170 results
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2. 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
- Subjects
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
3. 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
- Subjects
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
4. The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative
- Author
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Pushpendra Kumar and Vedat Suat Erturk
- Subjects
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
5. 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
- Subjects
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
6. Positive solutions for p-Kirchhoff type problems on.
- Author
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Cheng, Xiyou and Dai, Guowei
- Subjects
LAPLACIAN matrices ,KIRCHHOFF'S theory of diffraction ,APPLIED mathematics ,MATHEMATICS ,DYNAMICAL systems - Abstract
In this paper, we establish the existence and non-existence of positive solutions for p-Kirchhoff type problems with a parameter on [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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7. A nonlocal multi‐point singular fractional integro‐differential problem of Lane–Emden type
- Author
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Zoubir Dahmani, Yazid Gouari, Mehmet Zeki Sarikaya, and [Belirlenecek]
- Subjects
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
8. 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
9. Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting
- Author
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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
10. On the stability and nonexistence of turing patterns for the generalized Lengyel-Epstein model
- Author
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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
11. A novel simulation methodology of fractional order nuclear science model
- Author
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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
12. A data assimilation process for linear ill-posed problems
- Author
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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
13. Kantorovich variant of a new kind ofq-Bernstein-Schurer operators
- Author
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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
14. Uncertainty principles for images defined on the square
- Author
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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
15. Global exponential stability for interval general bidirectional associative memory (BAM) neural networks with proportional delays
- Author
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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
16. Almost periodic solutions for neutral-type neural networks with the delays in the leakage term on time scales
- Author
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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
17. On existence of solutions of differential-difference equations
- Author
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Hai-chou Li
- Subjects
Independent equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,Euler equations ,010101 applied mathematics ,Stochastic partial differential equation ,Examples of differential equations ,Theory of equations ,symbols.namesake ,Simultaneous equations ,symbols ,Applied mathematics ,0101 mathematics ,C0-semigroup ,Differential algebraic equation ,Mathematics - Abstract
This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
18. On global stability of an HIV pathogenesis model with cure rate
- Author
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Yoshiaki Muroya and Yoichi Enatsu
- Subjects
Lyapunov function ,Mathematical optimization ,General Mathematics ,General Engineering ,Human immunodeficiency virus (HIV) ,medicine.disease_cause ,Stability (probability) ,Upper and lower bounds ,Pathogenesis ,symbols.namesake ,Monotone polygon ,Stability theory ,medicine ,symbols ,Applied mathematics ,Logistic function ,Mathematics - Abstract
In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
19. Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays
- Author
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Yongkun Li and Lili Zhao
- Subjects
Almost periodic function ,Class (set theory) ,Artificial neural network ,General Mathematics ,Exponential dichotomy ,010102 general mathematics ,General Engineering ,02 engineering and technology ,01 natural sciences ,Periodic function ,Exponential stability ,Cellular neural network ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,Dynamic equation ,Mathematics - Abstract
In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
20. Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate
- Author
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Parvaiz Ahmad Naik, Muhammad Bilal Ghori, Zohre Eskandari, Jian Zu, and Mehraj-ud-din Naik
- Subjects
education.field_of_study ,General Mathematics ,Population ,Feasible region ,General Engineering ,Stability (probability) ,Fractional calculus ,Bounded function ,Applied mathematics ,education ,Epidemic model ,Basic reproduction number ,Bifurcation ,Mathematics - Abstract
The present paper studies a fractional-order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type-II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model’s analysis shows that two equilibria exist, namely, disease-free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0 < 1, then DFE is globally stable, leading to eradicating the population’s disease. If R0 > 1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters into disease transmission. The numerical solution of the model is obtained by the generalized Adams- Bashforth-Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results.
- Published
- 2022
21. New results for higher‐order Hadamard‐type fractional differential equations on the half‐line
- Author
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Fulya Yoruk Deren and Tugba Senlik Cerdik
- Subjects
General Mathematics ,General Engineering ,Positive Solutions ,Fixed-point theorem ,Existence ,Interval (mathematics) ,Type (model theory) ,Boundary-Value Problem ,Coupled System ,positive solution ,Hadamard transform ,Hadamard fractional derivative ,infinite interval ,Order (group theory) ,Applied mathematics ,Half line ,Boundary value problem ,Fractional differential ,Mathematics - Abstract
The purpose of this paper is to analyze a new kind of Hadamard fractional boundary value problem combining integral boundary condition and multipoint fractional integral boundary condition on an infinite interval. By the help of the Bai-Ge’s fixed point theorem, multiplicity results of positive solutions are derived for the Hadamard fractional boundary value problem. In the end, to illustrative the main result, an example is also presented.
