Axioms

16 pages, 274 KiB

Open AccessArticle

A New Perspective on Intuitionistic Fuzzy Structures in Sheffer Stroke BCK-Algebras

by Ravi Kumar Bandaru, Rajesh Neelamegarajan, Tahsin Oner and Amal S. Alali

Axioms 2025, 14(5), 347; https://doi.org/10.3390/axioms14050347 - 30 Apr 2025

This study introduces the concept of an intuitionistic fuzzy SBCK-subalgebra (SBCK-ideal) and explores the level set of an intuitionistic fuzzy set within the context of Sheffer stroke BCK-algebras. These newly defined concepts are crucial for understanding the interaction between intuitionistic logic and Sheffer [...] Read more.

This study introduces the concept of an intuitionistic fuzzy SBCK-subalgebra (SBCK-ideal) and explores the level set of an intuitionistic fuzzy set within the context of Sheffer stroke BCK-algebras. These newly defined concepts are crucial for understanding the interaction between intuitionistic logic and Sheffer stroke BCK-algebras. The paper establishes a connection between subalgebras and level sets in the framework of Sheffer stroke BCK-algebras, demonstrating that the level set of intuitionistic fuzzy SBCK-subalgebras corresponds precisely to their subalgebras, and conversely. Additionally, the study provides novel results regarding the structural properties of Sheffer stroke BCK-algebras under intuitionistic fuzzy logic, specifically focusing on the conditions under which fuzzy sets become SBCK-subalgebras or SBCK-ideals. This work contributes to the theoretical foundations of fuzzy logic in algebraic structures, offering a deeper understanding of the interplay between intuitionistic fuzzy sets and the algebraic operations within Sheffer stroke BCK-algebras. Full article

(This article belongs to the Section Algebra and Number Theory)

28 pages, 388 KiB

Open AccessArticle

Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity

by Sadeem Al-Harbi and Mohamed Ben Ayed

Axioms 2025, 14(5), 346; https://doi.org/10.3390/axioms14050346 - 30 Apr 2025

Abstract

In this paper, we consider the nonlinear Neumann problem

(Q_{ε}) : - Δ u + V (x) u = u^{\frac{n + 2}{n - 2} - ε}

, with

u > 0

in

Ω

and [...] Read more.

In this paper, we consider the nonlinear Neumann problem

(Q_{ε}) : - Δ u + V (x) u = u^{\frac{n + 2}{n - 2} - ε}

, with

u > 0

in

Ω

and

\partial u / \partial ν = 0

on

\partial Ω

, where

Ω

is a bounded regular domain in

R^{n}

, with

n \geq 4

,

ε

is a small positive parameter, and V is a non-constant smooth positive function on

\bar{Ω}

. Assuming the flatness of the boundary near the critical points of the restriction of the function V on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for

(Q_{ε})

. The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results. Full article

18 pages, 357 KiB

Open AccessArticle

Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs

by Liang Shan and Yang Liu

Axioms 2025, 14(5), 345; https://doi.org/10.3390/axioms14050345 - 30 Apr 2025

Abstract

This work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph

(V, E, μ, w)

: [...] Read more.

This work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph

(V, E, μ, w)

:

{(- Δ)}^{s} u = (K + λ) e^{2 u} - κ in V,

where the fraction

s \in (0, 1)

and real parameter

λ

are given, and the graph functions K and

κ

satisfy

\max_{x \in V} K (x) = 0

,

K ≢ 0

and

\int_{V} κ d μ < 0

. We derive the solvability characteristics of the above equation with the help of variational theory and the upper and lower solutions method. Full article

(This article belongs to the Section Mathematical Physics)

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18 pages, 266 KiB

Open AccessArticle

The Reverse Order Law for the {1,3M,4N}—The Inverse of Two Matrix Products

by Yingying Qin, Baifeng Qiu and Zhiping Xiong

Axioms 2025, 14(5), 344; https://doi.org/10.3390/axioms14050344 - 30 Apr 2025

Abstract

By using the maximal and minimal ranks of some generalized Schur complement, the equivalent conditions for the reverse order law

