A. Bensalem, Abdelkrim Salim, M. Benchohra, G. N’Guérékata
In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained.
{"title":"Approximate Controllability and Ulam Stability for Second-Order Impulsive Integrodifferential Evolution Equations with State-Dependent Delay","authors":"A. Bensalem, Abdelkrim Salim, M. Benchohra, G. N’Guérékata","doi":"10.1155/2024/8567425","DOIUrl":"https://doi.org/10.1155/2024/8567425","url":null,"abstract":"In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140366325","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}
S. Hussain, Fazal Haq, Abdullah Shah, D. Abduvalieva, Ali Shokri
Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.
由于存在立方项，艾伦-卡恩（AC）方程是一个高度非线性的方程，而且非常僵硬；因此，要找到其闭合形式的精确解析解并不容易。在本研究中，我们研究了 AC 方程的近似解析解。在这里，我们使用变分迭代法（VIM）来寻找交流方程的近似解析解。所得结果与双曲函数解法和行波解法进行了比较。我们还将所得结果与使用有限差分法（FDM）获得的数值解进行了比较。绝对误差分析表用于验证序列解。发现 VIM 获得的收敛级数解与分析解和数值解非常一致。
{"title":"Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation","authors":"S. Hussain, Fazal Haq, Abdullah Shah, D. Abduvalieva, Ali Shokri","doi":"10.1155/2024/8835138","DOIUrl":"https://doi.org/10.1155/2024/8835138","url":null,"abstract":"Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140210781","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}
Esam Y. Salah, Bahusaheb Sontakke, Mohammed S Abdo, W. Shatanawi, K. Abodayeh, M. D. Albalwi
The mathematical model of the dynamics of HIV/AIDS infection transmission is developed by adding the set of infected but noninfectious persons, using a conformable fractional derivative in the Liouville–Caputo sense. Some fixed point theorems are applied to this model to investigate the existence and uniqueness of the solutions. It is determined what the system’s fundamental reproduction number R0 is. The disease-free equilibrium displays the model’s stability and the local stability around the equilibrium. The study also examined the effects of different biological features on the system through numerical simulations using the Adams–Moulton approach. Additionally, varied values of fractional orders are simulated numerically, demonstrating that the results generated by the conformable fractional derivative-based model are more physiologically plausible than integer-order derivatives.
{"title":"Conformable Fractional-Order Modeling and Analysis of HIV/AIDS Transmission Dynamics","authors":"Esam Y. Salah, Bahusaheb Sontakke, Mohammed S Abdo, W. Shatanawi, K. Abodayeh, M. D. Albalwi","doi":"10.1155/2024/1958622","DOIUrl":"https://doi.org/10.1155/2024/1958622","url":null,"abstract":"The mathematical model of the dynamics of HIV/AIDS infection transmission is developed by adding the set of infected but noninfectious persons, using a conformable fractional derivative in the Liouville–Caputo sense. Some fixed point theorems are applied to this model to investigate the existence and uniqueness of the solutions. It is determined what the system’s fundamental reproduction number R0 is. The disease-free equilibrium displays the model’s stability and the local stability around the equilibrium. The study also examined the effects of different biological features on the system through numerical simulations using the Adams–Moulton approach. Additionally, varied values of fractional orders are simulated numerically, demonstrating that the results generated by the conformable fractional derivative-based model are more physiologically plausible than integer-order derivatives.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140250396","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}
Mohammed El Mokhtar Ould El Mokhtar, Saleh Fahad Aljurbua
This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.
本文试图建立具有符号变化权重、非线性奇异性和临界指数的奇异系统的非小解的存在性和多重性结果。通过使用变分法、Nehari 流形以及代表一些物理意义的参数 η 的充分条件，我们证明了一些现有结果，即在参数 α、β、μ 和 η 的一些充分条件下，研究临界点作为与 Nehari 流形定义的约束上所提问题 (2) 相关的能量函数的最小值，这些临界点就是我们系统的解。据我们所知，本文是对具有符号变化权重、非线性奇异性和临界指数的奇异系统研究的首次贡献之一。
{"title":"Multiple Solutions for Singular Systems with Sign-Changing Weight, Nonlinear Singularities and Critical Exponent","authors":"Mohammed El Mokhtar Ould El Mokhtar, Saleh Fahad Aljurbua","doi":"10.1155/2024/5582231","DOIUrl":"https://doi.org/10.1155/2024/5582231","url":null,"abstract":"This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139777882","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}
Mohammed El Mokhtar Ould El Mokhtar, Saleh Fahad Aljurbua
This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.
