This paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. The class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. The results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls. The results are based on the ad hoc use of Gronwall’s inequality.
{"title":"About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays","authors":"M. de La Sen","doi":"10.1155/2021/5593813","DOIUrl":"https://doi.org/10.1155/2021/5593813","url":null,"abstract":"This paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. The class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. The results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls. The results are based on the ad hoc use of Gronwall’s inequality.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47314992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper intends to investigate the impact of external computers and removable devices on virus spread in a network with heterogeneous immunity. For that purpose, a new dynamical model is presented and discussed. Theoretical analysis reveals the existence of a unique viral equilibrium that is locally and globally asymptotically stable with no criteria. This result implies that efforts to eliminate viruses are not possible. Therefore, sensitivity analysis is performed to have more insight into parameters’ impact on virus prevalence. As a result, strategies are suggested to contain virus spread to an acceptable level. Finally, to rationalize the analytical results, we execute some numerical simulations.
{"title":"Modeling the Effect of External Computers and Removable Devices on a Computer Network with Heterogeneous Immunity","authors":"Walaa S. Bahashwan, Salma M. Al-Tuwairqi","doi":"10.1155/2021/6694098","DOIUrl":"https://doi.org/10.1155/2021/6694098","url":null,"abstract":"This paper intends to investigate the impact of external computers and removable devices on virus spread in a network with heterogeneous immunity. For that purpose, a new dynamical model is presented and discussed. Theoretical analysis reveals the existence of a unique viral equilibrium that is locally and globally asymptotically stable with no criteria. This result implies that efforts to eliminate viruses are not possible. Therefore, sensitivity analysis is performed to have more insight into parameters’ impact on virus prevalence. As a result, strategies are suggested to contain virus spread to an acceptable level. Finally, to rationalize the analytical results, we execute some numerical simulations.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41243402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The COVID-19 pandemic has put the world in threat for a long time It was first identified in Wuhan, China, in December 2019 and has been declared a pandemic by the WHO This disease is mainly caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) So far, no vaccine or medicine has been developed for the proper treatment of this disease, so people are afraid of getting infected The pandemic has placed many nations at the door of socioeconomic emergencies Therefore, it is very important to predict the development trend of this epidemic, and we know mathematical modelling is a basic tool to research the dynamic behaviour of disease and predict the spreading trend of the disease In this study, we have formulated a mathematical model for the COVID-19 outbreak by introducing a quarantine class with media-induced fear in the disease transmission rate to analyze the dynamic behaviour of this epidemic We have calculated the basic reproduction number R0, and we observed that when R0 1, then the system is permanent and there exists a unique endemic equilibrium point Global stability of the endemic equilibrium point is developed by using Li and Muldowney's high-dimensional Bendixson criterion Finally, some numerical simulations are performed using MATLAB to verify our analytical results [ABSTRACT FROM AUTHOR] Copyright of International Journal of Differential Equations is the property of Hindawi Limited 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 abstract 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 abstract (Copyright applies to all Abstracts )
{"title":"Impact of Media-Induced Fear on the Control of COVID-19 Outbreak: A Mathematical Study","authors":"C. Maji","doi":"10.1155/2021/2129490","DOIUrl":"https://doi.org/10.1155/2021/2129490","url":null,"abstract":"The COVID-19 pandemic has put the world in threat for a long time It was first identified in Wuhan, China, in December 2019 and has been declared a pandemic by the WHO This disease is mainly caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) So far, no vaccine or medicine has been developed for the proper treatment of this disease, so people are afraid of getting infected The pandemic has placed many nations at the door of socioeconomic emergencies Therefore, it is very important to predict the development trend of this epidemic, and we know mathematical modelling is a basic tool to research the dynamic behaviour of disease and predict the spreading trend of the disease In this study, we have formulated a mathematical model for the COVID-19 outbreak by introducing a quarantine class with media-induced fear in the disease transmission rate to analyze the dynamic behaviour of this epidemic We have calculated the basic reproduction number R0, and we observed that when R0 1, then the system is permanent and there exists a unique endemic equilibrium point Global stability of the endemic equilibrium point is developed by using Li and Muldowney's high-dimensional Bendixson criterion Finally, some numerical simulations are performed using MATLAB to verify our analytical results [ABSTRACT FROM AUTHOR] Copyright of International Journal of Differential Equations is the property of Hindawi Limited 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 abstract 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 abstract (Copyright applies to all Abstracts )","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47500241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order .
{"title":"Fuzzy Conformable Fractional Differential Equations","authors":"A. Harir, S. Melliani, L. S. Chadli","doi":"10.1155/2021/6655450","DOIUrl":"https://doi.org/10.1155/2021/6655450","url":null,"abstract":"In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order .","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43253571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdelyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.