(A B) {1, 3 M, 4 K} = B {1, 3 N, 4 K} A {1, 3 M, 4 N}

are presented. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)

19 pages, 12884 KiB

Open AccessArticle

Evolutionary Search for Polynomial Lyapunov Functions: A Genetic Programming Method for Exponential Stability Certification

by Roman Pykhnivskyi, Anton Ryzhov, Andrii Sobchuk and Yurii Kravchenko

Axioms 2025, 14(5), 343; https://doi.org/10.3390/axioms14050343 - 30 Apr 2025

Abstract

This paper presents a method for constructing polynomial Lyapunov functions to analyze the stability of nonlinear dynamical systems. The approach is based on genetic programming, a variant of genetic algorithms where the search space consists of hierarchical tree structures. In our formulation, these [...] Read more.

This paper presents a method for constructing polynomial Lyapunov functions to analyze the stability of nonlinear dynamical systems. The approach is based on genetic programming, a variant of genetic algorithms where the search space consists of hierarchical tree structures. In our formulation, these polynomial functions are represented as binary trees. The Lyapunov conditions for exponential stability are interpreted as a minimax optimization problem, using a carefully designed fitness metric to ensure positivity and dissipation within a chosen domain. The genetic algorithm then evolves candidate polynomial trees, minimizing constraint violations and continuously refining stability guarantees. Numerical examples illustrate that this methodology can effectively identify and optimize Lyapunov functions for a wide range of systems, indicating a promising direction for automated stability proofs in engineering applications. Full article

(This article belongs to the Special Issue Modern Trends and Application of Decision-Making Theory, Stability and Control)

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21 pages, 331 KiB

Open AccessArticle

Optimality Conditions and Stability Analysis for the Second-Order Cone Constrained Variational Inequalities

by Li Wang, Yining Sun, Juhe Sun, Yanhong Yuan and Bin Wang

Axioms 2025, 14(5), 342; https://doi.org/10.3390/axioms14050342 - 29 Apr 2025

Abstract

In this paper, we study the optimality conditions and perform a stability analysis for the second-order cone constrained variational inequalities (SOCCVI) problem. The Lagrange function and Karush–Kuhn–Tucker (KKT) condition of the SOCCVI problem is given, and the optimality conditions for the SOCCVI problem [...] Read more.

In this paper, we study the optimality conditions and perform a stability analysis for the second-order cone constrained variational inequalities (SOCCVI) problem. The Lagrange function and Karush–Kuhn–Tucker (KKT) condition of the SOCCVI problem is given, and the optimality conditions for the SOCCVI problem are studied. Then, the second-order sufficient condition satisfying the constrained nondegenerate condition is proved. The strong second-order sufficient condition is defined. And the nonsingularity of Clarke’s generalized Jacobian of the KKT point, the strong regularity of the KKT point, the uniform second-order growth condition, the strong stability of the KKT point, and the local Lipschtiz homeomorphism of the KKT point for the SOCCVI problem are proved to be equivalent to each other. Then, the stability theorem of the SOCCVI problem is obtained. Full article

(This article belongs to the Section Mathematical Analysis)

21 pages, 678 KiB

Open AccessArticle

On the Bivariate Vector-Valued Rational Interpolation and Recovery Problems

by Lixia Xiao, Peng Xia and Shugong Zhang

Axioms 2025, 14(5), 341; https://doi.org/10.3390/axioms14050341 - 29 Apr 2025

Abstract

This paper investigates bivariate vector-valued rational interpolation and recovery problems with common divisors in their denominators. By leveraging these shared divisors and employing a component-wise rational interpolation approach, we reduce the degree of the interpolation problem. Furthermore, a new algorithm for recovering bivariate [...] Read more.

