Exponential Method Research Articles

Fiber and Antioxidant Activity of Spice Cookies with Local Flour as Functional Food for Elderly

Older people are vulnerable to infectious and degenerative diseases due to decreased organ function and lowered immune systems. This study aimed to analyze the quality and level of liking of the spice cookie formula and the fiber and antioxidant content of the selected cookie formula. The research method used a factorial complete randomized design with the treatment of wheat flour substitution using arrowroot flour and canna flour. There were three levels of substitution ratio, namely 50:50, 25:75, and 0:100, resulting in six formulas. The variables measured were sensory quality and level of liking using an organoleptic test by 30 semi-trained panelists. Organoleptic test data were statistically analyzed using the Kruskal Wallis test and Mann-Whitney further test. The two best formulas were selected using the exponential comparison method and then analyzed for dietary fiber content and antioxidant activity. The results showed that among the six formulas, there were significant differences in texture and aftertaste of the sensory quality and level of liking (p-value<0,05). The selected formulas were 50:50 substitution ratios of either arrowroot flour or canna flour. Cookies with canna flour substitution had a dietary fiber content of 9,6%, while cookies with arrowroot flour substitution were 7.8%. The antioxidant activity of cookies with canna flour substitution amounted to 9,5 mg/100g, while cookies with arrowroot flour substitution amounted to 12,0 mg/100g. In conclusion, spice cookies with a 50:50 ratio of canna flour and wheat flour substitution have the highest fiber content with strong antioxidant activity.

Generalized exponential rational function method to the fractional shallow water wave phenomena

In this study we introduce the fractional shallow water wave phenomena, namely, the new extended (2 + 1)-dimensional fractional Boussinesq equation and in the following we apply the generalized exponential rational function for extracting the soliton solutions to the Ex-FBe (extended (2 + 1)-dimensional fractional Boussinesq equation) by using Maple tools for calculating and Mathematica for graphical behaviors. Many types of exact soliton solutions for the fswwp equation have been derived by the erf method. Physical interpretations of some of the obtained solutions are also included. It can be easily concluded that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

Numerical simulation of one-dimensional nonlinear Schrodinger equation using PSO with exponential B-spline

Particle swarm optimization (PSO) is a widely adopted, rapid, efficient, and computer-friendly technique for optimization in various fields, including computer science, engineering, and mathematical modelling. In this paper, the nonlinear Schrodinger (NLS) equation has been solved numerically, which leads to soliton-type solutions, by utilizing the “Exponential cubic B-spline differential quadrature method (Expo-MCB-DQM) with PSO”. The exponential cubic B-spline basis functions involve a parameter that must be assigned a value to find the numerical solutions. Until now, the value of a parameter had been determined by the hit-and-trial approach, which leads to unstable results. By employing the PSO technique, the optimal value of the parameter is determined, which enhances the accuracy of the numerical solutions. This study introduces an innovative combination of Expo-MCB-DQM with PSO, which is an innovative combination that optimizes the parameters in the basis function, leading to improved outcomes. To validate the authenticity and effectiveness of this methodology, it has been applied to two problems to present the numerical results in comparison with the exact solution. Additionally, the paper provides insightful discussions on the significant application of the NLS equation in optical fiber communication.

Numerical analysis of the impact behaviour of a composite eco-structure

An explicit numerical model is developed for the impact behaviour of a new composite eco-structure made of a plywood core and flax/epoxy composite skin. A constitutive orthotropic material behaviour is imposed to model the elastic response of plies of plywood core and flax skin. A stress-based modified Hashin-3D damage model with four independent failure criteria in tensile and compressive mode is incorporated for predicting the damage initiation. A strain threshold based exponential method is adopted to predict the damage scaling and its evolution. The above proposed model is implemented through user material subroutine (VUMAT) in ABAQUS Explicit. The developed model is validated with experimental low-velocity impact test results and can accurately predict the stiffness degradation, plateau, and post-peak softening of wooden plies. A good agreement with the better accuracy is achieved between the predictions and the test results.