{"title":"Lie Symmetry Analysis and Explicit Solutions for the Time-Fractional Regularized Long-Wave Equation","authors":"N. Maarouf, Hicham Maadan, K. Hilal","doi":"10.1155/2021/6614231","DOIUrl":"https://doi.org/10.1155/2021/6614231","url":null,"abstract":"This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdelyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49264931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.
{"title":"Solutions of Conformable Fractional-Order SIR Epidemic Model","authors":"A. Harir, Said Malliani, Lalla Saadia Chandli","doi":"10.1155/2021/6636686","DOIUrl":"https://doi.org/10.1155/2021/6636686","url":null,"abstract":"In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44962120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several mathematicians have extensively investigated polynomials, their extensions, and their applications in various other research areas for a decade. Our paper aims to introduce another such polynomial, namely, Laguerre-based generalized Humbert polynomial, and investigate its properties. In particular, it derives elementary identities, recursive differential relations, additional symmetry identities, and implicit summation formulas.
{"title":"A Class of Laguerre-Based Generalized Humbert Polynomials","authors":"Saniya Batra, Prakriti Rai","doi":"10.1155/2021/4324466","DOIUrl":"https://doi.org/10.1155/2021/4324466","url":null,"abstract":"Several mathematicians have extensively investigated polynomials, their extensions, and their applications in various other research areas for a decade. Our paper aims to introduce another such polynomial, namely, Laguerre-based generalized Humbert polynomial, and investigate its properties. In particular, it derives elementary identities, recursive differential relations, additional symmetry identities, and implicit summation formulas.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64757722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval 0,1 . For small values of ε , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N .
{"title":"Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts","authors":"M. Woldaregay, G. Duressa","doi":"10.1155/2020/6661592","DOIUrl":"https://doi.org/10.1155/2020/6661592","url":null,"abstract":"This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter \u0000 \u0000 ε\u0000 \u0000 taking arbitrary values in the interval \u0000 \u0000 \u0000 \u0000 0,1\u0000 \u0000 \u0000 \u0000 . For small values of \u0000 \u0000 ε\u0000 \u0000 , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of \u0000 \u0000 ε\u0000 \u0000 and mesh number \u0000 \u0000 N\u0000 \u0000 .","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2020-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45024080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. The system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? This is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? This is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impulses, delays, memory, nonlocal conditions, and noises, which are intrinsic phenomena no taken into account by the mathematical model that is representing by a differential equation. For many control systems in real life, delays, impulses, and noises are natural properties that do not change their behavior. So, we conjecture that, under certain conditions, the abrupt changes, delays, and noises as perturbations of a system do not modify certain properties such as controllability. In this regard, we prove the interior S ∗ -controllability of the semilinear stochastic heat equation with impulses and delay on the state variable, and this is done by using new techniques avoiding fixed point theorems employed by Bashirov et al.
{"title":"Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay","authors":"H. Leiva, Miguel Narváez, Zoraida Sívoli","doi":"10.1155/2020/2515160","DOIUrl":"https://doi.org/10.1155/2020/2515160","url":null,"abstract":"LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. The system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? This is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? This is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impulses, delays, memory, nonlocal conditions, and noises, which are intrinsic phenomena no taken into account by the mathematical model that is representing by a differential equation. For many control systems in real life, delays, impulses, and noises are natural properties that do not change their behavior. So, we conjecture that, under certain conditions, the abrupt changes, delays, and noises as perturbations of a system do not modify certain properties such as controllability. In this regard, we prove the interior \u0000 \u0000 \u0000 \u0000 S\u0000 \u0000 \u0000 ∗\u0000 \u0000 \u0000 \u0000 -controllability of the semilinear stochastic heat equation with impulses and delay on the state variable, and this is done by using new techniques avoiding fixed point theorems employed by Bashirov et al.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/2515160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47125670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a fuzzy fractional semigroup of operators whose generator will be the fuzzy fractional derivative of the fuzzy semigroup at . We establish some of their proprieties and some results about the solution of fuzzy fractional Cauchy problem.
{"title":"Fuzzy Conformable Fractional Semigroups of Operators","authors":"A. Harir, S. Melliani, L. S. Chadli","doi":"10.1155/2020/8836011","DOIUrl":"https://doi.org/10.1155/2020/8836011","url":null,"abstract":"In this paper, we introduce a fuzzy fractional semigroup of operators whose generator will be the fuzzy fractional derivative of the fuzzy semigroup at . We establish some of their proprieties and some results about the solution of fuzzy fractional Cauchy problem.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/8836011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42321610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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