Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way.
{"title":"Some Results on Zinbiel Algebras and Rota–Baxter Operators","authors":"Jizhong Gao, Junna Ni, Jianhua Yu","doi":"10.3390/axioms13050314","DOIUrl":"https://doi.org/10.3390/axioms13050314","url":null,"abstract":"Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 44","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140993440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a combination of inverse probability weighting (IPW) and mean imputation is utilized to mitigate the bias caused by missing data. Under several mild conditions, it is demonstrated that the proposed estimators are consistent and possess oracle properties. The efficacy of the proposed parameter estimation process is assessed through Monte Carlo simulation studies. Finally, the applicability of the proposed method is further substantiated using the Boston Housing Dataset.
{"title":"Parameter Estimation in Spatial Autoregressive Models with Missing Data and Measurement Errors","authors":"Tengjun Li, Zhikang Zhang, Yunquan Song","doi":"10.3390/axioms13050315","DOIUrl":"https://doi.org/10.3390/axioms13050315","url":null,"abstract":"This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a combination of inverse probability weighting (IPW) and mean imputation is utilized to mitigate the bias caused by missing data. Under several mild conditions, it is demonstrated that the proposed estimators are consistent and possess oracle properties. The efficacy of the proposed parameter estimation process is assessed through Monte Carlo simulation studies. Finally, the applicability of the proposed method is further substantiated using the Boston Housing Dataset.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 18","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140990342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Rafiullah, Muhammad Asif, Dure Jabeen, M. A. Ibrahim
The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.
{"title":"Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method","authors":"M. Rafiullah, Muhammad Asif, Dure Jabeen, M. A. Ibrahim","doi":"10.3390/axioms13050311","DOIUrl":"https://doi.org/10.3390/axioms13050311","url":null,"abstract":"The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140997508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shi-Fan Cai, S. Chaubey, Xin Xu, Pan Zhang, Zhi-Heng Zhang
The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ)-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method.
{"title":"Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold","authors":"Shi-Fan Cai, S. Chaubey, Xin Xu, Pan Zhang, Zhi-Heng Zhang","doi":"10.3390/axioms13050312","DOIUrl":"https://doi.org/10.3390/axioms13050312","url":null,"abstract":"The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ)-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140995434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include the inverse transform method, acceptance–rejection method, and Box–Muller transformation, all of which rely on sampling from the uniform (0,1) distribution. A significant concept in statistics is the finite mixture model, characterized by a convex combination of multiple probability density functions. In this paper, we introduce a modified version of the composition method, a standard approach for sampling finite mixture models. Our modification offers the advantage of relying on sampling from the uniform (0,1) distribution, aligning with prevalent methods in computational statistics. This alignment simplifies teaching computational statistics courses, as well as having other benefits. We offer several examples to illustrate the approach.
{"title":"A Short Note on Generating a Random Sample from Finite Mixture Distributions","authors":"L. Al-Labadi, Anna Ly","doi":"10.3390/axioms13050307","DOIUrl":"https://doi.org/10.3390/axioms13050307","url":null,"abstract":"Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include the inverse transform method, acceptance–rejection method, and Box–Muller transformation, all of which rely on sampling from the uniform (0,1) distribution. A significant concept in statistics is the finite mixture model, characterized by a convex combination of multiple probability density functions. In this paper, we introduce a modified version of the composition method, a standard approach for sampling finite mixture models. Our modification offers the advantage of relying on sampling from the uniform (0,1) distribution, aligning with prevalent methods in computational statistics. This alignment simplifies teaching computational statistics courses, as well as having other benefits. We offer several examples to illustrate the approach.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":"5 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141001674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.
{"title":"Study on SEAI Model of COVID-19 Based on Asymptomatic Infection","authors":"Lidong Huang, Yue Xia, Wenjie Qin","doi":"10.3390/axioms13050309","DOIUrl":"https://doi.org/10.3390/axioms13050309","url":null,"abstract":"In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 45","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141000169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This Special Issue of the scientific journal Axioms, entitled “Recent Advances in Fractional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciences today [...]
{"title":"Recent Advances in Fractional Calculus","authors":"Péter Kórus, J. N. Nápoles Valdés","doi":"10.3390/axioms13050310","DOIUrl":"https://doi.org/10.3390/axioms13050310","url":null,"abstract":"This Special Issue of the scientific journal Axioms, entitled “Recent Advances in Fractional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciences today [...]","PeriodicalId":502355,"journal":{"name":"Axioms","volume":" 115","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141000699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. G. La Guardia, J. Chagas, E. Lenzi, L. Pires, Nicolás Zumelzu, Benjamín R. C. Bedregal
In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R0+. More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences.
