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Editor’s Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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20 pages, 611 KiB  
Article
A Novel FMEA Model Based on Rough BWM and Rough TOPSIS-AL for Risk Assessment
by Tai-Wu Chang, Huai-Wei Lo, Kai-Ying Chen and James J. H. Liou
Mathematics 2019, 7(10), 874; https://doi.org/10.3390/math7100874 - 20 Sep 2019
Cited by 49 | Viewed by 4120
Abstract
Failure mode and effects analysis (FMEA) is a risk assessment method that effectively diagnoses a product’s potential failure modes. It is based on expert experience and investigation to determine the potential failure modes of the system or product to develop improvement strategies to [...] Read more.
Failure mode and effects analysis (FMEA) is a risk assessment method that effectively diagnoses a product’s potential failure modes. It is based on expert experience and investigation to determine the potential failure modes of the system or product to develop improvement strategies to reduce the risk of failures. However, the traditional FMEA has many shortcomings that were proposed by many studies. This study proposes a hybrid FMEA and multi-attribute decision-making (MADM) model to establish an evaluation framework, combining the rough best worst method (R-BWM) and rough technique for order preference by similarity to an ideal solution technique (R-TOPSIS) to determine the improvement order of failure modes. In addition, this study adds the concept of aspiration level to R-TOPSIS technology (called R-TOPSIS-AL), which not only optimizes the reliability of the TOPSIS calculation program, but also obtains more potential information. This study then demonstrates the effectiveness and robustness of the proposed model through a multinational audio equipment manufacturing company. The results show that the proposed model can overcome many shortcomings of traditional FMEA, and effectively assist decision-makers and research and development (R&D) departments in improving the reliability of products. Full article
(This article belongs to the Special Issue Recent Advances in Modeling for Reliability Analysis)
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23 pages, 479 KiB  
Article
Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations
by Jean-Philippe Aguilar, Jan Korbel and Yuri Luchko
Mathematics 2019, 7(9), 796; https://doi.org/10.3390/math7090796 - 01 Sep 2019
Cited by 20 | Viewed by 5084
Abstract
In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on [...] Read more.
In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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23 pages, 372 KiB  
Article
Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays
by Grienggrai Rajchakit, Anbalagan Pratap, Ramachandran Raja, Jinde Cao, Jehad Alzabut and Chuangxia Huang
Mathematics 2019, 7(8), 759; https://doi.org/10.3390/math7080759 - 19 Aug 2019
Cited by 121 | Viewed by 4546
Abstract
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop [...] Read more.
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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19 pages, 8152 KiB  
Article
Development of Public Key Cryptographic Algorithm Using Matrix Pattern for Tele-Ultrasound Applications
by Seung-Hyeok Shin, Won-Sok Yoo and Hojong Choi
Mathematics 2019, 7(8), 752; https://doi.org/10.3390/math7080752 - 17 Aug 2019
Cited by 19 | Viewed by 4109
Abstract
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation [...] Read more.
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation during the encrypting process. Our proposed algorithm also provides a faster encoding/decoding speed when the patient’s images for tele-ultrasound applications are transmitted/received, compared to the RSA and ECC encrypting algorithms, because it encodes/decodes the plain memory block by simple addition and multiplication operations of n terms. However, the RSA and ECC algorithms encode/decode each memory block using complex mathematical exponentiation and congruence. To implement encrypting algorithms for tele-ultrasound applications, a streaming server was constructed to transmit the images to the systems using ultrasound machines. Using the obtained ultrasound images from a breast phantom, we compared our developed algorithm, utilizing a matrix pattern, with the RSA and ECC algorithms. The elapsed average time for our proposed algorithm is much faster than that for the RSA and ECC algorithms. Full article
(This article belongs to the Special Issue Information Theory, Cryptography, Randomness and Statistical Modeling)
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6 pages, 255 KiB  
Article
The Application of Fractional Calculus in Chinese Economic Growth Models
by Hao Ming, JinRong Wang and Michal Fečkan
Mathematics 2019, 7(8), 665; https://doi.org/10.3390/math7080665 - 25 Jul 2019
Cited by 28 | Viewed by 4119
Abstract
In this paper, we apply Caputo-type fractional order calculus to simulate China’s gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. Moreover, we compare the results for the fractional model with the [...] Read more.
