1,938 results
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2. (CMMSE paper) A finite‐difference model for indoctrination dynamics
- Author
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María G. Medina-Guevara, Héctor Vargas-Rodríguez, and Pedro B. Espinoza-Padilla
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Agent-based model ,Finite difference model ,Opinion dynamics ,General Mathematics ,Dynamics (mechanics) ,Indoctrination ,General Engineering ,Applied mathematics ,Mathematics - Published
- 2018
3. Comment on the paper 'Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions, Yu Zhang, Yanyan Zhang, Jinhuan Wang Mathematical Methods in the Applied Sciences , 2018; 41 :7678–7697'
- Author
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Asterios Pantokratoras
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Delta ,Shock wave ,symbols.namesake ,General Mathematics ,Mathematical analysis ,Zhàng ,General Engineering ,symbols ,Applied science ,Compressible fluid flow ,Mathematics ,Euler equations - Published
- 2020
4. On mistaken papers by Gouzheng Yan et al and related papers, and on a paper by Weibing Wang and Xuxin Yang
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Pavel A. Krutitskii
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General Mathematics ,MathematicsofComputing_GENERAL ,General Engineering ,Calculus ,Point (geometry) ,Boundary value problem ,Compact operator ,Integral equation ,Self-adjoint operator ,Mathematics - Abstract
The purpose of the present note is to point out mathematical mistakes in some published papers on boundary value problems to prevent usage of mistaken results and methods by mathematicians. Copyright © 2013 John Wiley & Sons, Ltd.
- Published
- 2013
5. A note on the paper ‘Analytical approach to heat and mass transfer in MHD free convection from a moving permeable vertical surface’ by A. Asgharian, D.D. Ganji, S. Soleimani, S. Asgharian, N. Sedaghatyzade and B. Mohammadi, Mathematical Methods in the App
- Author
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Tiegang Fang and Asterios Pantokratoras
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Surface (mathematics) ,Natural convection ,Combined forced and natural convection ,General Mathematics ,Mass transfer ,Heat transfer ,General Engineering ,Thermodynamics ,Magnetohydrodynamics ,Mathematics - Published
- 2014
6. A fractional‐order model of coronavirus disease 2019 (COVID‐19) with governmental action and individual reaction
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Jaouad Danane, Zakia Hammouch, Karam Allali, Saima Rashid, and Jagdev Singh
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78a70 ,Risk awareness ,2019-20 coronavirus outbreak ,Special Issue Papers ,Coronavirus disease 2019 (COVID-19) ,basic infection reproduction number ,General Mathematics ,Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ,General Engineering ,34a08 ,Fractional calculus ,37n25 ,Caputo fractional‐order derivative ,sensitivity analysis ,Action (philosophy) ,COVID‐19 ,Order (exchange) ,numerical simulation ,Special Issue Paper ,Econometrics ,26a33 ,Basic reproduction number ,Mathematics - Abstract
The deadly coronavirus disease 2019 (COVID-19) has recently affected each corner of the world. Many governments of different countries have imposed strict measures in order to reduce the severity of the infection. In this present paper, we will study a mathematical model describing COVID-19 dynamics taking into account the government action and the individuals reaction. To this end, we will suggest a system of seven fractional deferential equations (FDEs) that describe the interaction between the classical susceptible, exposed, infectious, and removed (SEIR) individuals along with the government action and individual reaction involvement. Both human-to-human and zoonotic transmissions are considered in the model. The well-posedness of the FDEs model is established in terms of existence, positivity, and boundedness. The basic reproduction number (BRN) is found via the new generation matrix method. Different numerical simulations were carried out by taking into account real reported data from Wuhan, China. It was shown that the governmental action and the individuals' risk awareness reduce effectively the infection spread. Moreover, it was established that with the fractional derivative, the infection converges more quickly to its steady state.
- Published
- 2021
7. A study on fractional COVID‐19 disease model by using Hermite wavelets
- Author
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Shaher Momani, Ranbir Kumar, Samir Hadid, and Sunil Kumar
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General Mathematics ,coronavirus ,Value (computer science) ,Derivative ,34a34 ,01 natural sciences ,Caputo derivative ,convergence analysis ,Wavelet ,Special Issue Paper ,operational matrix ,Applied mathematics ,0101 mathematics ,26a33 ,Hermite wavelets ,Mathematics ,Hermite polynomials ,Collocation ,Special Issue Papers ,Basis (linear algebra) ,010102 general mathematics ,General Engineering ,34a08 ,010101 applied mathematics ,Algebraic equation ,Scheme (mathematics) ,60g22 ,mathematical model - Abstract
The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.