- Published
- 2021
22. A higher order numerical scheme for solving fractional Bagley‐Torvik equation
- Author
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Patricia J. Y. Wong, Qinxu Ding, and School of Electrical and Electronic Engineering
- Subjects
Discrete Spline ,Fractional Bagley-Torvik Equation ,Order (business) ,General Mathematics ,Scheme (mathematics) ,Electrical and electronic engineering [Engineering] ,General Engineering ,Applied mathematics ,Mathematics - Abstract
In this paper, we develop a higher order numerical method for the fractional Bagley-Torvik equation. The main tools used include a new fourth-order approximation for the fractional derivative based on the weighted shifted Grünwald-Letnikov difference operator and a discrete cubic spline approach. We show that the theoretical convergence order improves those of previous work. Five examples are further presented to illustrate the efficiency of our method and to compare with other methods in the literature.
- Published
- 2021
23. Goodness‐of‐fit measures based on the Mellin transform for beta generalized lifetime data
- Author
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Abraão D. C. Nascimento, Renato J. Cintra, and Josimar Mendes de Vasconcelos
- Subjects
Mellin transform ,Class (set theory) ,Goodness of fit ,Heavy-tailed distribution ,General Mathematics ,Model selection ,General Engineering ,Order (ring theory) ,Applied mathematics ,Ellipse ,Statistic ,Mathematics - Abstract
In recent years various probability models have been proposed for describing lifetime data. Increasing model flexibility is often sought as a means to better describe asymmetric and heavy tail distributions. Such extensions were pioneered by the beta-G family. However, efficient goodness-of-fit (GoF) measures for the beta-G distributions are sought. In this paper, we combine probability weighted moments (PWMs) and the Mellin transform (MT) in order to furnish new qualitative and quantitative GoF tools for model selection within the beta-G class. We derive PWMs for the Fr\’{e}chet and Kumaraswamy distributions; and we provide expressions for the MT, and for the log-cumulants (LC) of the beta-Weibull, beta-Fr\’{e}chet, beta-Kumaraswamy, and beta-log-logistic distributions. Subsequently, we construct LC diagrams and, based on the Hotelling’s $T^2$ statistic, we derive confidence ellipses for the LCs. Finally, the proposed GoF measures are applied on five real data sets in order to demonstrate their applicability.
- Published
- 2021
24. A remark on the well‐posedness of the classical Green–Naghdi system
- Author
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Bashar Khorbatly
- Subjects
Operator (computer programming) ,General Mathematics ,Norm (mathematics) ,General Engineering ,Applied mathematics ,Natural energy ,Energy (signal processing) ,Well posedness ,Mathematics - Abstract
The aim of this paper is to give an alternative technique for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by S. Israwi in [2] but for the original system, in which the partial operator ∇× is not involved.
- Published
- 2021
25. Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise
- Author
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Jean Daniel Mukam and Antoine Tambue
- Subjects
General Mathematics ,Numerical analysis ,finite element method ,General Engineering ,White noise ,Exponential integrator ,VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 ,Noise (electronics) ,Finite element method ,strong convergence ,Stochastic partial differential equation ,Galerkin projection method ,Nonlinear system ,symbols.namesake ,Wiener process ,symbols ,Applied mathematics ,stochastic convection–reaction–diffusion equations ,additive noise ,exponential integrators ,Mathematics - Abstract
In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided
- Published
- 2021
26. On asymptotically statistical equivalent functions on time scales
- Author
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Selma Altundağ and Bayram Sözbir
- Subjects
Modulo operation ,General Mathematics ,General Engineering ,Applied mathematics ,Statistical convergence ,Mathematics - Abstract
In this paper, we introduce the concepts of asymptotically f-statistical equivalence, asymptotically f-lacunary statistical equivalence, and strong asymptotically f-lacunary equivalence for non-negative two delta measurable real-valued functions defined on time scales with the aid of modulus function f. Furthermore, the relationships between these new concepts are investigated. We also present some inclusion theorems.