In this paper, we apply Caputo-type fractional order calculus to simulate China’s gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. Moreover, we compare the results for the fractional model with the integer order model. In addition, we show the importance of variables according to the BIC criterion. The study shows that Caputo fractional order calculus can produce a better model and perform more accurately in predicting the GDP values from 2012–2016. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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9 pages, 254 KiB  
Article
Some New Oscillation Criteria for Second Order Neutral Differential Equations with Delayed Arguments
by Omar Bazighifan and Clemente Cesarano
Mathematics 2019, 7(7), 619; https://doi.org/10.3390/math7070619 - 11 Jul 2019
Cited by 41 | Viewed by 2251
Abstract
In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given. Full article
(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)
8 pages, 224 KiB  
Article
Growth Equation of the General Fractional Calculus
by Anatoly N. Kochubei and Yuri Kondratiev
Mathematics 2019, 7(7), 615; https://doi.org/10.3390/math7070615 - 11 Jul 2019
Cited by 41 | Viewed by 2821
Abstract
We consider the Cauchy problem ( D ( k ) u ) ( t ) = λ u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper [...] Read more.
We consider the Cauchy problem ( D ( k ) u ) ( t ) = λ u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583–600), λ > 0 . The solution is a generalization of the function t ↦ E α ( λ t α ) , where 0 < α < 1 , E α is the Mittag–Leffler function. The asymptotics of this solution, as t → ∞ , are studied. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
10 pages, 254 KiB  
Article
Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control
by Chao Song, Shumin Fei, Jinde Cao and Chuangxia Huang
Mathematics 2019, 7(7), 599; https://doi.org/10.3390/math7070599 - 05 Jul 2019
Cited by 68 | Viewed by 3139
Abstract
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the [...] Read more.
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the same time, the structures of a sliding mode surface and a sliding mode controller are also constructed. A sufficient criterion is presented to ensure the robust synchronization of FOCS according to the stability theory of the fractional calculus and sliding mode control technique. In addition, the result can be applied to identical or non-identical chaotic systems with fractional-order. In the end, we build two practical examples to illustrate the feasibility of our theoretical results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
12 pages, 831 KiB  
Article
Some Qualitative Behavior of Solutions of General Class of Difference Equations
by Osama Moaaz, Dimplekumar Chalishajar and Omar Bazighifan
Mathematics 2019, 7(7), 585; https://doi.org/10.3390/math7070585 - 01 Jul 2019
Cited by 36 | Viewed by 2717
Abstract
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, [...] Read more.
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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38 pages, 596 KiB  
Article
Long-Time Asymptotics of a Three-Component Coupled mKdV System
by Wen-Xiu Ma
Mathematics 2019, 7(7), 573; https://doi.org/10.3390/math7070573 - 27 Jun 2019
Cited by 76 | Viewed by 2803
Abstract
We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, [...] Read more.
We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method. Full article
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50 pages, 2766 KiB  
Review
Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models
by Vasily E. Tarasov
Mathematics 2019, 7(6), 554; https://doi.org/10.3390/math7060554 - 18 Jun 2019
Cited by 30 | Viewed by 3999
Abstract
This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We [...] Read more.
This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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16 pages, 1504 KiB  
Article
Back to Basics: Meaning of the Parameters of Fractional Order PID Controllers
by Inés Tejado, Blas M. Vinagre, José Emilio Traver, Javier Prieto-Arranz and Cristina Nuevo-Gallardo
Mathematics 2019, 7(6), 530; https://doi.org/10.3390/math7060530 - 11 Jun 2019
Cited by 49 | Viewed by 5491
Abstract
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the [...] Read more.
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order λ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with λ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, μ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of μ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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28 pages, 356 KiB  
Review
On History of Mathematical Economics: Application of Fractional Calculus
by Vasily E. Tarasov
Mathematics 2019, 7(6), 509; https://doi.org/10.3390/math7060509 - 04 Jun 2019
Cited by 136 | Viewed by 9352
Abstract
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At [...] Read more.
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
20 pages, 1109 KiB  
Article
Economic Machine-Learning-Based Predictive Control of Nonlinear Systems
by Zhe Wu and Panagiotis D. Christofides
Mathematics 2019, 7(6), 494; https://doi.org/10.3390/math7060494 - 01 Jun 2019
Cited by 40 | Viewed by 5478
Abstract
In this work, a Lyapunov-based economic model predictive control (LEMPC) method is developed to address economic optimality and closed-loop stability of nonlinear systems using machine learning-based models to make predictions. Specifically, an ensemble of recurrent neural network (RNN) models via a k-fold [...] Read more.