- Published
- 2021
8. Mathematical modeling of the spread of the coronavirus under strict social restrictions
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Khalid Dib, Kalyanasundaram Madhu, Mo'tassem Al-arydah, and Hailay Weldegiorgis Berhe
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Download ,General Mathematics ,media_common.quotation_subject ,coronavirus ,92Bxx ,Permission ,Unit (housing) ,COVID‐19 ,37Nxx ,Special Issue Paper ,Econometrics ,Quality (business) ,Mathematics ,media_common ,Notice ,Special Issue Papers ,Social distance ,Warranty ,General Engineering ,social distancing ,92b05 ,37n25 ,parameter estimations ,Order (business) ,variable transmission rate ,mathematical model - Abstract
We formulate a simple susceptible‐infectious‐recovery (SIR) model to describe the spread of the coronavirus under strict social restrictions. The transmission rate in this model is exponentially decreasing with time. We find a formula for basic reproduction function and estimate the maximum number of daily infected individuals. We fit the model to induced death data in Italy, United States, Germany, France, India, Spain, and China over the period from the first reported death to August 7, 2020. We notice that the model has excellent fit to the disease death data in these countries. We estimate the model's parameters in each of these countries with 95% confidence intervals. We order the strength of social restrictions in these countries using the exponential rate. We estimate the time needed to reduce the basic reproduction function to one unit and use it to order the quality of social restrictions in these countries. The social restriction in China was the strictest and the most effective and in India was the weakest and the least effective. Policy‐makers may apply the Chinese successful social restriction experiment and avoid the Indian unsuccessful one. [ FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)
- Published
- 2021
9. Analytical and qualitative investigation of COVID‐19 mathematical model under fractional differential operator
- Author
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Ali Ahmadian, Muhammad Sher, Kamal Shah, Soheil Salahshour, Bruno Antonio Pansera, and Hussam Rabai'ah
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Coronavirus disease 2019 (COVID-19) ,Special Issue Papers ,novel coronavirus mathematical models ,General Mathematics ,General Engineering ,65l05 ,Fractional differential operator ,34a12 ,analytical results ,graphical interpretation ,Special Issue Paper ,Applied mathematics ,fractional‐order derivative ,Adomian decomposition method ,26a33 ,Mathematics - Abstract
In the current article, we aim to study in detail a novel coronavirus (2019-nCoV or COVID-19) mathematical model for different aspects under Caputo fractional derivative. First, from analysis point of view, existence is necessary to be investigated for any applied problem. Therefore, we used fixed point theorem's due to Banach's and Schaefer's to establish some sufficient results regarding existence and uniqueness of the solution to the proposed model. On the other hand, stability is important in respect of approximate solution, so we have developed condition sufficient for the stability of Ulam-Hyers and their different types for the considered system. In addition, the model has also been considered for semianalytical solution via Laplace Adomian decomposition method (LADM). On Matlab, by taking some real data about Pakistan, we graph the obtained results. In the last of the manuscript, a detail discussion and brief conclusion are provided.
- Published
- 2021
10. Improving the performance of deep learning models using statistical features: The case study of COVID‐19 forecasting
- Author
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Hossein Abbasimehr, Reza Paki, and Aram Bahrini
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2019-20 coronavirus outbreak ,Coronavirus disease 2019 (COVID-19) ,62‐07 ,General Mathematics ,Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ,Context (language use) ,97r40 ,Machine learning ,computer.software_genre ,01 natural sciences ,Convolutional neural network ,Special Issue Paper ,0101 mathematics ,Combined method ,Mathematics ,Special Issue Papers ,business.industry ,Deep learning ,010102 general mathematics ,General Engineering ,deep learning ,COVID‐19 pandemic ,010101 applied mathematics ,hybrid methods ,Memory model ,Artificial intelligence ,business ,computer ,statistical features - Abstract
COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.
- Published
- 2021
11. The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative
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Pushpendra Kumar and Vedat Suat Erturk
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COVID‐19 epidemic ,Caputo fractional derivative ,Coronavirus disease 2019 (COVID-19) ,Special Issue Papers ,Banach fixed-point theorem ,General Mathematics ,fixed point theory ,34c60 ,General Engineering ,Fixed-point theorem ,predictor–corrector scheme ,Lipschitz continuity ,time delay ,SEIR model ,Fractional calculus ,92c60 ,Norm (mathematics) ,92d30 ,Special Issue Paper ,Applied mathematics ,Fractional differential ,Epidemic model ,26a33 ,Mathematics - Abstract
Novel coronavirus (COVID-19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID-19 SEIR epidemic model via Caputo fractional derivatives using a predictor-corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag-Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs.