- Published
- 2021
27. An iterative method for optimal control of bilateral free boundaries problem
- Author
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Youness El Yazidi and Abdellatif Ellabib
- Subjects
Computer science ,Iterative method ,General Mathematics ,010102 general mathematics ,General Engineering ,Inverse problem ,Optimal control ,01 natural sciences ,Regularization (mathematics) ,Finite element method ,010101 applied mathematics ,Robustness (computer science) ,Conjugate gradient method ,Applied mathematics ,Gravitational singularity ,Shape gradient ,Shape optimization ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to construct a numerical scheme for solving a class of bilateral free boundaries problem. First, using a shape functional and some regularization terms, an optimal control problem is formulated, in addition, we prove its solution existence's. The first optimality conditions and the shape gradient are computed. the proposed numerical scheme is a genetic algorithm guided conjugate gradient combined with the finite element method, at each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results, and to prove the robustness of the proposed scheme.
- Published
- 2021
28. Mathematical models for the improvement of detection techniques of industrial noise sources from acoustic images
- Author
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Gianluca Vinti, Giorgio Baldinelli, Francesco Bianchi, Francesco D'Alessandro, Marco Seracini, Danilo Costarelli, Francesco Asdrubali, Flavio Scrucca, Asdrubali, Francesco, Baldinelli, Giorgio, Bianchi, Francesco, Costarelli, Danilo, D'Alessandro, Francesco, Scrucca, Flavio, Seracini, Marco, and Vinti, Gianluca
- Subjects
Beamforming ,acoustic images, applied mathematics, beamforming, image reconstruction, industrial noise, sam-pling Kantorovich algorithm ,Mathematical model ,business.industry ,General Mathematics ,industrial noise ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,General Engineering ,acoustic images ,Industrial noise ,Iterative reconstruction ,image reconstruction ,sampling Kantorovich algorithm ,beamforming ,applied mathematics ,Computer vision ,Artificial intelligence ,business ,Mathematics - Abstract
In this paper, a procedure for the detection of the sources of industrial noise and the evaluation of their distances is introduced. The above method is based on the analysis of acoustic and optical data recorded by an acoustic camera. In order to improve the resolution of the data, interpolation and quasi interpolation algorithms for digital data processing have been used, such as the bilinear, bicubic, and sampling Kantorovich (SK). The experimental tests show that the SK algorithm allows to perform the above task more accurately than the other considered methods.
- Published
- 2021
29. A generalized fractional ( q , h )–Gronwall inequality and its applications to nonlinear fractional delay ( q , h )–difference systems
- Author
-
Feifei Du and Baoguo Jia
- Subjects
010101 applied mathematics ,Nonlinear system ,Uniqueness theorem for Poisson's equation ,Stability criterion ,General Mathematics ,Gronwall's inequality ,010102 general mathematics ,General Engineering ,Applied mathematics ,Uniqueness ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, a generalized fractional $(q,h)$-Gronwall inequality is investigated. Based on this inequality, we derive the uniqueness theorem and the finite-time stability criterion of nonlinear fractional delay $(q,h)$-difference systems. Several examples are given to illustrate our theoretical result.
- Published
- 2021
30. Existence and stability for a nonlinear hybrid differential equation of fractional order via regular Mittag–Leffler kernel
- Author
-
Ibrahim Slimane, Juan J. Nieto, Thabet Abdeljawad, and Zoubir Dahmani
- Subjects
Mathematics::Functional Analysis ,Nonlinear system ,Differential equation ,General Mathematics ,Kernel (statistics) ,Mathematics::Classical Analysis and ODEs ,General Engineering ,Order (group theory) ,Applied mathematics ,Contraction (operator theory) ,Stability (probability) ,Mathematics ,Fractional calculus - Abstract
This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed-point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem.