In this work, a Lyapunov-based economic model predictive control (LEMPC) method is developed to address economic optimality and closed-loop stability of nonlinear systems using machine learning-based models to make predictions. Specifically, an ensemble of recurrent neural network (RNN) models via a k-fold cross validation is first developed to capture process dynamics in an operating region. Then, the LEMPC using an RNN ensemble is designed to maintain the closed-loop state in a stability region and optimize process economic benefits simultaneously. Parallel computing is employed to improve computational efficiency of real-time implementation of LEMPC with an RNN ensemble. The proposed machine-learning-based LEMPC method is demonstrated using a nonlinear chemical process example. Full article
(This article belongs to the Special Issue Control and Optimization: From Complex Process to Systems Engineering Problems)
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20 pages, 984 KiB  
Article
Joint Inventory and Pricing Policy for an Online to Offline Closed-Loop Supply Chain Model with Random Defective Rate and Returnable Transport Items
by Biswajit Sarkar, Mehran Ullah and Seok-Beom Choi
Mathematics 2019, 7(6), 497; https://doi.org/10.3390/math7060497 - 01 Jun 2019
Cited by 25 | Viewed by 4222
Abstract
Environmental deterioration is one of the current hot topics of the business world. To cope with the negative environmental impacts of corporate activities, researchers introduced the concept of closed-loop supply chain (CLSC) management and remanufacturing. This paper studies joint inventory and pricing decisions [...] Read more.
Environmental deterioration is one of the current hot topics of the business world. To cope with the negative environmental impacts of corporate activities, researchers introduced the concept of closed-loop supply chain (CLSC) management and remanufacturing. This paper studies joint inventory and pricing decisions in a multi-echelon CLSC model that considers online to offline (O2O) business strategy. An imperfect production process is examined with a random defective rate that follows a probability distribution. The results show that the O2O channel increases the profit of the system. For the defective rate, three different distributions are considered and three examples are solved. The results of the three examples conclude that the highest profit is generated when the defective rate follows a uniform distribution. Furthermore, based on the salvage value of defective items, two cases were studied. Results and sensitivity analysis show that the increase in defective rate does not reduce total profit in every situation, as perceived by the existing literature. Sensitivity analysis and numerical examples are given to show robustness of the model and draw important managerial insights. Full article
(This article belongs to the Special Issue Application of Optimization in Production, Logistics, Inventory, Supply Chain Management and Block Chain)
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20 pages, 291 KiB  
Article
Positive Solutions for a Hadamard Fractional p-Laplacian Three-Point Boundary Value Problem
by Jiqiang Jiang, Donal O’Regan, Jiafa Xu and Yujun Cui
Mathematics 2019, 7(5), 439; https://doi.org/10.3390/math7050439 - 17 May 2019
Cited by 23 | Viewed by 2360
Abstract
This article is to study a three-point boundary value problem of Hadamard fractional p-Laplacian differential equation. When our nonlinearity grows ( p − 1 ) -superlinearly and ( p − 1 ) -sublinearly, the existence of positive solutions is obtained via fixed [...] Read more.
This article is to study a three-point boundary value problem of Hadamard fractional p-Laplacian differential equation. When our nonlinearity grows ( p − 1 ) -superlinearly and ( p − 1 ) -sublinearly, the existence of positive solutions is obtained via fixed point index. Moreover, using an increasing operator fixed-point theorem, the uniqueness of positive solutions and uniform convergence sequences are also established. Full article
21 pages, 425 KiB  
Review
Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial
by Roberto Garrappa, Eva Kaslik and Marina Popolizio
Mathematics 2019, 7(5), 407; https://doi.org/10.3390/math7050407 - 07 May 2019
Cited by 88 | Viewed by 6653
Abstract
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches [...] Read more.
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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35 pages, 3547 KiB  
Article
Enhancing Elephant Herding Optimization with Novel Individual Updating Strategies for Large-Scale Optimization Problems
by Jiang Li, Lihong Guo, Yan Li and Chang Liu
Mathematics 2019, 7(5), 395; https://doi.org/10.3390/math7050395 - 30 Apr 2019
Cited by 37 | Viewed by 3946
Abstract
Inspired by the behavior of elephants in nature, elephant herd optimization (EHO) was proposed recently for global optimization. Like most other metaheuristic algorithms, EHO does not use the previous individuals in the later updating process. If the useful information in the previous individuals [...] Read more.