This paper investigates bivariate vector-valued rational interpolation and recovery problems with common divisors in their denominators. By leveraging these shared divisors and employing a component-wise rational interpolation approach, we reduce the degree of the interpolation problem. Furthermore, a new algorithm for recovering bivariate vector-valued rational functions is proposed. Experimental results demonstrate that, compared to classical algorithms, our method achieves faster computation speed without compromising accuracy. This advantage is particularly evident in the recovery of bivariate vector-valued rational functions. Full article

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19 pages, 324 KiB

Open AccessArticle

Sparse Robust Weighted Expectile Screening for Ultra-High-Dimensional Data

by Xianjun Wu, Pingping Han and Mingqiu Wang

Axioms 2025, 14(5), 340; https://doi.org/10.3390/axioms14050340 - 28 Apr 2025

Abstract

This paper investigates robust feature screening for ultra-high dimensional data in the presence of outliers and heterogeneity. Considering the susceptibility of likelihood methods to outliers, we propose a Sparse Robust Weighted Expectile Regression (SRoWER) method that combines the

L_{2} E

criterion with [...] Read more.

This paper investigates robust feature screening for ultra-high dimensional data in the presence of outliers and heterogeneity. Considering the susceptibility of likelihood methods to outliers, we propose a Sparse Robust Weighted Expectile Regression (SRoWER) method that combines the

L_{2} E

criterion with expectile regression. By utilizing the IHT algorithm, our method effectively incorporates correlations of covariates and enables joint feature screening. The proposed approach demonstrates robustness against heavy-tailed errors and outliers in data. Simulation studies and a real data analysis are provided to demonstrate the superior performance of the SRoWER method when dealing with outlier-contaminated explanatory variables and/or heavy-tailed error distributions. Full article

(This article belongs to the Section Mathematical Physics)

12 pages, 1106 KiB

Open AccessArticle

A Penalized Orthogonal Kriging Method for Selecting a Global Trend

by Xituo Zhang, Guoxing Gao, Jianxin Zhao and Xinmin Li

Axioms 2025, 14(5), 339; https://doi.org/10.3390/axioms14050339 - 28 Apr 2025

Abstract

A kriging regression model is a popular and effective type of surrogate model in computer experiments. A significant challenge arises when the mean function of the model includes polynomial terms with unknown coefficients, leading to identifiability problems and potentially unreliable results. To overcome [...] Read more.

A kriging regression model is a popular and effective type of surrogate model in computer experiments. A significant challenge arises when the mean function of the model includes polynomial terms with unknown coefficients, leading to identifiability problems and potentially unreliable results. To overcome this problem, Plumlee and Joseph (2018) introduced an orthogonal kriging model. Variable selection for kriging models has been widely considered by researchers in computer experiments. In this paper, we introduce a new method for combining orthogonal kriging with penalized variable selection. Furthermore, an efficient algorithm is given to select the correct mean function. The simulation results and an example study with real data show that the proposed method is superior to others in variable recognition rate and prediction accuracy. Full article

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22 pages, 307 KiB

Open AccessArticle

An Investigation into Bipolar Fuzzy Hoop Algebras and Their Applications

by Tahsin Oner, Rajesh Neelamegarajan, Ravi Kumar Bandaru and Hashem Bordbar

Axioms 2025, 14(5), 338; https://doi.org/10.3390/axioms14050338 - 28 Apr 2025

Abstract

This paper introduces bipolar fuzzy sub-hoops and bipolar fuzzy filters within hoop algebras, extending fuzzy logic to incorporate both positive and negative membership degrees. We define these structures, explore their algebraic properties, and establish their interplay through rigorous theorems. Key results include characterizations [...] Read more.