- Published
- 2021
31. Nonoscillation of half‐linear dynamic equations on time scales
- Author
-
Michal Veselý, Petr Hasil, Michal Pospíšil, and Jozef Kiselak
- Subjects
Ideal (set theory) ,Oscillation ,General Mathematics ,010102 general mathematics ,General Engineering ,Qualitative theory ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Riccati equation ,Applied mathematics ,0101 mathematics ,Dynamic equation ,Linear equation ,Mathematics - Abstract
The research contained in this paper belongs to the qualitative theory of dynamic equations on time scales. Via the detailed analysis of solutions of the associated Riccati equation and an advanced averaging technique, we provide the description of domain of nonoscillation of very general equations. The results are formulated and proved for half-linear equations (i.e., equations connected to PDEs with one dimensional p-Laplacian) on time scales. Nevertheless, we obtain new results also for linear difference equations. Moreover, the combination of the presented results with previous ones shows that many useful equations are conditionally oscillatory. Such equations are ideal as testing and comparison equations in real-world models which are beyond capabilities of known oscillation and nonoscillation criteria often.
- Published
- 2021
32. The gradient descent method from the perspective of fractional calculus
- Author
-
Joel A. Rosenfeld and Pham Viet Hai
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Order (ring theory) ,Unconstrained optimization ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Perspective (geometry) ,Optimization and Control (math.OC) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Gradient descent ,Mathematics - Optimization and Control ,Gradient method ,Mathematics - Abstract
Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $\psi$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $\alpha$ and weight $\psi$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods., Comment: 27 pages
- Published
- 2020
33. Translation‐invariant generalized P ‐adic Gibbs measures for the Ising model on Cayley trees
- Author
-
Otabek Khakimov and Farrukh Mukhamedov
- Subjects
Phase transition ,General Mathematics ,010102 general mathematics ,General Engineering ,Fixed point ,Invariant (physics) ,01 natural sciences ,010101 applied mathematics ,Singularity ,Probability theory ,Physical phenomena ,Applied mathematics ,Ising model ,0101 mathematics ,Mathematics ,p-adic number - Abstract
Main aim of the present paper is explore certain physical phenomena by means of $p$-adic probability theory. To overcome this study, we deal with a more general setting to define $p$-adic Gibbs measures. For the sake of simplicity of explanations, we restrict ourselves to the Ising model on the Cayley tree, since such a model has broad theoretical and practical applications. To study $p$-adic quasi Gibbs measures, we reduce the problem to the description of the fixed points of the Ising-Potts mapping. Finding fixed points is not an easy job as in the real setting. Furthermore, the phase transition for the model is established. In the real case, the phase transition yields the the singularity of the limiting Gibbs measures. However, we show that the $p$-adic quasi Gibbs measures do not exhibit the mentioned type of singularity, such kind of phenomena is called strong phase transition. Finally, we deal with the solvability and the number of solutions of ceratin $p$-adic equation depending on several parameters. Such a description allows us to find all possible translation-invariant $p$a-adic quasi Gibbs measures.
- Published
- 2020
34. Generalized approximate boundary synchronization for a coupled system of wave equations
- Author
-
Yanyan Wang
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Boundary (topology) ,State (functional analysis) ,Kalman filter ,Wave equation ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Matrix (mathematics) ,symbols.namesake ,Synchronization (computer science) ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.
- Published
- 2020
35. Consensus of the Hegselmann–Krause opinion formation model with time delay
- Author
-
Cristina Pignotti, Alessandro Paolucci, and Young-Pil Choi
- Subjects
Particle system ,Partial differential equation ,Continuum (topology) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,General Engineering ,Infinity ,01 natural sciences ,Exponential function ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,Uniqueness ,Limit (mathematics) ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics ,media_common ,Opinion formation - Abstract
In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.