本文试图建立具有符号变化权重、非线性奇异性和临界指数的奇异系统的非小解的存在性和多重性结果。通过使用变分法、Nehari 流形以及代表一些物理意义的参数 η 的充分条件，我们证明了一些现有结果，即在参数 α、β、μ 和 η 的一些充分条件下，研究临界点作为与 Nehari 流形定义的约束上所提问题 (2) 相关的能量函数的最小值，这些临界点就是我们系统的解。据我们所知，本文是对具有符号变化权重、非线性奇异性和临界指数的奇异系统研究的首次贡献之一。
{"title":"Multiple Solutions for Singular Systems with Sign-Changing Weight, Nonlinear Singularities and Critical Exponent","authors":"Mohammed El Mokhtar Ould El Mokhtar, Saleh Fahad Aljurbua","doi":"10.1155/2024/5582231","DOIUrl":"https://doi.org/10.1155/2024/5582231","url":null,"abstract":"This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139837638","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 article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.
{"title":"Stability Results for Nonlinear Implicit ϑ-Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions","authors":"I. Kaddoura, Yahia Awad","doi":"10.1155/2023/5561399","DOIUrl":"https://doi.org/10.1155/2023/5561399","url":null,"abstract":"This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139133377","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 the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s effectiveness. The article concludes by discussing the biological implications of these outcomes.
{"title":"Spatiotemporal Dynamics of a Reaction Diffusive Predator-Prey Model: A Weak Nonlinear Analysis","authors":"N. B. Sharmila, C. Gunasundari, Mohammad Sajid","doi":"10.1155/2023/9190167","DOIUrl":"https://doi.org/10.1155/2023/9190167","url":null,"abstract":"In the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s effectiveness. The article concludes by discussing the biological implications of these outcomes.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294780","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}
Endale Ersino Bafe, Mitiku Daba Firdi, Lemi Guta Enyadene
Heat transfer in fluid mechanisms has a stronghold in everyday activities. To this end, nanofluids take a leading position in the advent of the betterment of thermal conductivity. The present study examines numerical investigations of incompressible magnetohydrodynamic (MHD) flow of Carreau nanofluid by considering nonlinear thermal radiation, Joule heating, temperature-dependent heat source/sink, and chemical reactions with attached Brownian movement and thermophoresis above a stretching sheet that saturates the porous medium. Pertaining similarity assumptions are used to change the flow equations into tractable forms of higher order nonlinear ordinary differential equations (ODEs). The continuation technique is adopted in the MATLAB bvp4c package for the numerical outcomes. The velocity, temperature, and nanoparticle concentration distributions in contrast to the leading parameters are availed in graphical and tabular descriptions. Among the many outcomes, increasing the radiation parameter from 0.2 to 0.8 surged the heat transfer rate by at n = 1.5 and lifted it only by at n = 0.5. By boosting the magnetic parameter from 0 to 1.5, respective and rises in local drag forces are achieved in shear-thickening and thinning regions. On top of that, chemical reactions and Brownian motion parameters decay the concentration field. The distinctiveness of this method is that a solution is secured for the problem, which is highly sensitive to initial and boundary conditions. It will be worth mentioning that these fluid flow models will be applicable in various fields, such as engineering, petroleum, nuclear safety processes, and medical science.