This paper introduces bipolar fuzzy sub-hoops and bipolar fuzzy filters within hoop algebras, extending fuzzy logic to incorporate both positive and negative membership degrees. We define these structures, explore their algebraic properties, and establish their interplay through rigorous theorems. Key results include characterizations of bipolar fuzzy filters via level sets and conditions under which they become implicative filters. These findings enhance the theoretical framework of many-valued logic and offer practical applications in decision-making, image processing, and spatial reasoning under uncertainty. Our work provides a foundation for advanced fuzzy systems handling complex, contradictory information. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition)

21 pages, 929 KiB

Open AccessArticle

Involute Partner-Ruled Surfaces Formed by Involutes of Spacelike Curves in Minkowski Three-Space

by Özgür Boyacıoğlu Kalkan, Süleyman Şenyurt, Davut Canlı and Luca Grilli

Axioms 2025, 14(5), 337; https://doi.org/10.3390/axioms14050337 - 28 Apr 2025

Abstract

We introduced the concept of involute partner-ruled surfaces, which are formed by the involutes of spacelike curves and additional conditions ensuring the presence of definite surface normals in Minkowski three-space. First, we provided the criteria for each couple of involute partner-ruled surfaces to [...] Read more.

We introduced the concept of involute partner-ruled surfaces, which are formed by the involutes of spacelike curves and additional conditions ensuring the presence of definite surface normals in Minkowski three-space. First, we provided the criteria for each couple of involute partner-ruled surfaces to be simultaneously developable and minimal. Then, we established the requirements for the coordinate curves lying on these surfaces to be geodesic, asymptotic, and lines of curvature. We also expanded this paper with an example by providing graphical illustrations of the involute partner-ruled surfaces. Full article

(This article belongs to the Section Geometry and Topology)

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13 pages, 280 KiB

Open AccessArticle

Exploring Geometrical Properties of Annihilator Intersection Graph of Commutative Rings

by Ali Al Khabyah and Moin A. Ansari

Axioms 2025, 14(5), 336; https://doi.org/10.3390/axioms14050336 - 27 Apr 2025

Abstract

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not [...] Read more.

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not directed in nature, where the vertex set is represented by

D {(Λ)}^{*}

. There is a connection in the form of an edge between two distinct vertices

ς

and

ϱ

in

AIG (Λ)

iff

A n n (ς ϱ) \neq A n n (ς) \cap A n n (ϱ)

. In this work, we begin by categorizing commutative rings

Λ

, which are finite in structure, so that

AIG (Λ)

forms a star graph/2-outerplanar graph, and we identify the inner vertex number of

AIG (Λ)

. In addition, a classification of the finite rings where the genus of

AIG (Λ)

is 2, meaning

AIG (Λ)

is a double-toroidal graph, is also investigated. Further, we determine

Λ

, having a crosscap 1 of

AIG (Λ),

indicating that

AIG (Λ)

is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two. Full article

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11 pages, 278 KiB

Open AccessArticle

On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields

by Abdulaziz Deajim

Axioms 2025, 14(5), 335; https://doi.org/10.3390/axioms14050335 - 27 Apr 2025

Abstract

Consider the modular group

PSL (2, Z) = ⟨ x, y | x^{2} = y^{3} = 1 ⟩

generated by the transformations

x : z \mapsto - 1 / z

and [...] Read more.

Consider the modular group

PSL (2, Z) = ⟨ x, y | x^{2} = y^{3} = 1 ⟩

generated by the transformations

x : z \mapsto - 1 / z

and

y : z \mapsto (z - 1) / z

. Let H be the proper subgroup

⟨ y, v | y^{3} = v^{3} = 1 ⟩

of

PSL (2, Z)

, where

v = x y x

. For a positive square-free integer n, this article studies the action of H on the subset

{\frac{a + \sqrt{- n}}{c} | a, b = \frac{a^{2} + n}{c}, c \in Z, c \neq 0}

of the imaginary quadratic number field

Q (\sqrt{- n})

where, in particular, the accurate estimate of the number of orbits arising from this action is given, correcting the estimate given in some of the relevant literature. Full article