- Published
- 2020
36. On Bernoulli series approximation for the matrix cosine
- Author
-
Jose M. Alonso, Emilio Defez, Javier Ibáñez, and Pedro Alonso-Jordá
- Subjects
Matrix exponential and similar functions of matrices ,Matrix (mathematics) ,Bernoulli's principle ,Polynomials and matrices ,General Mathematics ,CIENCIAS DE LA COMPUTACION E INTELIGENCIA ARTIFICIAL ,General Engineering ,Applied mathematics ,Trigonometric functions ,Series approximation ,MATEMATICA APLICADA ,Mathematics - Abstract
[EN] This paper presents a new series expansion based on Bernoulli matrix polynomials to approximate the matrix cosine function. An approximation based on this series is not a straightforward exercise since there exist different options to implement such a solution. We dive into these options and include a thorough comparative of performance and accuracy in the experimental results section that shows benefits and downsides of each one. Also, a comparison with the Pade approximation is included. The algorithms have been implemented in MATLAB and in CUDA for NVIDIA GPUs., Spanish Ministerio de Economia y Competitividad and European Regional Development Fund, Grant/Award Number: TIN2017-89314-P; Universitat Politecnica de Valencia, Grant/Award Number: SP20180016
- Published
- 2020
37. Structured singular value of implicit systems
- Author
-
George Halikias, Nicos Karcanias, and Olga Limantseva
- Subjects
Matrix (mathematics) ,Singular value ,Systems theory ,General Mathematics ,General Engineering ,Stability (learning theory) ,Structure (category theory) ,Scalar (physics) ,Applied mathematics ,Algebraic number ,Robust control ,QA ,Mathematics - Abstract
Implicit systems provide a general framework in which many important properties of dynamic systems can be studied. Implicit systems are especially relevant to behavioural systems theory, the analysis and synthesis of complex interconnected systems, systems identification and robust control. By incorporating algebraic constraints, implicit models provide additional versatility relative to the standard input–output framework. Problems of robust stability in implicit systems lead in a natural way to non‐standard structured singular value (μ) formulations. In this note, it is shown that for a class of uncertainty structures involving repeated scalar parameters, these problems reduce to a standard μ problem which is well studied and for the solution of which several numerical algorithms are available. Our results are based on a matrix dilation technique and the redefinition of the uncertainty structure of the transformed problem. The main results of the paper are illustrated with a numerical example.
- Published
- 2020
38. Analysis and numerical simulation of novel coronavirus (COVID‐19) model with Mittag‐Leffler Kernel
- Author
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V. Padmavathi, Amit Prakash, K. Alagesan, and Nanjundan Magesh
- Subjects
Current (mathematics) ,Coronavirus disease 2019 (COVID-19) ,Computer simulation ,General Mathematics ,Homotopy ,010102 general mathematics ,General Engineering ,01 natural sciences ,Fractional operator ,Nonlinear differential equations ,Fractional calculus ,010101 applied mathematics ,Kernel (statistics) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Every now and then, there has been natural or man-made calamities Such adversities instigate various institutions to find solutions for them The current study attempts to explore the disaster caused by the micro enemy called coronavirus for the past few months and aims at finding the solution for the system of nonlinear ordinary differential equations to which q?homotopy analysis transform method (q?HATM) has been applied to arrive at effective results In this paper, there are eight nonlinear ordinary differential equations considered and to solve them the advanced fractional operator Atangana-Baleanu (AB) fractional derivative has been applied to produce better understanding The outcomes have been presented in terms of plots Ultimately, the present study assists in examining the real-world models and aids in predicting their behavior corresponding to the parameters considered in the models
- Published
- 2020
39. A critical Kirchhoff‐type problem driven by ap (·)‐fractional Laplace operator with variables (·) ‐order
- Author
-
Alessio Fiscella, Tianqing An, Jiabin Zuo, Zuo, J, An, T, and Fiscella, A
- Subjects
Kirchhoff coefficient ,variable exponent ,Variable exponent ,Kirchhoff type ,General Mathematics ,p(·)-fractional Laplacian ,General Engineering ,Applied mathematics ,Order (group theory) ,critical nonlinearity ,Laplace operator ,Mathematics ,Variable (mathematics) - Abstract
The paper deals with the following Kirchhoff-type problem (Formula presented.) where M models a Kirchhoff coefficient, (Formula presented.) is a variable s(·)-order p(·)-fractional Laplace operator, with (Formula presented.) and (Formula presented.). Here, (Formula presented.) is a bounded smooth domain with N > p(x, y)s(x, y) for any (Formula presented.), μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent (Formula presented.), given with (Formula presented.) and (Formula presented.) for (Formula presented.). We prove the existence and asymptotic behavior of at least one non-trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb-type lemma for fractional Sobolev spaces with variable order and variable exponent.