Inspired by the behavior of elephants in nature, elephant herd optimization (EHO) was proposed recently for global optimization. Like most other metaheuristic algorithms, EHO does not use the previous individuals in the later updating process. If the useful information in the previous individuals were fully exploited and used in the later optimization process, the quality of solutions may be improved significantly. In this paper, we propose several new updating strategies for EHO, in which one, two, or three individuals are selected from the previous iterations, and their useful information is incorporated into the updating process. Accordingly, the final individual at this iteration is generated according to the elephant generated by the basic EHO, and the selected previous elephants through a weighted sum. The weights are determined by a random number and the fitness of the elephant individuals at the previous iteration. We incorporated each of the six individual updating strategies individually into the basic EHO, creating six improved variants of EHO. We benchmarked these proposed methods using sixteen test functions. Our experimental results demonstrated that the proposed improved methods significantly outperformed the basic EHO. Full article
(This article belongs to the Special Issue Evolutionary Computation)
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25 pages, 512 KiB  
Article
A Two-Echelon Supply Chain Management With Setup Time and Cost Reduction, Quality Improvement and Variable Production Rate
by Bikash Koli Dey, Biswajit Sarkar and Sarla Pareek
Mathematics 2019, 7(4), 328; https://doi.org/10.3390/math7040328 - 03 Apr 2019
Cited by 39 | Viewed by 5319
Abstract
This model investigates the variable production cost for a production house; under a two-echelon supply chain management where a single vendor and multi-retailers are involved. This production system goes through a long run system and generates an out-of-control state due to different issues [...] Read more.
This model investigates the variable production cost for a production house; under a two-echelon supply chain management where a single vendor and multi-retailers are involved. This production system goes through a long run system and generates an out-of-control state due to different issues and produces defective items. This model considers the reduction of the defective rate and setup cost through investment. A discrete investment for setup cost reduction and a continuous investment is considered to reduce the defective rate and to increase the quality of products. Setup and processing time are dependent on lead time in this model. The model is solved analytically to find the optimal values of the production rate, safety factors, optimum quantity, lead time length, investment for setup cost reduction, and the probability of the production process going out-of-control. An efficient algorithm is constructed to find the optimal solution numerically and sensitivity analysis is given to show the impact of different parameters. A case study and different cases are also given to validate the model. Full article
(This article belongs to the Special Issue Application of Optimization in Production, Logistics, Inventory, Supply Chain Management and Block Chain)
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10 pages, 273 KiB  
Article
The Bounds of Vertex Padmakar–Ivan Index on k-Trees
by Shaohui Wang, Zehui Shao, Jia-Bao Liu and Bing Wei
Mathematics 2019, 7(4), 324; https://doi.org/10.3390/math7040324 - 01 Apr 2019
Cited by 18 | Viewed by 2687
Abstract
The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u [...] Read more.
The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
22 pages, 3000 KiB  
Article
The Multivariate Theory of Connections
by Daniele Mortari and Carl Leake
Mathematics 2019, 7(3), 296; https://doi.org/10.3390/math7030296 - 22 Mar 2019
Cited by 37 | Viewed by 4423
Abstract
This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called [...] Read more.
This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g ( x , y ) , satisfy all constraints no matter what the g ( x , y ) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. Full article
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
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18 pages, 2526 KiB  
Article
An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation
by Pundikala Veeresha, Doddabhadrappla Gowda Prakasha and Dumitru Baleanu
Mathematics 2019, 7(3), 265; https://doi.org/10.3390/math7030265 - 14 Mar 2019
Cited by 50 | Viewed by 3164
Abstract
The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with [...] Read more.
The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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24 pages, 5440 KiB  
Article
Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials
by Harendra Singh, Rajesh K. Pandey and Hari Mohan Srivastava
Mathematics 2019, 7(3), 224; https://doi.org/10.3390/math7030224 - 27 Feb 2019
Cited by 41 | Viewed by 3394
Abstract
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to [...] Read more.