Dynamics of novel exact soliton solutions to Stochastic Chiral Nonlinear Schrödinger Equation

The core objective of this study is to explore the some novel stochastic solutions. For this purpose, we consider the stochastic (2+1)-dimensional Chiral nonlinear Schrödinger equation (2D-SCNLSE) which is derived with multiplicative noise in the Itô sense. To achieve novel stochastic solutions, we employ two modified techniques as modified generalized exponential rational function method (mGERFM) and the modified rational sine-cosine and sinh-cosh methods. We extract exponential, periodic, bright, dark, and singular in single and combo forms. Due to the applications of the Chiral nonlinear Schrödinger equation in soliton theory, these solutions are extremely viable to exemplify some sensational complicated physical phenomena and applicable in diversified fields of applied sciences. This study enhances the theory of Itô calculus by directly performing it into analytical approaches for the solution of differential equations. To examine the impact of multiplicative noise on the results, several graphs have been plotted. We comprehend that the noise destroys the symmetry of the solutions of adopted model. The evaluated achievements suggested that the proposed methods are categorical, efficacious, reliable, and robust and can be the best way to handle other complex equations arising in applied sciences.

Effects of Fertilization Mode on the Growth of Lactuca sativa L. and Soil Nutrients in Facilitated Cultivation

This paper aims to find a fertilization method that better matches the growth characteristics of vegetables and reduces the soil and environmental problems caused by unreasonable fertilization methods, in view of the strong buffering of the clayey black soil in the north and the mismatch between the amount of chemical fertilizer applied and the growth characteristics of vegetables during cultivation. In this experiment, Lactuca sativa L. (Lactuca sativa) was studied in a randomized complete block design in greenhouse and five different fertilizer application methods were arranged: exponential fertilization (EF), linear fertilization (LF), average fertilization (AF), one-time fertilization (OF), and no fertilization as a control group (CG). The effects of different fertilization methods on soil nutrients, agronomic traits of Lactuca sativa growth, and related quality were investigated, and the pattern of effects of different fertilization methods on growth and soil nutrients was obtained. The results of the experiment showed that the exponential fertilization method gradually took advantage as the growth time extended. The plant height (PLH), leaf length (LL), leaf width (LW), yield, soluble protein (SP), soluble sugar (SS), vitamin C (VC), and elemental nitrogen (EN) and potassium (EK) of Lactuca sativa were significantly improved under the exponential fertilization method compared with other fertilization methods, by 29.9 cm, 51.5 cm, 5.96 cm, 22, 2.32 kg/m2, 0.23%, 0.44%, 3.93%, 94.66 mg/kg, 1.58 g/kg, and 1.94 g/kg, respectively. The alkali-hydrolyzed nitrogen (SAN), available phosphorus (SAP), and available potassium (SAK) in the soil after fertilization were 139.69 mg/kg, 50.23 mg/kg, and 180.30 mg/kg, respectively. The above results showed that the exponential fertilization method not only improved the quality of Lactuca sativa and thus the quality of the crop growth traits, but also changed the soil nutrients favorably after fertilization, which is of some importance for the protection of black soils.

Distribution of probabilistic design fires for single residential items

When designing scenarios for fire safety in residential buildings, one should consider the natural variability of heat release rate of a single item that can possibly influence fire spread between multiple items. This study intends to represent experimentally measured design fire for single residential items using a combination of t-squared growth method and Ingason exponential decay method. The characterizations required three representative data from experiments including maximum heat release rate, time to reach the maximum heat release rate and total heat released. The methods have established good representations of the experimental design fires with more than 86% similarity. Distribution statistics are considered to characterize the design fires in a probabilistic manner which develop possible design fires from an average curve as well as within a minimum/maximum limit and a 5th/95th percentile limit. Comparisons denote that the probabilistic design fires are reliable to represent the possible heat release rate for single residential items.

Exponential method of estimation in sampling theory under robust quantile regression methods

–In the regression analysis, ordinary least square techniques is commonly used. However, the data’s outcomes may be untrustworthy if there is an outliers in it. In order to deal with the outliers problem, robust quantile regression methods have been frequently presented as alternatives to OLS for a long time. In this article, primarily a exponential ratio-type estimators is suggested. After that, robust quantile regression estimators are proposed, that is a useful strategy. The application of robust quantile regression empowered the efficiency of the estimators especially for outliers in the data. The MSE equations of the various estimators are computed and compared to OLS approaches. Numerical illustration and simulations studies are performed to support our theoretical findings.