在本文中，我们扩展了非负实数半域 R0+ 上的半向量空间和半代数理论。更确切地说，我们证明了有关这些理论的几个新结果。我们向文献介绍了半线性算子的特征值和特征向量的概念，描述了如何计算它们。我们还研究了半矢量空间的拓扑特性，如完备性和可分性。我们展示了由半度量、半规范和半内积等衍生出的半矢量空间新族。此外，我们还展示了有关半代数的几个新结果。在这种理论方法之后，我们将这种理论应用于模糊自动机。更确切地说，我们描述了 A-模糊正则表达式语言的半代数，并将模糊自动机理论应用于 DNA 序列的计数模式。
{"title":"On Semi-Vector Spaces and Semi-Algebras with Applications in Fuzzy Automata","authors":"G. G. La Guardia, J. Chagas, E. Lenzi, L. Pires, Nicolás Zumelzu, Benjamín R. C. Bedregal","doi":"10.3390/axioms13050308","DOIUrl":"https://doi.org/10.3390/axioms13050308","url":null,"abstract":"In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R0+. More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":"14 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141001375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joaquín Borrego-Díaz, Andrés Cordón-Franco, Francisco Félix Lara-Martín
In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form α(v)→β(v). Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by T+E) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by T+E–Rule). The theory T+E will be stronger than T+E–Rule, but how much stronger can T+E be? More specifically, is T+E conservative over T+E–Rule for theorems of some fixed syntactical complexity Γ? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas Π and in the applications of E–Rule side formulas from the class Π and can be eliminated (in a certain precise sense), then T+E is ∀B(Π)-conservative over T+E–Rule. Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of E–Rule of a depth at most of m suffice to obtain ∀B(Π) conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework.
在本文中，我们将在经典一阶逻辑的框架内探讨以下一般性问题。假设某个数学原理可以通过一组形式为 α(v)→β(v) 的条件式 E 在一阶语言中形式化。给定一个基础理论 T，我们可以用条件式集 E 以两种自然的方式扩展基础理论。要么把 E 中的每个公式作为新公理加到 T 中（从而得到一个用 T+E 表示的理论），要么把 E 中的公式作为推理规则的实例来扩展 T（从而得到一个用 T+E-Rule 表示的理论）。理论 T+E 将比 T+E-Rule 更强，但 T+E 能强到什么程度呢？更具体地说，对于某些固定句法复杂度 Γ 的定理，T+E 比 T+E-Rule 保守吗？在条件公式集 E 的非常一般的假设下，我们在这方面得到了两个主要的守恒结果。首先，如果 E 中的公式相对于某种规定的公式类 Π 具有较低的语法复杂性，并且在 E 规则的应用中，来自类 Π 的边公式可以被消除（在某种精确的意义上），那么 T+E 对 T+E 规则是∀B(Π)保守的。其次，如果 E 是一个包含 m 个条件句的有限集，那么 E-Rule 深度最多为 m 的嵌套应用足以获得∀B(Π) 保守性。这些公理与推理规则之间的守恒结果将著名的一阶算术学片段守恒定理扩展到了一般的纯逻辑框架。
{"title":"On Conditional Axioms and Associated Inference Rules","authors":"Joaquín Borrego-Díaz, Andrés Cordón-Franco, Francisco Félix Lara-Martín","doi":"10.3390/axioms13050306","DOIUrl":"https://doi.org/10.3390/axioms13050306","url":null,"abstract":"In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form α(v)→β(v). Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by T+E) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by T+E–Rule). The theory T+E will be stronger than T+E–Rule, but how much stronger can T+E be? More specifically, is T+E conservative over T+E–Rule for theorems of some fixed syntactical complexity Γ? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas Π and in the applications of E–Rule side formulas from the class Π and can be eliminated (in a certain precise sense), then T+E is ∀B(Π)-conservative over T+E–Rule. Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of E–Rule of a depth at most of m suffice to obtain ∀B(Π) conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":"99 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141003890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.
{"title":"Construction of Fractional Pseudospectral Differentiation Matrices with Applications","authors":"Wenbin Li, Hongjun Ma, Tinggang Zhao","doi":"10.3390/axioms13050305","DOIUrl":"https://doi.org/10.3390/axioms13050305","url":null,"abstract":"Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.","PeriodicalId":502355,"journal":{"name":"Axioms","volume":"36 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141013876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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