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":"2021 1","pages":"1-7"},"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":"1 1","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":"2020 1","pages":"1-6"},"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}
In this paper, the effects of magnetic field, thermal radiation, buoyancy force, and internal heat generation on the laminar boundary layer flow about a vertical plate in the presence of a convective surface boundary condition have been investigated. In the analysis, it is assumed that the left surface of the plate is in contact with a hot fluid, whereas a stream of cold fluid flows steadily over the right surface, and the heat source decays exponentially outwards from the surface of the plate. The governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations with the help of similarity transformation which were solved analytically by applying the optimal homotopy asymptotic method. The variations of fluid velocity and surface temperature for different values of the Grashof number, magnetic parameter, Prandtl number, internal heat generation parameter, Biot number, and radiation absorption parameter are tabulated, graphed, and interpreted in physical terms. A comparison with previously published results on similar special cases of the problem shows an excellent agreement.
{"title":"Analytical Analysis of Effects of Buoyancy, Internal Heat Generation, Magnetic Field, and Thermal Radiation on a Boundary Layer over a Vertical Plate with a Convective Surface Boundary Condition","authors":"Solomon Bati Kejela, Mitiku Daba Firdi","doi":"10.1155/2020/8890510","DOIUrl":"https://doi.org/10.1155/2020/8890510","url":null,"abstract":"In this paper, the effects of magnetic field, thermal radiation, buoyancy force, and internal heat generation on the laminar boundary layer flow about a vertical plate in the presence of a convective surface boundary condition have been investigated. In the analysis, it is assumed that the left surface of the plate is in contact with a hot fluid, whereas a stream of cold fluid flows steadily over the right surface, and the heat source decays exponentially outwards from the surface of the plate. The governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations with the help of similarity transformation which were solved analytically by applying the optimal homotopy asymptotic method. The variations of fluid velocity and surface temperature for different values of the Grashof number, magnetic parameter, Prandtl number, internal heat generation parameter, Biot number, and radiation absorption parameter are tabulated, graphed, and interpreted in physical terms. A comparison with previously published results on similar special cases of the problem shows an excellent agreement.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":"2020 1","pages":"1-16"},"PeriodicalIF":1.6,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/8890510","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43541583","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 aim to investigate optimal control to a new mathematical model that describes agree-disagree opinions during polls, which we presented and analyzed in Bidah et al., 2020. We first present the model and recall its different compartments. We formulate the optimal control problem by supplementing our model with a objective functional. Optimal control strategies are proposed to reduce the number of disagreeing people and the cost of interventions. We prove the existence of solutions to the control problem, we employ Pontryagin’s maximum principle to find the necessary conditions for the existence of the optimal controls, and Runge–Kutta forward-backward sweep numerical approximation method is used to solve the optimal control system, and we perform numerical simulations using various initial conditions and parameters to investigate several scenarios. Finally, a global sensitivity analysis is carried out based on the partial rank correlation coefficient method and the Latin hypercube sampling to study the influence of various parameters on the objective functional and to identify the most influential parameters.
{"title":"Modeling and Control of the Public Opinion: An Agree-Disagree Opinion Model","authors":"S. Bidah, O. Zakary, M. Rachik","doi":"10.1155/2020/5864238","DOIUrl":"https://doi.org/10.1155/2020/5864238","url":null,"abstract":"In this paper, we aim to investigate optimal control to a new mathematical model that describes agree-disagree opinions during polls, which we presented and analyzed in Bidah et al., 2020. We first present the model and recall its different compartments. We formulate the optimal control problem by supplementing our model with a objective functional. Optimal control strategies are proposed to reduce the number of disagreeing people and the cost of interventions. We prove the existence of solutions to the control problem, we employ Pontryagin’s maximum principle to find the necessary conditions for the existence of the optimal controls, and Runge–Kutta forward-backward sweep numerical approximation method is used to solve the optimal control system, and we perform numerical simulations using various initial conditions and parameters to investigate several scenarios. Finally, a global sensitivity analysis is carried out based on the partial rank correlation coefficient method and the Latin hypercube sampling to study the influence of various parameters on the objective functional and to identify the most influential parameters.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/5864238","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46599161","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}
C. Derbazi, Z. Baitiche, M. Benchohra, G. N’Guérékata
Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.
{"title":"Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving \u0000 ψ\u0000 -Caputo Derivative in Banach and Fréchet Spaces","authors":"C. Derbazi, Z. Baitiche, M. Benchohra, G. N’Guérékata","doi":"10.1155/2020/6383916","DOIUrl":"https://doi.org/10.1155/2020/6383916","url":null,"abstract":"Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving \u0000 \u0000 ψ\u0000 \u0000 -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/6383916","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49253140","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}
Ghassane Benrhmach, Khalil Namir, J. Bouyaghroumni
The World Health Organization declared that the total number of confirmed cases tested positive for SARS‐CoV‐2, affecting 210 countries, exceeded 3 million on 29 April 2020, with more than 207,973 deaths. In order to end the global COVID‐19 pandemic, public authorities have put in place multiple strategies like testing, contact tracing, and social distancing. Predictive mathematical models for epidemics are fundamental to understand the development of the epidemic and to plan effective control strategies. Some hosts may carry SARS‐CoV‐2 and transmit it to others, yet display no symptoms themselves. We propose applying a model (SELIAHRD) taking in consideration the number of asymptomatic infected people. The SELIAHRD model consists of eight stages: Susceptible, Exposed, Latent, Symptomatic Infected, Asymptomatic Infected, Hospitalized, Recovered, and Dead. The asymptomatic carriers contribute to the spread of disease, but go largely undetected and can therefore undermine efforts to control transmission. The simulation of possible scenarios of the implementation of social distancing shows that if we rigorously follow the social distancing rule then the healthcare system will not be overloaded.
{"title":"Modelling and Simulating the Novel Coronavirus with Implications of Asymptomatic Carriers","authors":"Ghassane Benrhmach, Khalil Namir, J. Bouyaghroumni","doi":"10.1155/2020/5487147","DOIUrl":"https://doi.org/10.1155/2020/5487147","url":null,"abstract":"The World Health Organization declared that the total number of confirmed cases tested positive for SARS‐CoV‐2, affecting 210 countries, exceeded 3 million on 29 April 2020, with more than 207,973 deaths. In order to end the global COVID‐19 pandemic, public authorities have put in place multiple strategies like testing, contact tracing, and social distancing. Predictive mathematical models for epidemics are fundamental to understand the development of the epidemic and to plan effective control strategies. Some hosts may carry SARS‐CoV‐2 and transmit it to others, yet display no symptoms themselves. We propose applying a model (SELIAHRD) taking in consideration the number of asymptomatic infected people. The SELIAHRD model consists of eight stages: Susceptible, Exposed, Latent, Symptomatic Infected, Asymptomatic Infected, Hospitalized, Recovered, and Dead. The asymptomatic carriers contribute to the spread of disease, but go largely undetected and can therefore undermine efforts to control transmission. The simulation of possible scenarios of the implementation of social distancing shows that if we rigorously follow the social distancing rule then the healthcare system will not be overloaded.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/5487147","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44252601","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 aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. The existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.
{"title":"Boundary Value Problem of Nonlinear Hybrid Differential Equations with Linear and Nonlinear Perturbations","authors":"S. Melliani, A. El Allaoui, L. S. Chadli","doi":"10.1155/2020/9850924","DOIUrl":"https://doi.org/10.1155/2020/9850924","url":null,"abstract":"The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. The existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/9850924","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48188085","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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