Static and dynamic analysis of couple-stress elastic structures by using peridynamic differential operator

In this paper, a peridynamic differential operator (PDDO) method is developed to analyze the static and dynamic responses of size-dependent elastic structures in the frame of a consistent couple stress theory (CCST). The displacement governing equations together with the boundary conditions of the couple-stress static problems are transformed from the local differential form to the nonlocal integral form through a fourth-order PDDO and then solved by a meshfree scheme. For dynamic cases, the time-domain problems are first transformed into frequency domain by Fourier transform, then solved by the PDDO meshless method in the frequency domain. The time-domain results are finally obtained by using an exponential window method (EWM). The size effects on the static and dynamic responses of some typical elastic problems are investigated by the developed PDDO method. A very high level of computational accuracy is achieved by comparing with the analytical solutions and the existing numerical results.

Solitary wave structures for the stochastic Nizhnik–Novikov–Veselov system via modified generalized rational exponential function method

In this article, a significant stochastic Nizhnik–Novikov–Veselov (SNNV) system with truncated M-fractional derivative (TMD) is investigated. This mathematical model is considered an isotropic Lax extension of the one-dimensional Kortewegde Vries equation which has diverse applications in the areas of warm ions, shallow-water waves, electromagnetic signals, and river irrigation flows. To collect different solitary wave solutions of the SNNV system, a modification of generalized rational exponential function method is employed. Some new trigonometric, exponential and hyperbolic stochastic solutions are obtained by using the modified generalized rational exponential function method (mGERFM). Furthermore, some graphs of the formulated solutions are presented to depict the physical configuration of stochastic solutions by changing the parameters. The method used in this research is effective, precise, competent, and reliable to compute the soliton solutions for nonlinear models. We predict that the performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

Frequency‐domain fundamental solutions and boundary element method for consistent couple stress elastodynamic problems

AbstractIn this article, the elastodynamic problems are investigated within the framework of a consistent couple‐stress theory (CCST). The basic equations for the CCST elastodynamic problems are derived at first. These dynamic formulas are then analyzed in the frequency domain by the Fourier transform. The frequency‐domain fundamental solutions to the transformed elastodynamic problems are subsequently derived for the two‐dimensional (2D) plane‐strain state. Based on the fundamental solutions and the elastodynamic reciprocal theorem, the frequency‐domain boundary integral equations (BIEs) are established. Then, the BIEs are numerically solved by a collocation method in conjunction with analytical and numerical treatments of the arising singular integrals. To obtain the time‐domain dynamic responses, an exponential window method (EWM) is adopted to transform the frequency‐domain results to the time‐domain solutions. By using the developed boundary element method (BEM), several typical couple‐stress elastodynamic problems are numerically investigated and the size‐effects are studied by using different values of the length‐scale parameter in the CCST. Good agreements between the results obtained by the developed BEM for a tiny length‐scale parameter and the corresponding classical BEM results or analytical solutions are achieved, which validates the high accuracy of the present BEM.

Exact solutions of time fractional generalized Burgers–Fisher equation using exp and exponential rational function methods

Exact solutions of time fractional generalized Burgers–Fisher equation using exp and exponential rational function methods

ON NEW HYBRID ROOT-FINDING ALGORITHMS FOR SOLVING TRANSCENDENTAL EQUATIONS USING EXPONENTIAL AND HALLEY'S METHODS

The objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple.

MATHEMATICAL ANALYSIS AND STUDY OF THE NUMEROUS TRAVELING WAVE BEHAVIOR FOR DIFFERENT WAVE VELOCITIES OF THE SOLITON SOLUTIONS FOR THE NONLINEAR LANDAU-GINSBERG-HIGGS MODEL IN NONLINEAR MEDIA

In this study, the nonlinear Landau-Ginsberg-Higgs (LGH) model is proposed and examined. The stated model is applied to analyze superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. This is undeniably a robust mathematical model in real-world applications. The generalized exponential rational function method (GERFM) is utilized to extract the suitable, useful, and further general solitary wave solutions of the LGH model via the traveling wave transformation. Furthermore, we investigate the effects of wave velocity in a particular time limit through a graphical representation of the examined solutions of the model to understand the dynamic behavior of the system. The attained results confirm the effectiveness and reliability of the considered scheme

Efficacy of exponentiation method with a convolutional neural network for classifying lung nodules on CT images by malignancy level.