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature. Full article
(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)
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17 pages, 307 KiB  
Article
Dynamic Keynesian Model of Economic Growth with Memory and Lag
by Vasily E. Tarasov and Valentina V. Tarasova
Mathematics 2019, 7(2), 178; https://doi.org/10.3390/math7020178 - 15 Feb 2019
Cited by 20 | Viewed by 5351
Abstract
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the [...] Read more.
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
15 pages, 2766 KiB  
Article
A Partial-Consensus Posterior-Aggregation FAHP Method—Supplier Selection Problem as an Example
by Yu-Cheng Wang and Tin-Chih Toly Chen
Mathematics 2019, 7(2), 179; https://doi.org/10.3390/math7020179 - 15 Feb 2019
Cited by 37 | Viewed by 2939
Abstract
Existing fuzzy analytic hierarchy process (FAHP) methods usually aggregate the fuzzy pairwise comparison results produced by multiple decision-makers (DMs) rather than the fuzzy weights estimations. This is problematic because fuzzy pairwise comparison results are subject to uncertainty and lack consensus. To address this [...] Read more.
Existing fuzzy analytic hierarchy process (FAHP) methods usually aggregate the fuzzy pairwise comparison results produced by multiple decision-makers (DMs) rather than the fuzzy weights estimations. This is problematic because fuzzy pairwise comparison results are subject to uncertainty and lack consensus. To address this problem, a partial-consensus posterior-aggregation FAHP (PCPA-FAHP) approach is proposed in this study. The PCPA-FAHP approach seeks a partial consensus among most DMs instead of an overall consensus among all DMs, thereby increasing the possibility of reaching a consensus. Subsequently, the aggregation result is defuzzified using the prevalent center-of-gravity method. The PCPA-FAHP approach was applied to a supplier selection problem to validate its effectiveness. According to the experimental results, the PCPA-FAHP approach not only successfully found out the partial consensus among the DMs, but also shrunk the widths of the estimated fuzzy weights to enhance the precision of the FAHP analysis. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
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15 pages, 280 KiB  
Article
Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain
by Hari M. Srivastava, Qazi Zahoor Ahmad, Nasir Khan, Nazar Khan and Bilal Khan
Mathematics 2019, 7(2), 181; https://doi.org/10.3390/math7020181 - 15 Feb 2019
Cited by 104 | Viewed by 3478
Abstract
By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel [...] Read more.
By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results. Full article
(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)
14 pages, 254 KiB  
Article
New Integral Inequalities via the Katugampola Fractional Integrals for Functions Whose Second Derivatives Are Strongly η-Convex
by Seth Kermausuor, Eze R. Nwaeze and Ana M. Tameru
Mathematics 2019, 7(2), 183; https://doi.org/10.3390/math7020183 - 15 Feb 2019
Cited by 17 | Viewed by 2681
Abstract
In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly η -convex functions via the Katugampola fractional integrals. Full article
(This article belongs to the Special Issue Inequalities)
26 pages, 747 KiB  
Article
Lexicographic Orders of Intuitionistic Fuzzy Values and Their Relationships
by Feng Feng, Meiqi Liang, Hamido Fujita, Ronald R. Yager and Xiaoyan Liu
Mathematics 2019, 7(2), 166; https://doi.org/10.3390/math7020166 - 13 Feb 2019
Cited by 45 | Viewed by 2664
Abstract
Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a [...] Read more.
Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a number of lexicographic orders by means of several measures such as the membership, non-membership, score, accuracy and expectation score functions. Some equivalent characterizations and illustrative examples are provided, from which the relationships among these lexicographic orders are ascertained. We also propose three different compatible properties of preorders with respect to the algebraic sum and scalar product operations of intuitionistic fuzzy values, and apply them to the investigation of compatible properties of various lexicographic orders. In addition, a benchmark problem regarding risk investment is further explored to give a comparative analysis of different lexicographic orders and highlight the practical value of the obtained results for solving real-world decision-making problems. Full article
(This article belongs to the Section Engineering Mathematics)
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35 pages, 924 KiB  
Article
Analysis of General Humoral Immunity HIV Dynamics Model with HAART and Distributed Delays
by A. M. Elaiw and E. Kh. Elnahary
Mathematics 2019, 7(2), 157; https://doi.org/10.3390/math7020157 - 09 Feb 2019
Cited by 52 | Viewed by 2456
Abstract
This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4 + T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. [...] Read more.
