Research

Jump to: Review

24 pages, 660 KiB

Open AccessArticle

A Nonclassical Stefan Problem with Nonlinear Thermal Parameters of General Order and Heat Source Term

by Ammar Khanfer, Lazhar Bougoffa and Smail Bougouffa

Axioms 2024, 13(1), 14; https://doi.org/10.3390/axioms13010014 - 25 Dec 2023

Viewed by 841

The analytic solution for a general form of the Stefan problem with nonlinear temperature-dependent thermal parameters and a heat source the term is obtained. We prove the existence and uniqueness of the solution to the problem in the absence of a heat source [...] Read more.

The analytic solution for a general form of the Stefan problem with nonlinear temperature-dependent thermal parameters and a heat source the term is obtained. We prove the existence and uniqueness of the solution to the problem in the absence of a heat source

(β = 0

), and in the presence of a heat source

β (x) = exp (- x^{2})

. Then, we establish lower and upper bounds for the solutions of the homogeneous equation and the nonhomogeneous equation, for different values of

δ_{i}

and

γ_{i}

. It was found that the lower bounds exhibit an excellent alignment with the numerical solutions of the homogeneous and nonhomogeneous equations, so the lower bounds can serve as approximate analytic solutions to the problem. This is a generalization to the open problem proposed by Cho and Sunderland in 1974 and also generalizes the problem proposed by Oliver and Sunderland in 1987, in addition to the problems investigated recently. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

22 pages, 354 KiB

Open AccessArticle

Some Axioms and Identities of L-Moments from Logistic Distribution with Generalizations

by Khalaf S. Sultan and Nashmiah R. AL-Shamari

Axioms 2023, 12(10), 928; https://doi.org/10.3390/axioms12100928 - 28 Sep 2023

Viewed by 557

Abstract

In this paper, we derive the L-moments for some distributions, such as logistic, generalized logistic, doubly truncated logistic, and doubly truncated generalized logistic distributions. We also establish some new axioms and identities, including recurrence relations satisfied by the L-moment from the underlying derivations. [...] Read more.

In this paper, we derive the L-moments for some distributions, such as logistic, generalized logistic, doubly truncated logistic, and doubly truncated generalized logistic distributions. We also establish some new axioms and identities, including recurrence relations satisfied by the L-moment from the underlying derivations. In addition, we establish some new general recurrence relations satisfied by the L-moment from any distribution. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

15 pages, 589 KiB

Open AccessArticle

An Analytic Method to Determine the Optimal Time for the Induction Phase of Anesthesia

by Mohamed A. Zaitri, Cristiana J. Silva and Delfim F. M. Torres

Axioms 2023, 12(9), 867; https://doi.org/10.3390/axioms12090867 - 08 Sep 2023

Cited by 1 | Viewed by 629

Abstract

We obtain an analytical solution for the time-optimal control problem in the induction phase of anesthesia. Our solution is shown to align numerically with the results obtained from the conventional shooting method. The induction phase of anesthesia relies on a pharmacokinetic/pharmacodynamic (PK/PD) model [...] Read more.

We obtain an analytical solution for the time-optimal control problem in the induction phase of anesthesia. Our solution is shown to align numerically with the results obtained from the conventional shooting method. The induction phase of anesthesia relies on a pharmacokinetic/pharmacodynamic (PK/PD) model proposed by Bailey and Haddad in 2005 to regulate the infusion of propofol. In order to evaluate our approach and compare it with existing results in the literature, we examine a minimum-time problem for anesthetizing a patient. By applying the Pontryagin minimum principle, we introduce the shooting method as a means to solve the problem at hand. Additionally, we conducted numerical simulations using the MATLAB computing environment. We solve the time-optimal control problem using our newly proposed analytical method and discover that the optimal continuous infusion rate of the anesthetic and the minimum required time for transition from the awake state to an anesthetized state exhibit similarity between the two methods. However, the advantage of our new analytic method lies in its independence from unknown initial conditions for the adjoint variables. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

17 pages, 1587 KiB

Open AccessArticle

Reliability Estimation of Inverse Weibull Distribution Based on Intuitionistic Fuzzy Lifetime Data

by Xue Hu and Haiping Ren

Axioms 2023, 12(9), 838; https://doi.org/10.3390/axioms12090838 - 29 Aug 2023

Viewed by 733

Abstract

As a commonly used model in reliability analysis, the inverse Weibull distribution (IWD) is widely applied in various scientific fields. This paper considers the reliability estimation of the IWD based on intuitionistic fuzzy lifetime data. Firstly, the related concepts of the fuzzy set [...] Read more.