- Published
- 2020
12. (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications
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Ana Navarro-Quiles, M.-D. Roselló, José Vicente Romero, and Juan Carlos Cortés
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education.field_of_study ,Differential equation ,Stochastic process ,General Mathematics ,Computation ,010102 general mathematics ,Population ,General Engineering ,Probability density function ,01 natural sciences ,010101 applied mathematics ,Transformation (function) ,Applied mathematics ,0101 mathematics ,Logistic function ,education ,Random variable ,Mathematics - Abstract
The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.
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13. A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives
- Author
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Pushpendra Kumar and Vedat Suat Erturk
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Covid‐19 epidemic ,General Mathematics ,Banach space ,Fixed-point theorem ,new generalised Caputo non‐integer order derivative ,01 natural sciences ,92c60 ,Special Issue Paper ,Applied mathematics ,Uniform boundedness ,Uniqueness ,0101 mathematics ,26a33 ,Mathematics ,Special Issue Papers ,fixed point theory ,010102 general mathematics ,34c60 ,General Engineering ,Equicontinuity ,Fractional calculus ,010101 applied mathematics ,Norm (mathematics) ,92d30 ,Predictor‐Corrector scheme ,Epidemic model ,mathematical model - Abstract
The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.
- Published
- 2020
14. Wastewater bioremediation using white rot fungi: Validation of a dynamical system with real data obtained in laboratory
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Iulia Martina Bulai, Giovanna Cristina Varese, Ezio Venturino, and Federica Spina
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parameter fitting ,010304 chemical physics ,General Mathematics ,General Engineering ,010501 environmental sciences ,Biodegradation ,Pulp and paper industry ,Dynamical system ,biodegradation ,01 natural sciences ,dynamical system ,fungi ,wastewater ,Bioremediation ,Wastewater ,0103 physical sciences ,White rot ,0105 earth and related environmental sciences ,Mathematics - Published
- 2018
15. A scattering problem for a local perturbation of an open periodic waveguide
- Author
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Kirsch A
- Subjects
Physics ,Optics ,Scattering ,business.industry ,General Mathematics ,General Engineering ,Perturbation (astronomy) ,Waveguide (acoustics) ,ddc:510 ,business ,Mathematics - Abstract
In this paper, we consider the propagation of waves in an open waveguide in ℝ$^{2}$ where the index of refraction is a local perturbation of a function which is periodic along the axis of the waveguide (which we choose to be the x$_{1}$ axis) and equal to one for |x$_{2}$| > h$_{0}$ for some h$_{0}$ > 0. Motivated by the limiting absorption principle (proven in an earlier paper by the author), we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution under the assumption that no bound states exist. In the second part, we determine the order of decay of the radiating part of the solution in the direction of the layer and in the direction orthogonal to it. Finally, we show that it satisfies the classical Sommerfeld radiation condition and allows the definition of a far field pattern.
- Published
- 2022
16. Addendum to 'On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space', Math Meth Appl Sci. 2020; 1–8
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Humberto Rafeiro and Stefan Samko
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Pure mathematics ,Variable exponent ,Riesz potential ,General Mathematics ,Operator (physics) ,General Engineering ,Addendum ,Variable exponent Campanato spaces ,Space (mathematics) ,Variable exponent Morrey spaces ,Order (group theory) ,Fractional integral ,BMO ,Mathematics ,Variable (mathematics) - Abstract
In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order 𝛼(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and 𝜆(x) of the Morrey space. Assumptions on the exponents were different depending on whether 𝛼(x)p(x)−n+𝜆(x) p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0 ⩽ 𝛼(x)p(x)−n+𝜆(x) p(x) ⩽ 1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. info:eu-repo/semantics/publishedVersion
- Published
- 2021
17. A nonlocal multi‐point singular fractional integro‐differential problem of Lane–Emden type
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Zoubir Dahmani, Yazid Gouari, Mehmet Zeki Sarikaya, and [Belirlenecek]
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Equation ,General Mathematics ,General Engineering ,Positive Solutions ,Existence ,Type (model theory) ,multi-point problem ,Caputo derivative ,Lane-Emden equation ,singular equation ,Applied mathematics ,Lane–Emden equation ,Singular equation ,existence of solution ,Differential (mathematics) ,Multi point ,Model ,Mathematics - Abstract
In this paper, using Riemann-Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane-Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results. DGRSDT, Direction Generale de la Recherche Scientifique et du Developpement Technologique, Algeria The authors express a special thanks to the associate editor and referees for their motivated comments that made the original manuscript significant and improved. This paper is supported by DGRSDT, Direction Generale de la Recherche Scientifique et du Developpement Technologique, Algeria. WOS:000526582000001 2-s2.0-85084038419
- Published
- 2020
18. Solutions of fractional gas dynamics equation by a new technique
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Alicia Cordero Barbero, Juan Ramón Torregrosa Sánchez, and Ali Akgül
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Fractional gas dynamics equation ,General Mathematics ,Operators ,010102 general mathematics ,Hilbert space ,General Engineering ,Gas dynamics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,MATEMATICA APLICADA ,Mathematics ,Mathematical physics - Abstract
[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.
