of presented methods is demonstrated by solving example problems. Full article