The aim of this study was to examine the performance of a convolutional neural network (CNN) combined with exponentiating each pixel value in classifying benign and malignant lung nodules on computed tomography (CT) images. Images in the Lung Image Database Consortium-Image Database Resource Initiative (LIDC-IDRI) were analyzed. Four CNN models were then constructed to classify the lung nodules by malignancy level (malignancy level 1 vs. 2, malignancy level 1 vs. 3, malignancy level 1 vs. 4, and malignancy level 1 vs. 5). The exponentiation method was applied for exponent values of 1.0 to 10.0 in increments of 0.5. Accuracy, sensitivity, specificity, and area under the curve of receiver operating characteristics (AUC-ROC) were calculated. These statistics were compared between an exponent value of 1.0 and all other exponent values in each model by the Mann-Whitney U-test. In malignancy 1 vs. 4, maximum test accuracy (MTA; exponent value = 2.0, 3.0, 3.5, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, and 10.0) and specificity (6.5, 7.0, and 9.0) were improved by up to 0.012 and 0.037, respectively. In malignancy 1 vs. 5, MTA (6.5 and 7.0) and sensitivity (1.5) were improved by up to 0.030 and 0.0040, respectively. The exponentiation method improved the performance of the CNN in the task of classifying lung nodules on CT images as benign or malignant. The exponentiation method demonstrated two advantages: improved accuracy, and the ability to adjust sensitivity and specificity by selecting an appropriate exponent value. Adjustment of sensitivity and specificity by selecting an exponent value enables the construction of proper CNN models for screening, diagnosis, and treatment processes among patients with lung nodules. • The exponentiation method improved the performance of the convolutional neural network. • Contrast accentuation by the exponentiation method may derive features of lung nodules. • Sensitivity and specificity can be adjusted by selecting an exponent value.

Conformable mathematical modeling of the COVID‐19 transmission dynamics: A more general study

Many challenges are still faced in bridging the gap between mathematical modeling and biological sciences. Measuring population immunity to assess the epidemiology of health and disease is a challenging task and is currently an active area of research. However, to meet these challenges, mathematical modeling is an effective technique in shaping the population dynamics that can help disease control. In this paper, we introduce a susceptible–infected–recovered (SIR) model and a susceptible–exposed–infected–recovered–deceased (SEIRD) model based on conformable space‐time partial differential equations (PDEs) for the coronavirus disease 2019 (COVID‐19) pandemic. As efficient analytical tools, we present new modifications based on the fractional exponential rational function method (ERFM) and an analytical technique based on the Adomian decomposition method for obtaining the solutions for the proposed models. These analytical approaches are more efficacious for obtaining analytical solutions for nonlinear systems of PDEs with conformable derivatives. The interesting result of this paper is that it yields new exact and approximate solutions to the proposed COVID‐19 pandemic models with conformable space‐time partial derivatives.

Application of the Single Exponential Smoothing Method For Flood Disaster Prediction

The country of Indonesia is seen as one that is particularly vulnerable to natural catastrophes as well as calamities brought on by human activity. A disaster that may be brought on by both natural and human sources is a flood. Disasters brought on by flooding are unpredictable occurrences that frequently cause losses in the form of property damage, the theft of assets, and lost productivity at work and in school. Through this prediction information system, the people can find out the level of risk of flooding through excessive rainfall. in order to better anticipate and prepare for all possibilities that occur before the flood, the method used is Single Exponential Smoothing. This method was chosen because of the simple way the system works to find predictive values ??through past data. With this system, researchers can input rainfall data taken from the Meteorology, Climatology and Geophysics Agency, then the data is processed through the system and if rainfall gets high results. The risk of flooding will also be very high and a warning will be given to the public so that better prepared for the risk of flooding. The results obtained from this study are the results of an analysis of the exponential method single to obtain accurate rainfall prediction information with data MAD, MSE and MAPE.