- Published
- 2020
40. Global stability of a multistrain SIS model with superinfection and patch structure
- Author
-
Gergely Röst, Attila Dénes, and Yoshiaki Muroya
- Subjects
Sequence ,General Mathematics ,010102 general mathematics ,Patch model ,General Engineering ,Structure (category theory) ,medicine.disease_cause ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Nonlinear system ,Superinfection ,medicine ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
We study the global stability of a multistrain SIS model with superinfection and patch structure. We establish an iterative procedure to obtain a sequence of threshold parameters. By a repeated application of a result by Takeuchi et al. [Nonlinear Anal Real World Appl. 2006 7:235-247], we show that these parameters completely determine the global dynamics of the system: for any number of patches and strains with different infectivities, any subset of the strains can stably coexist depending on the particular choice of the parameters. Finally, we return to the special case of one patch examined in [Math Biosci Eng. 2017 14:421-35] and give a correction to the proof of Theorem 2.2 of that paper.
- Published
- 2020
41. Nontrivial solutions for impulsive fractional differential systems through variational methods
- Author
-
Shapour Heidarkhani and Amjad Salari
- Subjects
Class (set theory) ,General Mathematics ,Weak solution ,010102 general mathematics ,General Engineering ,Term (logic) ,01 natural sciences ,Critical point (mathematics) ,010101 applied mathematics ,Nonlinear system ,Mountain pass theorem ,Applied mathematics ,0101 mathematics ,Algebraic number ,Fractional differential ,Mathematics - Abstract
This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two weak solutions, applying the mountain pass theorem we establish the existence of third weak solution for our system.
- Published
- 2020
42. Hierarchic control of a linear heat equation with missing data
- Author
-
Romario Gildas Foko Tiomela, Gaston M. N’Guérékata, Gisèle Mophou, Laboratoire de Mathématiques Informatique et Applications (LAMIA), Université des Antilles (UA), Department of Mathematics, Morgan State University, Baltimore, and MOPHOU LOUDJOM, GISELE
- Subjects
Rest (physics) ,General Mathematics ,010102 general mathematics ,Control (management) ,General Engineering ,[MATH] Mathematics [math] ,State (functional analysis) ,Missing data ,Optimal control ,01 natural sciences ,010101 applied mathematics ,Stackelberg strategy ,Stackelberg competition ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Applied mathematics ,Heat equation ,[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP] ,[MATH]Mathematics [math] ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
The paper is devoted to the Stackelberg control of a linear parabolic equation with missing initial conditions. The strategy involves two controls called follower and leader. The objective of the follower is to bring the state to a desired state while the leader has to bring the system to rest at the final time. The results are obtained by means of Fenchel-Legendre transform and appropriate Carleman inequalities.
- Published
- 2020
43. A proper generalized decomposition approach for optical flow estimation
- Author
-
B. Denis de Senneville, Abdallah El Hamidi, Nicolas Papadakis, Marwan Saleh, Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356 (LaSIE), Université de La Rochelle (ULR)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Optimization ,Weak convergence ,General Mathematics ,Computation ,media_common.quotation_subject ,010102 general mathematics ,General Engineering ,Optical flow ,Fidelity ,Proper Generalized Decomposition (PGD) ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,Sobolev space ,Quadratic equation ,Optical Flow ,[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing ,Applied mathematics ,[MATH]Mathematics [math] ,0101 mathematics ,Image resolution ,media_common ,Mathematics - Abstract
International audience; This paper introduces the use of the Proper Generalized Decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, i.e. optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularisation term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (i) the weak convergence based on the Brézis-Lieb decomposition, (ii) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely low image resolution and/or low Signal-To-Noise Ratio (SNR).
- Published
- 2020
44. Extending the choice of starting points for Newton's method
- Author
-
Argyros, Ioannis Konstantinos, Ezquerro, José Antonio, Hernández-Verón, Miguel Ángel, Kim, Young Ik, Magreñán, Ángel Alberto, and 0000-0002-6991-5706
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Lipschitz continuity ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Order (group theory) ,Applied mathematics ,Center (algebra and category theory) ,0101 mathematics ,Newton's method ,Second derivative ,Mathematics - Abstract
In this paper, we propose a center Lipschitz condition for the second derivative together with the use of restricted domains in order to improve the starting points for Newton's method when compared with previous results. Moreover, we present some numerical examples validating the theoretical results.