- Published
- 2019
19. Fractional-order backstepping strategy for fractional-order model of COVID-19 outbreak
- Author
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Hadi Delavari and Amir Veisi
- Subjects
Transmission (telecommunications) ,Operations research ,Download ,Control theory ,General Mathematics ,Backstepping ,Warranty ,Control (management) ,General Engineering ,Permission ,Sliding mode control ,Mathematics - Abstract
The coronavirus disease (COVID‐19) pandemic has impacted many nations around the world. Recently, new variant of this virus has been identified that have a much higher rate of transmission. Although vaccine production and distribution are currently underway, non‐pharmacological interventions are still being implemented as an important and fundamental strategy to control the spread of the virus in countries around the world. To realize and forecast the transmission dynamics of this disease, mathematical models can be very effective. Various mathematical modeling methods have been proposed to investigate the transmission patterns of this new infection. In this paper, we utilized the fractional‐order dynamics of COVID‐19. The goal is to control the prevalence of the disease using non‐pharmacological interventions. In this paper, a novel fractional‐order backstepping sliding mode control (FOBSMC) is proposed for non‐pharmacological decisions. Recently, new variant of this virus have been identified that have a much higher rate of transmission, so finally the effectiveness of the proposed controller in the presence of new variant of COVID‐19 is investigated. [ FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)
- Published
- 2021
20. Generalized form of fractional order COVID-19 model with Mittag-Leffler kernel
- Author
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Ali Akgül, Muhammad Aslam, Aqeel Ahmad, Meng Sun, and Muhammad Farman
- Subjects
Sumudu transform ,Dynamical systems theory ,General Mathematics ,Type (model theory) ,93b07 ,01 natural sciences ,93b05 ,Mittag–Leffler kernel ,COVID‐19 ,numerical methods ,Applied mathematics ,0101 mathematics ,Research Articles ,Mathematics ,37c75 ,Numerical analysis ,010102 general mathematics ,General Engineering ,Parity (physics) ,Function (mathematics) ,65l07 ,Fractional calculus ,010101 applied mathematics ,Kernel (statistics) ,Unit (ring theory) ,Research Article - Abstract
An important advantage of fractional derivatives is that we can formulate models describing much better systems with memory effects. Fractional operators with different memory are related to the different type of relaxation process of the nonlocal dynamical systems. Therefore, we investigate the COVID-19 model with the fractional derivatives in this paper. We apply very effective numerical methods to obtain the numerical results. We also use the Sumudu transform to get the solutions of the models. The Sumudu transform is able to keep the unit of the function, the parity of the function, and has many other properties that are more valuable. We present scientific results in the paper and also prove these results by effective numerical techniques which will be helpful to understand the outbreak of COVID-19.
- Published
- 2020
21. Fractional powers of the noncommutative Fourier's law by theS‐spectrum approach
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Stefano Pinton, Samuele Mongodi, Marco M. Peloso, Fabrizio Colombo, Colombo, F, Mongodi, S, Peloso, M, and Pinton, S
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S-spectrum ,General Mathematics ,fractional diffusion processe ,General Engineering ,fractional diffusion processes ,Spectrum (topology) ,Noncommutative geometry ,fractional Fourier's law ,symbols.namesake ,Engineering (all) ,Fourier transform ,the S-spectrum approach ,symbols ,Mathematics (all) ,fractional powers of vector operators ,fractional powers of vector operator ,Mathematical physics ,Mathematics - Abstract
Let e ℓ , for ℓ = 1,2,3, be orthogonal unit vectors in R 3 and let Ω ⊂ R 3 be a bounded open set with smooth boundary ∂Ω. Denoting by x a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T = a(x)∂xe1 + b(x)∂ye2 + c(x)∂ze3, into the conservation of energy law, here a, b, c ∶ Ω → R are given functions. With the S-spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c ∶ Ω → R the fractional powers of T exist in the sense of the S-spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.