This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4 + T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The model is incorporated with distributed time delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The effect of antibody on HIV infection is analyzed. The production and removal rates of the ten compartments of the model are given by general nonlinear functions which satisfy reasonable conditions. Nonnegativity and ultimately boundedness of the solutions are proven. Using the Lyapunov method, the global stability of the equilibria of the model is proven. Numerical simulations of the system are provided to confirm the theoretical results. We have shown that the antibodies can play a significant role in controlling the HIV infection, but it cannot clear the HIV particles from the plasma. Moreover, we have demonstrated that the intracellular time delay plays a similar role as the Highly Active Antiretroviral Therapies (HAAT) drugs in eliminating the HIV particles. Full article
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14 pages, 358 KiB  
Article
Fractional Derivatives: The Perspective of System Theory
by Manuel Duarte Ortigueira and José Tenreiro Machado
Mathematics 2019, 7(2), 150; https://doi.org/10.3390/math7020150 - 05 Feb 2019
Cited by 45 | Viewed by 3695
Abstract
This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify [...] Read more.
This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify this important topic in Fractional Calculus. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
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5 pages, 250 KiB  
Article
Desiderata for Fractional Derivatives and Integrals
by Rudolf Hilfer and Yuri Luchko
Mathematics 2019, 7(2), 149; https://doi.org/10.3390/math7020149 - 04 Feb 2019
Cited by 87 | Viewed by 5196
Abstract
The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead [...] Read more.
The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
17 pages, 2881 KiB  
Article
A Novel Bat Algorithm with Multiple Strategies Coupling for Numerical Optimization
by Yechuang Wang, Penghong Wang, Jiangjiang Zhang, Zhihua Cui, Xingjuan Cai, Wensheng Zhang and Jinjun Chen
Mathematics 2019, 7(2), 135; https://doi.org/10.3390/math7020135 - 01 Feb 2019
Cited by 105 | Viewed by 11538
Abstract
A bat algorithm (BA) is a heuristic algorithm that operates by imitating the echolocation behavior of bats to perform global optimization. The BA is widely used in various optimization problems because of its excellent performance. In the bat algorithm, the global search capability [...] Read more.
A bat algorithm (BA) is a heuristic algorithm that operates by imitating the echolocation behavior of bats to perform global optimization. The BA is widely used in various optimization problems because of its excellent performance. In the bat algorithm, the global search capability is determined by the parameter loudness and frequency. However, experiments show that each operator in the algorithm can only improve the performance of the algorithm at a certain time. In this paper, a novel bat algorithm with multiple strategies coupling (mixBA) is proposed to solve this problem. To prove the effectiveness of the algorithm, we compared it with CEC2013 benchmarks test suits. Furthermore, the Wilcoxon and Friedman tests were conducted to distinguish the differences between it and other algorithms. The results prove that the proposed algorithm is significantly superior to others on the majority of benchmark functions. Full article
(This article belongs to the Special Issue Evolutionary Computation)
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6 pages, 746 KiB  
Article
Calculating Nodal Voltages Using the Admittance Matrix Spectrum of an Electrical Network
by Ioannis Dassios, Andrew Keane and Paul Cuffe
Mathematics 2019, 7(1), 106; https://doi.org/10.3390/math7010106 - 20 Jan 2019
Cited by 10 | Viewed by 4016
Abstract
Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These [...] Read more.
Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These insights should permit the functioning of electrical networks to be understood in the context of spectral analysis. Full article
(This article belongs to the Special Issue Mathematical Methods in Applied Sciences)
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8 pages, 235 KiB  
Article
ω-Interpolative Ćirić-Reich-Rus-Type Contractions
by Hassen Aydi, Erdal Karapinar and Antonio Francisco Roldán López de Hierro
Mathematics 2019, 7(1), 57; https://doi.org/10.3390/math7010057 - 08 Jan 2019
Cited by 97 | Viewed by 3402
Abstract
In this paper, using the concept of ω -admissibility, we prove some fixed point results for interpolate Ćirić-Reich-Rus-type contraction mappings. We also present some consequences and a useful example. Full article
(This article belongs to the Special Issue Recent Advances in Fixed Point Theory and Its Applications)
15 pages, 793 KiB  
Article
An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems
by Yonghong Yao, Mihai Postolache and Jen-Chih Yao
Mathematics 2019, 7(1), 61; https://doi.org/10.3390/math7010061 - 08 Jan 2019
Cited by 98 | Viewed by 5138
Abstract
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested [...] Read more.
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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