As a commonly used model in reliability analysis, the inverse Weibull distribution (IWD) is widely applied in various scientific fields. This paper considers the reliability estimation of the IWD based on intuitionistic fuzzy lifetime data. Firstly, the related concepts of the fuzzy set theory are reviewed, and the concepts of the intuitionistic fuzzy conditional density, intuitionistic fuzzy likelihood function, and intuitionistic fuzzy conditional expectation are obtained by extension. In classical estimations, the maximum likelihood estimators of parameters and reliability are derived. Due to the nonlinearity, the EM algorithm is used to obtain the maximum likelihood estimates. In the Bayesian estimation, the gamma prior is selected, and the Bayesian estimation of the parameters and reliability is conducted under the symmetric entropy and the scale square error loss function, respectively. Since the integrals are complicated, the Lindley approximation is used to approximate the Bayesian estimates. Then, the performance of these estimators is evaluated by the Monte Carlo simulation. The simulation results show that the Bayesian estimation is more suitable than the maximum likelihood estimation for the reliability estimation. Finally, a set of real data is used to prove the effectiveness of these proposed methods. Through these methods, the reliability of the intuitive fuzzy life data is accurately estimated, which provides an important reference for the reliability analysis in the scientific field. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

20 pages, 440 KiB

Open AccessArticle

A Flexible Dispersed Count Model Based on Bernoulli Poisson–Lindley Convolution and Its Regression Model

by Hassan S. Bakouch, Christophe Chesneau, Radhakumari Maya, Muhammed Rasheed Irshad, Sreedeviamma Aswathy and Najla Qarmalah

Axioms 2023, 12(9), 813; https://doi.org/10.3390/axioms12090813 - 24 Aug 2023

Viewed by 854

Abstract

Count data are encountered in real-life dealings. More understanding of such data and the extraction of important information about the data require some statistical analysis or modeling. One innovative technique to increase the modeling flexibility of well-known distributions is to use the convolution [...] Read more.

Count data are encountered in real-life dealings. More understanding of such data and the extraction of important information about the data require some statistical analysis or modeling. One innovative technique to increase the modeling flexibility of well-known distributions is to use the convolution of random variables. This study examines the distribution that results from adding two independent random variables, one with the Bernoulli distribution and the other with the Poisson–Lindley distribution. The considered distribution is named as the two-parameter Bernoulli–Poisson–Lindley distribution. Many of its statistical properties are investigated, such as moments, survival and hazard rate functions, mode, dispersion behavior, mean deviation about the mean, and parameter inference based on the maximum likelihood method. To evaluate the effectiveness of the bias and mean square error of the produced estimates, a simulation exercise is carried out. Then, applications to two practical data sets are given. Finally, we construct a flexible count data regression model based on the proposed distribution with two practical examples. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

15 pages, 906 KiB

Open AccessArticle

An Extended AHP-Based Corpus Assessment Approach for Handling Keyword Ranking of NLP: An Example of COVID-19 Corpus Data

by Liang-Ching Chen and Kuei-Hu Chang

Axioms 2023, 12(8), 740; https://doi.org/10.3390/axioms12080740 - 28 Jul 2023

Cited by 4 | Viewed by 918

Abstract

The use of corpus assessment approaches to determine and rank keywords for corpus data is critical due to the issues of information retrieval (IR) in Natural Language Processing (NLP), such as when encountering COVID-19, as it can determine whether people can rapidly obtain [...] Read more.