- Published
- 2019
45. On the impulsive implicit Ψ‐Hilfer fractional differential equations with delay
- Author
-
Jyoti P. Kharade and Kishor D. Kucche
- Subjects
Mathematics::Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics::Probability ,Mathematics::Complex Variables ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,General Engineering ,Applied mathematics ,Mathematics - Dynamical Systems ,Fractional differential ,Stability (probability) ,Mathematics - Abstract
In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $\Psi$--Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam--Hyers and generalized Ulam--Hyers stability are the specific cases of Ulam--Hyers--Mittag--Leffler stability. Extended version of Gronwall inequality, abstract Gronwall lemma and Picard operator theory are the primary devices in our investigation. We give an example to illustrate the obtained results., Comment: 15
- Published
- 2019
46. Domain of existence for the solution of some IVP's and BVP's by using an efficient ninth‐order iterative method
- Author
-
José L. Hueso, Eulalia Martínez, Fabricio Cevallos, and Cory L. Howk
- Subjects
Ninth ,Iterative methods ,Iterative method ,General Mathematics ,010102 general mathematics ,General Engineering ,Nonlinear equations ,01 natural sciences ,Domain (mathematical analysis) ,Computational efficiency ,010101 applied mathematics ,Semilocal convergence ,Rate of convergence ,Order of convergence ,Applied mathematics ,Order (group theory) ,Christian ministry ,0101 mathematics ,MATEMATICA APLICADA ,Mathematics - Abstract
[EN] In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite-dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton-type method with frozen first derivative in Banach spaces. We impose only the assumption that the Frechet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems., Spanish Ministry of Science and Innovation. Grant Number: MTM2014- 52016-C2-2-P Generalitat Valenciana Prometeo. Grant Number: 2016/089
- Published
- 2019
47. The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus
- Author
-
Hamed Taghavian
- Subjects
Laplace inversion ,Laplace transform ,General Mathematics ,Composite number ,General Engineering ,Fractional calculus ,symbols.namesake ,Mittag-Leffler function ,symbols ,Partition (number theory) ,Applied mathematics ,Polynomial series ,Laplace transform inversion ,Mathematics - Abstract
This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact soluti ...
- Published
- 2019
48. On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation
- Author
-
Daniela Oliveira, Edmundo Capelas de Oliveira, and José Vanterler da Costa Sousa
- Subjects
Mathematics::Functional Analysis ,26A33, 34A08, 34A12, 34K20, 34G20 ,General Mathematics ,010102 general mathematics ,General Engineering ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, by means of Banach fixed point theorem, we investigate the existence and Ulam--Hyers--Rassias stability of the non-instantaneous impulsive integrodifferential equation by means of $\psi$-Hilfer fractional derivative. In this sense, some examples are presented, in order to consolidate the results obtained., Comment: 15 pages
- Published
- 2018
49. Fractional h‐differences with exponential kernels and their monotonicity properties
- Author
-
Shahd Owies, Thabet Abdeljawad, and Iyad Suwan
- Subjects
010101 applied mathematics ,General Mathematics ,010102 general mathematics ,General Engineering ,Applied mathematics ,Monotonic function ,0101 mathematics ,01 natural sciences ,Exponential function ,Mathematics - Abstract
SPECIAL ISSUE PAPER Fractional ℎ ‐differences with exponential kernels and their monotonicity properties Iyad Suwan Shahd Owies Thabet Abdeljawad Mathematical Methods in the Applied Sciences, Early View First published: 25 January 2020
- Published
- 2020
50. Effective numerical evaluation of the double Hilbert transform
- Author
-
Min Ku, Xiaoyun Sun, Ieng Tak Leong, and Pei Dang
- Subjects
Pointwise ,General Mathematics ,010102 general mathematics ,General Engineering ,01 natural sciences ,010101 applied mathematics ,Periodic function ,Quadratic formula ,symbols.namesake ,symbols ,Applied mathematics ,Nyström method ,Hilbert transform ,0101 mathematics ,Remainder ,Energy (signal processing) ,Mathematics ,Trigonometric interpolation - Abstract
In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.
- Published
- 2020
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