- Published
- 2019
22. On integral operators in weighted grand Lebesgue spaces of Banach-valued functions
- Author
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Alexander Meskhi and Vakhtang Kokilashvili
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Mathematics::Functional Analysis ,Pure mathematics ,Weight function ,General Mathematics ,010102 general mathematics ,Diagonal ,Mathematics::Classical Analysis and ODEs ,General Engineering ,Singular integral ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Multiplier (Fourier analysis) ,Maximal function ,0101 mathematics ,Lp space ,Constant (mathematics) ,Mathematics - Abstract
The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator we rely on the reasonable bound of the sharp constant in the Buckley type theorem which is also derived in the paper.
- Published
- 2020
23. Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion
- Author
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Takeshi Fukao, Shunsuke Kurima, and Tomomi Yokota
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General Mathematics ,010102 general mathematics ,General Engineering ,Stefan problem ,Cauchy distribution ,Monotonic function ,Subderivative ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Mathematics - Analysis of PDEs ,Domain (ring theory) ,FOS: Mathematics ,Nonlinear diffusion ,Uniqueness ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.
- Published
- 2018
24. Time-harmonic and asymptotically linear Maxwell equations in anisotropic media
- Author
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Xianhua Tang and Dongdong Qin
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Lipschitz domain ,Maxwell's equations ,Bounded function ,Homogeneous space ,symbols ,Tensor ,Boundary value problem ,0101 mathematics ,Perfect conductor ,Nehari manifold ,Mathematics - Abstract
This paper is focused on following time-harmonic Maxwell equation: ∇×(μ−1(x)∇×u)−ω2e(x)u=f(x,u),inΩ,ν×u=0,on∂Ω, where Ω⊂R3 is a bounded Lipschitz domain, ν:∂Ω→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as |u|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor e∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.
- Published
- 2017
25. The Cauchy problem of a fluid-particle interaction model with external forces
- Author
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Zaihong Jiang, Ning Zhong, and Li Li
- Subjects
Cauchy problem ,Picard–Lindelöf theorem ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Interaction model ,01 natural sciences ,010101 applied mathematics ,Fluid particle ,Nonlinear system ,Decomposition (computer science) ,Initial value problem ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we consider the Cauchy problem of a fluid-particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro-micro decomposition.
- Published
- 2017
26. Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term
- Author
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Masakazu Kato and Yoshihiro Ueda
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Lower order ,Damped wave ,Space (mathematics) ,01 natural sciences ,Burgers' equation ,Term (time) ,010101 applied mathematics ,Initial value problem ,Nonlinear convection ,0101 mathematics ,Representation (mathematics) ,Mathematics - Abstract
This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.
- Published
- 2017
27. Computation of periodic orbits in three-dimensional Lotka-Volterra systems
- Author
-
Rubén Poveda and Juan F. Navarro
- Subjects
Series (mathematics) ,General Mathematics ,Computation ,Mathematical analysis ,General Engineering ,Periodic sequence ,010103 numerical & computational mathematics ,Systems modeling ,Symbolic computation ,01 natural sciences ,Poincaré–Lindstedt method ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,symbols ,Periodic orbits ,0101 mathematics ,Mathematics - Abstract
This paper deals with an adaptation of the Poincare-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.
- Published
- 2017
28. Bounds for Shannon and Zipf-Mandelbrot entropies
- Author
-
Muhammad Adil Khan, Đilda Pečarić, and Josip Pečarić
- Subjects
convex function ,Jensen inequality ,Shannon entropy ,Zipf-Mandelbrot entropy ,Discrete mathematics ,Mathematics::Dynamical Systems ,Shannon's source coding theorem ,General Mathematics ,010102 general mathematics ,General Engineering ,Maximum entropy thermodynamics ,Min entropy ,Entropy in thermodynamics and information theory ,01 natural sciences ,010101 applied mathematics ,Rényi entropy ,Entropy power inequality ,Combinatorics ,0101 mathematics ,Entropic uncertainty ,Limiting density of discrete points ,Mathematics - Abstract
Shannon and Zipf-Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well-know Jensen inequality and obtain different bounds for Shannon and Zipf-Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf-Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf-Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandelbrot entropy.