(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)

17 pages, 524 KiB

Open AccessArticle

Closed-Form Meromorphic Solutions of High-Order and High-Dimensional Differential Equations

by Hongqiang Tu and Yongyi Gu

Axioms 2025, 14(5), 334; https://doi.org/10.3390/axioms14050334 - 27 Apr 2025

Abstract

In this paper, we investigate closed-form meromorphic solutions of the fifth-order Sawada-Kotera (fSK) equation and (3+1)-dimensional generalized shallow water (gSW) equation. The study of high-order and high-dimensional differential equations is pivotal for modeling complex nonlinear phenomena in physics and engineering, where higher-order dispersion, [...] Read more.

In this paper, we investigate closed-form meromorphic solutions of the fifth-order Sawada-Kotera (fSK) equation and (3+1)-dimensional generalized shallow water (gSW) equation. The study of high-order and high-dimensional differential equations is pivotal for modeling complex nonlinear phenomena in physics and engineering, where higher-order dispersion, dissipation, and multidimensional dynamics govern system behavior. Constructing explicit solutions is of great significance for the study of these equations. The elliptic, hyperbolic, rational, and exponential function solutions for these high-order and high-dimensional differential equations are achieved by proposing the extended complex method. The planar dynamics behavior of the (3+1)-dimensional gSW equation and its phase portraits are analyzed. Using computational simulation, the chaos behaviors of the high-dimensional differential equation under noise perturbations are examined. The dynamic structures of some obtained solutions are revealed via some 2D and 3D graphs. The results show that the extended complex method is an efficient and straightforward approach to solving diverse differential equations in mathematical physics. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Related Topics)

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21 pages, 382 KiB

Open AccessArticle

Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications

by Mona Hosny

Axioms 2025, 14(5), 333; https://doi.org/10.3390/axioms14050333 - 27 Apr 2025

Abstract

Several different topologies utilizing ideals are created and compared with previous topologies. The results show that the previous ones are weaker than the current ones and that the current ones are stronger. The merits of these topologies are proposed, and the smallest and [...] Read more.

Several different topologies utilizing ideals are created and compared with previous topologies. The results show that the previous ones are weaker than the current ones and that the current ones are stronger. The merits of these topologies are proposed, and the smallest and largest among them are identified; this merit distinguishes the present study from previous ones. Afterwards, these topologies are employed to conduct more in-depth investigations on broadened rough sets. The proposed approximate models are particularly significant as applied to rough sets because they diminish vagueness and uncertainty compared to prior models. Moreover, the proposed models stand out from their predecessors because they can compare all types of approximations, display all the features described by Pawlak, and possess the property of monotonicity across any relations. Furthermore, a medical application is showcased to emphasize the significance of the current findings. Additionally, the advantages of the adopted approach are examined, alongside an evaluation of its limitations. The paper wraps up with the essential features of the proposed manner and recommend avenues for future research. Full article

(This article belongs to the Special Issue Topics in General Topology and Applications)

16 pages, 416 KiB

Open AccessArticle

Compositional Scheduling in Industry 4.0 Cyber-Physical Systems

by Fernando Tohmé and Daniel Rossit

Axioms 2025, 14(5), 332; https://doi.org/10.3390/axioms14050332 - 27 Apr 2025

Abstract

Cyber-physical systems (CPSs) are fundamental components of Industry 4.0 production environments. Their interconnection is crucial for the successful implementation of distributed and autonomous production plans. A particularly relevant challenge is the optimal scheduling of tasks that require the collaboration of multiple CPSs. To [...] Read more.