The use of corpus assessment approaches to determine and rank keywords for corpus data is critical due to the issues of information retrieval (IR) in Natural Language Processing (NLP), such as when encountering COVID-19, as it can determine whether people can rapidly obtain knowledge of the disease. The algorithms used for corpus assessment have to consider multiple parameters and integrate individuals’ subjective evaluation information simultaneously to meet real-world needs. However, traditional keyword-list-generating approaches are based on only one parameter (i.e., the keyness value) to determine and rank keywords, which is insufficient. To improve the evaluation benefit of the traditional keyword-list-generating approach, this paper proposed an extended analytic hierarchy process (AHP)-based corpus assessment approach to, firstly, refine the corpus data and then use the AHP method to compute the relative weights of three parameters (keyness, frequency, and range). To verify the proposed approach, this paper adopted 53 COVID-19-related research environmental science research articles from the Web of Science (WOS) as an empirical example. After comparing with the traditional keyword-list-generating approach and the equal weights (EW) method, the significant contributions are: (1) using the machine-based technique to remove function and meaningless words for optimizing the corpus data; (2) being able to consider multiple parameters simultaneously; and (3) being able to integrate the experts’ evaluation results to determine the relative weights of the parameters. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

13 pages, 821 KiB

Open AccessArticle

Solutions of (2+1)-D & (3+1)-D Burgers Equations by New Laplace Variational Iteration Technique

by Gurpreet Singh, Inderdeep Singh, Afrah M. AlDerea, Agaeb Mahal Alanzi and Hamiden Abd El-Wahed Khalifa

Axioms 2023, 12(7), 647; https://doi.org/10.3390/axioms12070647 - 29 Jun 2023

Viewed by 695

Abstract

The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into [...] Read more.

The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into an algebraic form using the Laplace transform, and then the algebraic equations are iteratively solved using the modified variational iterative approach. By utilizing this technique, the Burgers equations can be solved both numerically and analytically. The study demonstrates the effectiveness of the new Laplace variational iterative approach through three specific examples. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

14 pages, 749 KiB

Open AccessArticle

Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory

by Fahad Al Basir, Jahangir Chowdhury and Delfim F. M. Torres

Axioms 2023, 12(4), 391; https://doi.org/10.3390/axioms12040391 - 18 Apr 2023

Cited by 1 | Viewed by 1135

Abstract

We formulate an integrated pest management model to control natural pests of the crop through the periodic application of biopesticide and chemical pesticides. In a theoretical analysis of the system pest eradication, a periodic solution is found and established. All the system variables [...] Read more.

We formulate an integrated pest management model to control natural pests of the crop through the periodic application of biopesticide and chemical pesticides. In a theoretical analysis of the system pest eradication, a periodic solution is found and established. All the system variables are proved to be bounded. Our main goal is then to ensure that pesticides are optimized, in terms of pesticide concentration and pesticide application frequency, and that the optimum combination of pesticides is found to provide the most benefit to the crop. By using Floquet theory and the small amplitude perturbation method, we prove that the pest eradication periodic solution is locally and globally stable. The acquired results establish a threshold time limit for the impulsive release of various controls as well as some valid theoretical conclusions for effective pest management. Furthermore, after a numerical comparison, we conclude that integrated pest management is more effective than single biological or chemical controls. Finally, we illustrate the analytical results through numerical simulations. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

18 pages, 2678 KiB

Open AccessArticle

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

by Ali Fareed Jameel, Dulfikar Jawad Hashim, Nidal Anakira, Osama Ababneh, Ahmad Qazza, Abedel-Karrem Alomari and Khamis S. Al Kalbani

Axioms 2023, 12(4), 387; https://doi.org/10.3390/axioms12040387 - 17 Apr 2023

Cited by 1 | Viewed by 1040

Abstract

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and [...] Read more.