Constructive exponential tracking control for mechanical systems via Hamiltonian realization and contraction analysis method

The conventional tracking control of mechanical systems is generally based on the stabilization of error dynamics and most of the results can only guarantee asymptotic tracking of the desired trajectories. Since error dynamics are generally time-varying, it is very difficult to find appropriate Lyapunov functions or control Lyapunov functions to complete stability analysis and controller design. To address this limitation, the novel constructive exponential tracking control method is proposed to mechanical systems by utilizing the Hamiltonian realization and contraction analysis in this paper. Firstly, based on the Hamiltonian realization and use the structural characteristics of port-Hamiltonian systems, the exponential tracking controllers are constructed for fully actuated and under-actuated mechanical systems by combining the pre-feedback with feedback control. The proposed tracking control strategies can be used to discuss fully actuated and under-actuated mechanical systems in a unified framework. Then the exponential decay-rate of tracking controllers and procedure for selecting control parameters for fully actuated and under-actuated mechanical systems are given. Finally, comparative simulations and experiments are carried out to illustrate the effectiveness and robustness of the proposed control strategy.

A new method for assessment of electro-osmotic permeability through the integration of theoretical and experimental ion flux in electrokinetic processes

Electrokinetic (EK) technology is promising for removing heavy metals from contaminated unsaturated soils. It is crucial to accurately determine the unsaturated electro-osmotic permeability for predicting the efficiency of EK treatment, optimizing treatment strategies, and accurately predicting the distribution of contaminant concentrations. However, the current approach of estimating unsaturated electro-osmotic permeability, which involves measuring effective voltage, drainage volume, and performing exponential fitting, fails to address the issue of uneven voltage gradient distribution during EK treatment. Herein, a novel method was presented for estimating the electro-osmotic permeability of unsaturated porous media. This method quantifies the electro-osmotic flow in an unsaturated porous medium by considering the difference in mass-transfer efficiency (MTE) between real (with electro-osmotic flow) and hypothetical cases (without electro-osmotic flow). This difference serves as a metric for estimating the electro-osmotic permeability. Results revealed a linear relationship between the electro-osmotic permeability and the product of volumetric moisture content and tortuosity, with the slope related to the ionic mobility of target ions, hypothetical and actual MTE. To validate this method, hexavalent Cr (Cr(VI)) was selected as the target contaminant and six EK experiments were conducted with varying initial volumetric moisture content. The feasibility of the method was evaluated by fitting the results of these experiments to obtain the specific slope of the porous medium used. Compared to the existing effective voltage–drainage volume–exponential fitting method, the proposed method offers several advantages. First, it effectively addressed the issue of nonuniform voltage distribution during EK treatment in the unsaturated porous medium. Second, it overcame the problem of a nonzero electro-osmotic permeability at zero volumetric moisture content in the exponential empirical formula. Third, the proposed method was based on theoretical derivations instead of relying solely on empirical fitting. Finally, the proposed method does not require a prior estimate of the saturated electro-osmotic permeability of the porous medium.

Enhancing the accuracy of the scale of fluctuation estimating for tunnel surrounding rock mass properties

The scale of fluctuation (SOF) is used to evaluate the correlation between two locations of geotechnical parameter, which is important for mineral exploration and engineering construction. In this study, the performances of conventional moment methods are firstly estimated, and then the exponential autocorrelation function method that can achieve the best performance in estimating SOF is selected. After that, the Torque Penetration Index (TPI) and the Field Penetration Index (FPI) obtained from the Yinsong Water Diversion Project are used to detect the scale effect of SOF and to quantify the influences of rock types and surrounding rock mass grades on SOF. As the spatial scale increases, the SOF values increase, and the uncertainty in the SOF of the surrounding rock mass properties increases. The SOF calculated using TPI is slightly higher than using FPI. The mean values of SOF for granite, tuff, diorite, and limestone are quite close at the 50 m–200 m scales, with coefficients of variation (COVs) of SOF range from about 0.2 to 0.4. There are differences in SOF between grade III and grade IV surrounding rock masses at the 50 m–500 m scales. The mean values of SOF in grade III surrounding rock mass are about 4%–15% higher than that in grade IV surrounding rock mass, and this difference in general increases as scale increases. The rock mass with better conditions has a stronger spatial correlation. The COVs of SOF for the two rock grades reached about 0.3–0.5, owning to the inherent spatial variability.