Cyber-physical systems (CPSs) are fundamental components of Industry 4.0 production environments. Their interconnection is crucial for the successful implementation of distributed and autonomous production plans. A particularly relevant challenge is the optimal scheduling of tasks that require the collaboration of multiple CPSs. To ensure the feasibility of optimal schedules, two primary issues must be addressed: (1) The design of global systems emerging from the interconnection of CPSs; (2) The development of a scheduling formalism tailored to interconnected Industry 4.0 settings. Our approach is based on a Category Theory formalization of interconnections as compositions. This framework aims to guarantee that the emergent behaviors align with the intended outcomes. Building upon this foundation, we introduce a formalism that captures the assignment of operations to cyber-physical systems. Full article

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28 pages, 333 KiB

Open AccessArticle

Bipolar Fuzzy Sheffer Stroke in BCK-Algebras

by Tahsin Oner, Rajesh Neelamegarajan, Ravi Kumar Bandaru and Amal S. Alali

Axioms 2025, 14(5), 331; https://doi.org/10.3390/axioms14050331 - 26 Apr 2025

Abstract

In this study, we examine bipolar fuzzy SBCK-subalgebras and their corresponding level sets of bipolar fuzzy sets in the setting of Sheffer stroke BCK-algebras. These concepts contribute significantly to the analysis of bipolar logical structures within this algebraic context. We demonstrate a bidirectional [...] Read more.

In this study, we examine bipolar fuzzy SBCK-subalgebras and their corresponding level sets of bipolar fuzzy sets in the setting of Sheffer stroke BCK-algebras. These concepts contribute significantly to the analysis of bipolar logical structures within this algebraic context. We demonstrate a bidirectional relationship between SBCK-subalgebras and their level sets, proving that each level set derived from a bipolar fuzzy SBCK-subalgebra constitutes a subalgebra, and, conversely, each such subalgebra defines an associated level set. This duality emphasizes the structural interplay between bipolar fuzzy logic and the Sheffer stroke operation in BCK-algebras. Full article

(This article belongs to the Section Algebra and Number Theory)

30 pages, 515 KiB

Open AccessArticle

Parameter Estimation of the Lomax Lifetime Distribution Based on Middle-Censored Data: Methodology, Applications, and Comparative Analysis

by Peiyao Ren, Wenhao Gui and Shan Liang

Axioms 2025, 14(5), 330; https://doi.org/10.3390/axioms14050330 - 26 Apr 2025

Abstract

The Lomax distribution has important applications in survival analysis, reliability engineering, insurance, finance, and other fields. Middle-censoring is an important censoring scheme, and data with middle-censoring will produce censoring in random intervals. This paper studies the parameter estimation of the Lomax distribution based [...] Read more.

The Lomax distribution has important applications in survival analysis, reliability engineering, insurance, finance, and other fields. Middle-censoring is an important censoring scheme, and data with middle-censoring will produce censoring in random intervals. This paper studies the parameter estimation of the Lomax distribution based on middle-censored data. The expectation–maximization algorithm is employed to compute the maximum likelihood estimates of the two unknown parameters of the Lomax distribution. After processing the data using the midpoint approach estimation, the parameter estimates are obtained by two computational methods: the Newton–Raphson iteration method and the fixed-point method. Moreover, the calculation methods for the asymptotic confidence intervals of the two parameters are provided, with the confidence interval coverage rate serving as one of the criteria for evaluating the estimation performance. In the Bayesian estimation aspect, the shape parameter is estimated using a Gamma prior distribution, and the Gibbs sampling method is employed for the solution. Finally, both simulation data and real data are used to compare the accuracy of the various estimation methods. Full article

(This article belongs to the Special Issue Computational Statistics and Its Applications)

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18 pages, 293 KiB

Open AccessArticle

Existence and Controls for Fractional Evolution Equations

by Ying Chen and Yong Zhou

Axioms 2025, 14(5), 329; https://doi.org/10.3390/axioms14050329 - 24 Apr 2025

Abstract

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals [...] Read more.

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals is studied, and the existence of optimal control pairs is proven. Finally, by constructing a corresponding Gramian controllability operator using the solution operator, a sufficient condition is provided for the existence of approximate controllability of the corresponding problem. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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