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

23 pages, 15949 KiB

Open AccessArticle

Reliability Estimation under Normal Operating Conditions for Progressively Type-II XLindley Censored Data

by Refah Alotaibi, Mazen Nassar and Ahmed Elshahhat

Axioms 2023, 12(4), 352; https://doi.org/10.3390/axioms12040352 - 02 Apr 2023

Viewed by 1159

Abstract

This paper assumes constant-stress accelerated life tests when the lifespan of the test units follows the XLindley distribution. In addition to the maximum likelihood estimation, the Bayesian estimation of the model parameters is acquired based on progressively Type-II censored samples. The point and [...] Read more.

This paper assumes constant-stress accelerated life tests when the lifespan of the test units follows the XLindley distribution. In addition to the maximum likelihood estimation, the Bayesian estimation of the model parameters is acquired based on progressively Type-II censored samples. The point and interval estimations of the model parameters and some reliability indices under normal operating conditions at mission time are derived using both estimation methods. Using the Markov chain Monte Carlo algorithm, the Bayes estimates are calculated using the squared error loss function. Simulating the performances of the different estimation methods is performed to illustrate the proposed methodology. As an example of how the proposed methods can be applied, we look at two real-life accelerated life test cases. According to the numerical outcomes and based on some criteria, including the root of the mean square error and interval length, we can conclude that the Bayesian estimation method based on the Markov chain Monte Carlo procedure performs better than the classical methods in evaluating the XLindley parameters and some of its reliability measures when a constant-stress accelerated life test is applied with progressively Type-II censoring. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

16 pages, 364 KiB

Open AccessArticle

Tolerance Limits and Sample-Size Determination Using Weibull Trimmed Data

by Arturo J. Fernández

Axioms 2023, 12(4), 351; https://doi.org/10.3390/axioms12040351 - 02 Apr 2023

Viewed by 1074

Abstract

Guaranteed-coverage and expected-coverage tolerance limits for Weibull models are derived when, owing to restrictions on data collection, experimental difficulties, the presence of outliers, or some other extraordinary reasons, certain proportions of the extreme sample values have been censored or disregarded. Unconditional and conditional [...] Read more.

Guaranteed-coverage and expected-coverage tolerance limits for Weibull models are derived when, owing to restrictions on data collection, experimental difficulties, the presence of outliers, or some other extraordinary reasons, certain proportions of the extreme sample values have been censored or disregarded. Unconditional and conditional tolerance bounds are presented and compared when some of the smallest observations have been discarded. In addition, the related problem of determining minimum sample sizes for setting Weibull tolerance limits from trimmed data is discussed when the numbers or proportions of the left and right trimmed observations are fixed. Step-by-step procedures for determining optimal sampling plans are also presented. Several numerical examples are included for illustrative purposes. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

18 pages, 1651 KiB

Open AccessArticle

An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System

by Luyang Han, Yongjun He, Bolin Liao and Cheng Hua

Axioms 2023, 12(3), 287; https://doi.org/10.3390/axioms12030287 - 09 Mar 2023

Cited by 3 | Viewed by 1162

Abstract

The dynamic Sylvester equation (DSE) is frequently encountered in engineering and mathematics fields. The original zeroing neural network (OZNN) can work well to handle DSE under a noise-free environment, but may not work in noise. Though an integral-enhanced zeroing neural network (IEZNN) can [...] Read more.

The dynamic Sylvester equation (DSE) is frequently encountered in engineering and mathematics fields. The original zeroing neural network (OZNN) can work well to handle DSE under a noise-free environment, but may not work in noise. Though an integral-enhanced zeroing neural network (IEZNN) can be employed to solve the DSE under multiple-noise, it may fall flat under linear noise, and its convergence speed is unsatisfactory. Therefore, an accelerated double-integral zeroing neural network (ADIZNN) is proposed based on an innovative design formula to resist linear noise and accelerate convergence. Besides, theoretical proofs verify the convergence and robustness of the ADIZNN model. Moreover, simulation experiments indicate that the convergence rate and anti-noise ability of the ADIZNN are far superior to the OZNN and IEZNN under linear noise. Finally, chaos control of the sine function memristor (SFM) chaotic system is provided to suggest that the controller based on the ADIZNN has a smaller amount of error and higher accuracy than other ZNNs. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