- Published
- 2017
29. Monotonicity, uniqueness, and stability of traveling waves in a nonlocal reaction-diffusion system with delay
- Author
-
Hai-Qin Zhao
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Monotonic function ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Transmission (telecommunications) ,Stability theory ,Reaction–diffusion system ,Traveling wave ,Uniqueness ,0101 mathematics ,Epidemic model ,Mathematics - Abstract
The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾c∗. Then we show that the traveling wave fronts with speed c>c∗ are exponentially asymptotically stable.
- Published
- 2017
30. Partial affine system-based frames and dual frames
- Author
-
Yu Tian and Yun-Zhang Li
- Subjects
Harris affine region detector ,General Mathematics ,010102 general mathematics ,General Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Affine plane ,Affine coordinate system ,Affine shape adaptation ,Affine combination ,Affine hull ,Affine group ,Affine transformation ,0101 mathematics ,Algorithm ,Mathematics - Abstract
In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.
- Published
- 2017
31. Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting
- Author
-
Lei Kong and Changrong Zhu
- Subjects
Cusp (singularity) ,Phase portrait ,General Mathematics ,010102 general mathematics ,General Engineering ,Codimension ,Type (model theory) ,01 natural sciences ,Nonlinear Sciences::Chaotic Dynamics ,010101 applied mathematics ,Control theory ,Limit cycle ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Homoclinic bifurcation ,Limit (mathematics) ,Homoclinic orbit ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.
- Published
- 2017
32. The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem
- Author
-
Manuel Arana-Jiménez and Tadeusz Antczak
- Subjects
Mathematical optimization ,021103 operations research ,Optimization problem ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,General Engineering ,Constrained optimization ,02 engineering and technology ,01 natural sciences ,Global optimal ,Local optimum ,Random optimization ,Differentiable function ,0101 mathematics ,Global optimization ,Equivalence (measure theory) ,Mathematics - Abstract
In the paper, a necessary and sufficient criterion it provided such that any local optimal solution is also global in a not necessarily differentiable constrained optimization problem. This criterion is compared to others earlier appeared in the literature, which are sufficient but not necessary for a local optimal solution to be global. The importance of the established criterion is illustrated by suitable examples of nonconvex optimization problems presented in the paper.
- Published
- 2017
33. On the stability and nonexistence of turing patterns for the generalized Lengyel-Epstein model
- Author
-
Salem Abdelmalek, Samir Bendoukha, and Belgacem Rebiai
- Subjects
010101 applied mathematics ,Lyapunov functional ,Turing patterns ,General Mathematics ,010102 general mathematics ,General Engineering ,Stability (learning theory) ,Applied mathematics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Abstract
This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.
- Published
- 2017
34. A novel simulation methodology of fractional order nuclear science model
- Author
-
Ali Akgül and Yasir Khan
- Subjects
Scheme (programming language) ,Mathematical optimization ,Computer simulation ,Process (engineering) ,General Mathematics ,010102 general mathematics ,General Engineering ,Order (ring theory) ,010103 numerical & computational mathematics ,Nuclear reactor ,01 natural sciences ,law.invention ,Fractional calculus ,law ,Simple (abstract algebra) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,0101 mathematics ,Nuclear science ,computer ,Mathematics ,computer.programming_language - Abstract
In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
35. A data assimilation process for linear ill-posed problems
- Author
-
X.-M. Yang and Z.-L. Deng
- Subjects
Well-posed problem ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,Bayesian probability ,Posterior probability ,General Engineering ,Markov chain Monte Carlo ,Inverse problem ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Data assimilation ,symbols ,Applied mathematics ,Ensemble Kalman filter ,0101 mathematics ,Randomness ,Mathematics - Abstract
In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
36. Kantorovich variant of a new kind ofq-Bernstein-Schurer operators
- Author
-
Nurhayat Ispir, Ruchi Ruchi, and Purshottam Narain Agrawal
- Subjects
Constant coefficients ,General Mathematics ,010102 general mathematics ,General Engineering ,Microlocal analysis ,010103 numerical & computational mathematics ,Spectral theorem ,Operator theory ,Lipschitz continuity ,01 natural sciences ,Fourier integral operator ,Algebra ,Rate of convergence ,Applied mathematics ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu etal. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright (c) 2017 John Wiley & Sons, Ltd.