{"title":"Numerical Investigation of MHD Carreau Nanofluid Flow with Nonlinear Thermal Radiation and Joule Heating by Employing Darcy–Forchheimer Effect over a Stretching Porous Medium","authors":"Endale Ersino Bafe, Mitiku Daba Firdi, Lemi Guta Enyadene","doi":"10.1155/2023/5495140","DOIUrl":"https://doi.org/10.1155/2023/5495140","url":null,"abstract":"Heat transfer in fluid mechanisms has a stronghold in everyday activities. To this end, nanofluids take a leading position in the advent of the betterment of thermal conductivity. The present study examines numerical investigations of incompressible magnetohydrodynamic (MHD) flow of Carreau nanofluid by considering nonlinear thermal radiation, Joule heating, temperature-dependent heat source/sink, and chemical reactions with attached Brownian movement and thermophoresis above a stretching sheet that saturates the porous medium. Pertaining similarity assumptions are used to change the flow equations into tractable forms of higher order nonlinear ordinary differential equations (ODEs). The continuation technique is adopted in the MATLAB bvp4c package for the numerical outcomes. The velocity, temperature, and nanoparticle concentration distributions in contrast to the leading parameters are availed in graphical and tabular descriptions. Among the many outcomes, increasing the radiation parameter from 0.2 to 0.8 surged the heat transfer rate by <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"> <mn>47.78</mn> <mo>%</mo> </math> at n = 1.5 and lifted it only by <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"> <mn>8.5</mn> <mo>%</mo> </math> at n = 0.5. By boosting the magnetic parameter from 0 to 1.5, respective <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\"> <mn>37.64</mn> <mo>%</mo> </math> and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\"> <mn>20.17</mn> <mo>%</mo> </math> rises in local drag forces are achieved in shear-thickening and thinning regions. On top of that, chemical reactions and Brownian motion parameters decay the concentration field. The distinctiveness of this method is that a solution is secured for the problem, which is highly sensitive to initial and boundary conditions. It will be worth mentioning that these fluid flow models will be applicable in various fields, such as engineering, petroleum, nuclear safety processes, and medical science.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093530","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}
Despite the recent progress of global control efforts, tuberculosis (TB) remains a significant public health threat worldwide, especially in developing countries, including Ethiopia. Furthermore, the emergence of multidrug-resistant tuberculosis (MDR-TB) has further complicated the situation. This study aims at identifying the most effective strategies for combating MDR-TB in Ethiopia. We first present a compartmental model of MDR-TB transmission dynamics in Ethiopia. The model is shown to have positive solutions, and the stability of the equilibrium points is analyzed. Then, we extend the model by incorporating time-dependent control variables. These control variables are vaccination, distancing, and treatment for DS-TB and MDR-TB. Finally, the optimality system is numerically simulated by considering different combinations of the strategies, and their cost effectiveness is analysed. Our finding shows that, among single control strategies, the successful treatment of drug-susceptible tuberculosis (DS-TB) is the most effective control factor for eliminating MDR-TB transmission in Ethiopia. Furthermore, within the six dual control strategies, the combination of distancing and successful treatment of DS-TB is less costly and more effective than other strategies. Finally, out of the triple control strategies, the combination of distancing, successful treatment for DS-TB, and treatment for MDR-TB is the most efficient strategy for curbing the MDR-TB disease in Ethiopia. Thus, to reduce MDR-TB efficiently, it is recommended that authorities focus on treating MDR-TB, effective treatment of DS-TB, and promoting social distancing through public health education and awareness programs.
{"title":"Cost-Effectiveness Analysis of the Optimal Control Strategies for Multidrug-Resistant Tuberculosis Transmission in Ethiopia","authors":"Ashenafi Kelemu Mengistu, Peter J. Witbooi","doi":"10.1155/2023/8822433","DOIUrl":"https://doi.org/10.1155/2023/8822433","url":null,"abstract":"Despite the recent progress of global control efforts, tuberculosis (TB) remains a significant public health threat worldwide, especially in developing countries, including Ethiopia. Furthermore, the emergence of multidrug-resistant tuberculosis (MDR-TB) has further complicated the situation. This study aims at identifying the most effective strategies for combating MDR-TB in Ethiopia. We first present a compartmental model of MDR-TB transmission dynamics in Ethiopia. The model is shown to have positive solutions, and the stability of the equilibrium points is analyzed. Then, we extend the model by incorporating time-dependent control variables. These control variables are vaccination, distancing, and treatment for DS-TB and MDR-TB. Finally, the optimality system is numerically simulated by considering different combinations of the strategies, and their cost effectiveness is analysed. Our finding shows that, among single control strategies, the successful treatment of drug-susceptible tuberculosis (DS-TB) is the most effective control factor for eliminating MDR-TB transmission in Ethiopia. Furthermore, within the six dual control strategies, the combination of distancing and successful treatment of DS-TB is less costly and more effective than other strategies. Finally, out of the triple control strategies, the combination of distancing, successful treatment for DS-TB, and treatment for MDR-TB is the most efficient strategy for curbing the MDR-TB disease in Ethiopia. Thus, to reduce MDR-TB efficiently, it is recommended that authorities focus on treating MDR-TB, effective treatment of DS-TB, and promoting social distancing through public health education and awareness programs.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135344980","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, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.
{"title":"Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation","authors":"Maria Ihsane El Bahi, K. Hilal","doi":"10.1155/2023/8241804","DOIUrl":"https://doi.org/10.1155/2023/8241804","url":null,"abstract":"In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64800183","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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