22 pages, 1334 KiB

Open AccessArticle

Multiplicative Mixed-Effects Modelling of Dengue Incidence: An Analysis of the 2019 Outbreak in the Dominican Republic

by Adelaide Freitas, Helena Sofia Rodrigues, Natália Martins, Adela Iutis, Michael A. Robert, Demian Herrera and Manuel Colomé-Hidalgo

Axioms 2023, 12(2), 150; https://doi.org/10.3390/axioms12020150 - 01 Feb 2023

Cited by 1 | Viewed by 1616

Abstract

Dengue is a vector-borne disease that is endemic to several countries, including the Dominican Republic, which has experienced dengue outbreaks for over four decades. With outbreaks growing in incidence in recent years, it is becoming increasingly important to develop better tools to understand [...] Read more.

Dengue is a vector-borne disease that is endemic to several countries, including the Dominican Republic, which has experienced dengue outbreaks for over four decades. With outbreaks growing in incidence in recent years, it is becoming increasingly important to develop better tools to understand drivers of dengue transmission. Such tools are critical for providing timely information to assist healthcare authorities in preparing human, material, and medical resources for outbreaks. Here, we investigate associations between meteorological variables and dengue transmission in the Dominican Republic in 2019, the year in which the country’s largest outbreak to date ocurred. We apply generalized linear mixed modelling with gamma family and log link to model the weekly dengue incidence rate. Because correlations in lags between climate variables and dengue cases exhibited different behaviour among provinces, a backward-type selection method was executed to find a final model with lags in the explanatory variables. We find that in the best models, meteorological conditions such as temperature and rainfall have an impact with a delay of 2–5 weeks in the development of an outbreak, ensuring breeding conditions for mosquitoes. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

10 pages, 4478 KiB

Open AccessArticle

On Λ-Fractional Derivative and Human Neural Network

by D. Karaoulanis, A. K. Lazopoulos, N. Lazopoulou and K. Lazopoulos

Axioms 2023, 12(2), 136; https://doi.org/10.3390/axioms12020136 - 29 Jan 2023

Cited by 1 | Viewed by 956

Abstract

Fractional derivatives can express anomalous diffusion in brain tissue. Various brain diseases such as Alzheimer’s disease, multiple sclerosis, and Parkinson’s disease are attributed to the accumulation of proteins in axons. Discrete swellings along the axons cause other neuro diseases. To model the propagation [...] Read more.

Fractional derivatives can express anomalous diffusion in brain tissue. Various brain diseases such as Alzheimer’s disease, multiple sclerosis, and Parkinson’s disease are attributed to the accumulation of proteins in axons. Discrete swellings along the axons cause other neuro diseases. To model the propagation of voltage in axons with all those causes, a fractional cable geometry has been adopted. Although a fractional cable model has already been presented, the non-existence of fractional differential geometry based on the well-known fractional derivatives raises questions. These minute parts of the human neural system are modeled as cables that function with a non-uniform cross-section in the fractional realm based upon the Λ-fractional derivative (Λ-FD). That derivative is considered the unique fractional derivative generating differential geometry. Examples are presented so that fruitful conclusions can be made. The present work is going to help medical and bioengineering scientists in controlling various brain diseases. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

12 pages, 4389 KiB

Open AccessArticle

A Study on the Beech Wood Machining Parameters Optimization Using Response Surface Methodology

by Sajjad Pakzad, Siamak Pedrammehr and Mahsa Hejazian

Axioms 2023, 12(1), 39; https://doi.org/10.3390/axioms12010039 - 29 Dec 2022

Cited by 2 | Viewed by 1151

Abstract

The surface quality of wooden products is of great importance to production industries. The best surface quality requires a thorough understanding of the cutting parameters’ effects on the wooden material. In this paper, response surface methodology, which is one of the conventional statistical [...] Read more.