- Published
- 2017
37. Some inequalities involving Hadamard-type k -fractional integral operators
- Author
-
Praveen Agarwal
- Subjects
Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,General Engineering ,Riemann integral ,Type (model theory) ,01 natural sciences ,Fourier integral operator ,Fractional calculus ,010101 applied mathematics ,Algebra ,symbols.namesake ,Hadamard transform ,Improper integral ,symbols ,Daniell integral ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2017
38. Two homoclinic solutions for a nonperiodic fourth-order differential equation without coercive condition
- Author
-
Shiping Lu and Tao Zhong
- Subjects
Class (set theory) ,Differential equation ,General Mathematics ,Open problem ,Mathematical analysis ,General Engineering ,01 natural sciences ,010101 applied mathematics ,Fourth order ,Variational method ,0103 physical sciences ,Mountain pass theorem ,Homoclinic orbit ,0101 mathematics ,010306 general physics ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations u(4)+wu′′+a(x)u=f(x,u), where w is a constant, a∈C(R,R) and f∈C(R×R,R). By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well-known (AR)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
39. Controllability of a class of heat equations with memory in one dimension
- Author
-
Xiuxiang Zhou and Hang Gao
- Subjects
0209 industrial biotechnology ,General Mathematics ,010102 general mathematics ,Null (mathematics) ,Mathematical analysis ,General Engineering ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Volterra integral equation ,Controllability ,symbols.namesake ,020901 industrial engineering & automation ,Dimension (vector space) ,symbols ,Initial value problem ,State space ,Heat equation ,0101 mathematics ,Mathematics - Abstract
This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
40. Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line
- Author
-
Renkun Shi
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Half-space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Green's function ,symbols ,Boundary value problem ,Half line ,0101 mathematics ,Exponential decay ,Stationary solution ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line x > as follows: ut−lux−uxx=−β(uvx)x,x>0,t>0,λv−vxx=u,x>0,t>0,lu(0,t)+ux(0,t)=vx(0,t)=0,t>0,u(x,0)=u0(x),x>0, where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l
- Published
- 2016
41. Uncertainty principles for images defined on the square
- Author
-
Pei Dang and Shujuan Wang
- Subjects
Uncertainty principle ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Phase (waves) ,020206 networking & telecommunications ,Torus ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,Square (algebra) ,Set (abstract data type) ,Amplitude ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
42. Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equations on the whole line
- Author
-
Tian-jun Wang
- Subjects
Condensed Matter::Quantum Gases ,Laguerre's method ,General Mathematics ,Mathematical analysis ,General Engineering ,Relaxation (iterative method) ,Domain decomposition methods ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Convergence (routing) ,Laguerre polynomials ,Fokker–Planck equation ,0101 mathematics ,Spectral method ,Mathematics - Abstract
In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
43. Global exponential stability for interval general bidirectional associative memory (BAM) neural networks with proportional delays
- Author
-
Changjin Xu, Yicheng Pang, and Peiluan Li
- Subjects
Equilibrium point ,0209 industrial biotechnology ,Artificial neural network ,General Mathematics ,General Engineering ,Fixed-point theorem ,02 engineering and technology ,Interval (mathematics) ,Nonlinear system ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,020201 artificial intelligence & image processing ,Bidirectional associative memory ,Uniqueness ,Mathematics - Abstract
This paper is concerned with interval general bidirectional associative memory (BAM) neural networks with proportional delays. Using appropriate nonlinear variable transformations, the interval general BAM neural networks with proportional delays can be equivalently transformed into the interval general BAM neural networks with constant delays. The sufficient condition for the existence and uniqueness of equilibrium point of the model is established by applying Brouwer's fixed point theorem. By constructing suitable delay differential inequalities, some sufficient conditions for the global exponential stability of the model are obtained. Two examples are given to illustrate the effectiveness of the obtained results. This paper ends with a brief conclusion. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
44. Finite-region stability and finite-region boundedness for 2D Roesser models
- Author
-
Weiqun Wang and Guangchen Zhang
- Subjects
0209 industrial biotechnology ,General Mathematics ,General Engineering ,Stability (learning theory) ,010103 numerical & computational mathematics ,02 engineering and technology ,State (functional analysis) ,Linear matrix ,01 natural sciences ,020901 industrial engineering & automation ,Control theory ,Transient (oscillation) ,0101 mathematics ,Analysis method ,Mathematics - Abstract
In this paper, we establish finite-region stability (FRS) and finite-region boundedness analysis methods to investigate the transient behavior of discrete two-dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite-region stabilization issue is analyzed for the corresponding two-dimensional closed-loop system. Similar to FRS analysis, the finite-region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
45. Qualitative analysis of a new Lorenz-type chaotic system and its simulation
- Author
-
Guangyun Zhang, Chunlai Mu, Kunqiong Li, and Fuchen Zhang
- Subjects