The surface quality of wooden products is of great importance to production industries. The best surface quality requires a thorough understanding of the cutting parameters’ effects on the wooden material. In this paper, response surface methodology, which is one of the conventional statistical methods in experiment design, has been used to design experiments and investigate the effect of different machining parameters as feed rate, spindle speed, step over, and depth of cut on surface quality of the beech wood. The mathematical model of the examined parameters and the surface roughness have also been obtained by the method. Finally, the optimal machining parameters have been obtained to achieve the best quality of the machined surface, which reduced the surface roughness up to 4.2 (µm). Each of the machining parameters has a considerable effect on surface quality, although it is noted that the feed rate has the greatest effect. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

18 pages, 606 KiB

Open AccessArticle

Extending Normality: A Case of Unit Distribution Generated from the Moments of the Standard Normal Distribution

by Miguel S. Concha-Aracena, Leonardo Barrios-Blanco, David Elal-Olivero, Paulo Henrique Ferreira da Silva and Diego Carvalho do Nascimento

Axioms 2022, 11(12), 666; https://doi.org/10.3390/axioms11120666 - 24 Nov 2022

Cited by 2 | Viewed by 958

Abstract

This paper presents an important theorem, which shows that, heading from the moments of the standard normal distribution, one can generate density functions originating a family of models. Additionally, we discussed that different random variable domains are achieved with transformations. For instance, we [...] Read more.

This paper presents an important theorem, which shows that, heading from the moments of the standard normal distribution, one can generate density functions originating a family of models. Additionally, we discussed that different random variable domains are achieved with transformations. For instance, we adopted the moment of order two, from the proposed theorem, and transformed it, which enabled us to exemplify this class as a unit distribution. We named it as Alpha-Unit (AU) distribution, which contains a single positive parameter

α

(

AU (α) \in [0, 1]

). We presented its properties and demonstrated two estimation methods for the

α

parameter, the maximum likelihood estimator (MLE) and uniformly minimum-variance unbiased estimator (UMVUE) methods. In order to analyze the statistical consistency of the estimators, a Monte Carlo simulation study was carried out, in which the robustness was demonstrated. As a real-world application, we adopted two sets of unit data, the first regarding the dynamics of Chilean inflation in the post-military period, and the other one regarding the daily maximum relative humidity of the air in the Atacama Desert. In both cases presented, the AU model is competitive, whenever the data present a range greater than 0.4 and extremely heavy asymmetric tail. We compared our model with other commonly used unit models, such as the beta, Kumaraswamy, logit-normal, simplex, unit-half-normal, and unit-Lindley distributions. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

Review

Jump to: Research

22 pages, 2316 KiB

Open AccessReview

Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review

by Abida Hussain, Ibrahima Faye, Mohana Sundaram Muthuvalu, Tong Boon Tang and Mudasar Zafar

Axioms 2023, 12(4), 326; https://doi.org/10.3390/axioms12040326 - 28 Mar 2023

Cited by 4 | Viewed by 1459

Abstract

In the field of biomedical image reconstruction, functional near infra-red spectroscopy (fNIRs) is a promising technology that uses near infra-red light for non-invasive imaging and reconstruction. Reconstructing an image requires both forward and backward problem-solving in order to figure out what the image’s [...] Read more.

In the field of biomedical image reconstruction, functional near infra-red spectroscopy (fNIRs) is a promising technology that uses near infra-red light for non-invasive imaging and reconstruction. Reconstructing an image requires both forward and backward problem-solving in order to figure out what the image’s optical properties are from the boundary data that has been measured. Researchers are using a variety of numerical methods to solve both the forward and backward problems in depth. This study will show the latest improvements in numerical methods for solving forward and backward problems in fNIRs. The physical interpretation of the forward problem is described, followed by the explanation of the state-of-the-art numerical methods and the description of the toolboxes. A more in-depth discussion of the numerical solution approaches for the inverse problem for fNIRs is also provided. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Mathematical Methods in the Applied Sciences

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (17 papers)

Research

Review

Further Information

Guidelines

MDPI Initiatives

Follow MDPI