Correctness ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,General Engineering ,Chaotic ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Algebra ,Qualitative analysis ,Stability theory ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
46. On holonomic mathematical F ‐ pendulum
- Author
-
Alexander P. Buslaev and Marina V. Yashina
- Subjects
Double pendulum ,Holonomic ,General Mathematics ,General Engineering ,Pendulum ,Discrete dynamical system ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,010101 applied mathematics ,Algebra ,Graph power ,Control theory ,0101 mathematics ,Mathematics ,Real number - Abstract
A discrete dynamical system named real - valued pendulum is presented in the paper. The system is developed for modeling of particles relocations on abstract graph with formalized rules of behavior (competition for space) and movement plans logistics. Logistics is given in a formal statement by the real numbers of the unit closed interval, presented in calculus with base equals to the graph power N. The central question is to study the behavior of the pendulum depending on its architecture. In the previous papers, with investigation of logistic pendulum, we have studied chaotic pendulum, where the particle plan for “tomorrow” is played “today”. A particular case of logistic pendulum, such that particle plans are time shifts of N−ary digital representation of single generating number, is called phase pendulum. In this paper, we are considering a dynamical system with generation of particles plans using a some mapping F, defined on the set of system states. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
47. Global well-posedness of the nonhomogeneous incompressible liquid crystals systems
- Author
-
Xiaoyu Xi and Dongjuan Niu
- Subjects
Small data ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Space (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Strong solutions ,Liquid crystal ,Bounded function ,Compressibility ,Calculus ,0101 mathematics ,Well posedness ,Mathematics - Abstract
This paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
48. Almost periodic solutions for neutral-type neural networks with the delays in the leakage term on time scales
- Author
-
Yuan Ye, Pan Wang, and Yongkun Li
- Subjects
0209 industrial biotechnology ,Class (set theory) ,Artificial neural network ,Banach fixed-point theorem ,General Mathematics ,General Engineering ,02 engineering and technology ,Type (model theory) ,Term (time) ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,020201 artificial intelligence & image processing ,Leakage (electronics) ,Mathematics - Abstract
In this paper, a class of neutral-type neural networks with delays in the leakage term on time scales are considered. By using the Banach fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. The results of this paper are new and complementary to the previously known results. Finally, an example is presented to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
49. A quadratic convex combination approach on robust dissipativity and passivity analysis for Takagi-Sugeno fuzzy Cohen-Grossberg neural networks with time-varying delays
- Author
-
T. Radhika and G. Nagamani
- Subjects
0209 industrial biotechnology ,Artificial neural network ,General Mathematics ,General Engineering ,Linear matrix inequality ,Monotonic function ,02 engineering and technology ,Lipschitz continuity ,Fuzzy logic ,020901 industrial engineering & automation ,Quadratic equation ,Control theory ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Convex combination ,Differentiable function ,Mathematics - Abstract
In this paper, the robust dissipativity and passivity criteria for Takagi–Sugeno fuzzy Cohen–Grossberg neural networks with time-varying delays have been investigated. The delay is of the time-varying nature, and the activation functions are assumed to be neither differentiable nor strictly monotonic. Furthermore, the description of the activation functions is more general than the commonly used Lipschitz conditions. By using a Lyapunov–Krasovskii functional and employing the quadratic convex combination approach, a set of sufficient conditions are established to ensure the dissipativity of the proposed model. The obtained conditions are presented in terms of linear matrix inequalities, so that its feasibility can be checked easily via standard numerical toolboxes. The quadratic convex combination approach used in our paper gives a reduced conservatism without using Jensen's inequality. In addition to that, numerical examples with simulation results are given to show the effectiveness of the obtained linear matrix inequality conditions. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
50. Fuzzy games for a general Bayesian abstract fuzzy economy model of product measurable spaces
- Author
-
Poom Kumam and Plern Saipara
- Subjects
Fuzzy classification ,Fuzzy measure theory ,Neuro-fuzzy ,General Mathematics ,010102 general mathematics ,General Engineering ,Type-2 fuzzy sets and systems ,01 natural sciences ,Defuzzification ,Fuzzy logic ,010101 applied mathematics ,Fuzzy set operations ,Fuzzy number ,0101 mathematics ,Mathematical economics ,Mathematics - Abstract
In this paper, we introduce a general Bayesian abstract fuzzy economy model of product measurable spaces, and we prove the existence of Bayesian fuzzy equilibrium for this model. Our results extend and improve the corresponding recent results announced by Patriche and many authors from the literature. It captures the idea that the uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities. In this paper, the uncertainties can be described by using random fuzzy mappings. Further attention is needed for the study of applications of the established result in the game theory and the fuzzy economic field.Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
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