Pub Date : 2024-11-12DOI: 10.1016/j.padiff.2024.100978
Rasha M. Yaseen , Nidal F. Ali , Ahmed A. Mohsen , Aziz Khan , Thabet Abdeljawad
In this study, a cholera model with asymptomatic carriers was examined. A Holling type-II functional response function was used to describe disease transmission. For analyzing the dynamical behavior of cholera disease, a fractional-order model was developed. First, the positivity and boundedness of the system’s solutions were established. The local stability of the equilibrium points was also analyzed. Second, a Lyapunov function was used to construct the global asymptotic stability of the system for both endemic and disease-free equilibrium points. Finally, numerical simulations and sensitivity analysis were carried out using matlab software to demonstrate the accuracy and validate the obtained results.
本研究对无症状带菌者的霍乱模型进行了研究。霍林 II 型功能响应函数被用来描述疾病的传播。为分析霍乱疾病的动力学行为，建立了一个分数阶模型。首先，确定了系统解的实在性和有界性。还分析了平衡点的局部稳定性。其次，利用 Lyapunov 函数构建了该系统在地方病和无病平衡点上的全局渐进稳定性。最后，利用 matlab 软件进行了数值模拟和敏感性分析，以证明所获结果的准确性和有效性。
{"title":"The modeling and mathematical analysis of the fractional-order of Cholera disease: Dynamical and Simulation","authors":"Rasha M. Yaseen , Nidal F. Ali , Ahmed A. Mohsen , Aziz Khan , Thabet Abdeljawad","doi":"10.1016/j.padiff.2024.100978","DOIUrl":"10.1016/j.padiff.2024.100978","url":null,"abstract":"<div><div>In this study, a cholera model with asymptomatic carriers was examined. A Holling type-II functional response function was used to describe disease transmission. For analyzing the dynamical behavior of cholera disease, a fractional-order model was developed. First, the positivity and boundedness of the system’s solutions were established. The local stability of the equilibrium points was also analyzed. Second, a Lyapunov function was used to construct the global asymptotic stability of the system for both endemic and disease-free equilibrium points. Finally, numerical simulations and sensitivity analysis were carried out using matlab software to demonstrate the accuracy and validate the obtained results.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100978"},"PeriodicalIF":0.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.padiff.2024.100988
Subhajit Panda , Titilayo M Agbaje , Rupa Baithalu , S.R. Mishra
The time-dependent thermal management along with convective flows are vital in various heat transfer applications in engineering systems such as in automotive cooling systems, and industrial heat exchangers. Because of enhanced thermal conductivity, nanofluids are widely considered for advanced cooling systems such as electronics, aerospace, geothermal energy extraction, etc. The current analysis presents comparative results of the radiative, time-dependent flow of water/kerosene-based Copper nanofluids between squeezing Riga plates focusing on heat dissipation. Both the plates are embedded within a porous matrix and the influence of non-uniform heat source/sink and thermal convective boundary conditions is examined. Riga plates, generally utilized for their ability to generate electromagnetic fields provide greater control over the fluid flow. The problem designed with the inclusion of aforesaid factors is transformed into a non-dimensional form for the utilization of appropriate similarity functions. Further, the impacts of several effective terms on the flow profiles are presented followed by the numerical solution of the profile obtained using the spectral quasilinearization method. Moreover, some of the outstanding findings are; an increase in the fluid velocity is marked for the separation of the plates but the rate of enhancement in the case of kerosene is more pronounced than that of water. Further, irrespective to the type of fluids, the heat transfer rate enhances for the increasing heat fluid which provides the variation of thermal radiation.
{"title":"Comparative analysis on the radiative time-dependent water/kerosene-based Cu nanofluid through squeezing Riga plates with heat dissipation: Spectral quasilinearization technique","authors":"Subhajit Panda , Titilayo M Agbaje , Rupa Baithalu , S.R. Mishra","doi":"10.1016/j.padiff.2024.100988","DOIUrl":"10.1016/j.padiff.2024.100988","url":null,"abstract":"<div><div>The time-dependent thermal management along with convective flows are vital in various heat transfer applications in engineering systems such as in automotive cooling systems, and industrial heat exchangers. Because of enhanced thermal conductivity, nanofluids are widely considered for advanced cooling systems such as electronics, aerospace, geothermal energy extraction, etc. The current analysis presents comparative results of the radiative, time-dependent flow of water/kerosene-based Copper nanofluids between squeezing Riga plates focusing on heat dissipation. Both the plates are embedded within a porous matrix and the influence of non-uniform heat source/sink and thermal convective boundary conditions is examined. Riga plates, generally utilized for their ability to generate electromagnetic fields provide greater control over the fluid flow. The problem designed with the inclusion of aforesaid factors is transformed into a non-dimensional form for the utilization of appropriate similarity functions. Further, the impacts of several effective terms on the flow profiles are presented followed by the numerical solution of the profile obtained using the spectral quasilinearization method. Moreover, some of the outstanding findings are; an increase in the fluid velocity is marked for the separation of the plates but the rate of enhancement in the case of kerosene is more pronounced than that of water. Further, irrespective to the type of fluids, the heat transfer rate enhances for the increasing heat fluid which provides the variation of thermal radiation.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100988"},"PeriodicalIF":0.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.padiff.2024.100987
K. Hosseini , F. Alizadeh , S. Kheybari , E. Hinçal , B. Kaymakamzade , M.S. Osman
Ginzburg–Landau (GL) equations describe a wide range of phenomena involving superconductivity, superfluidity, etc. In the present paper, Ginzburg–Landau equations involving distinct laws are considered, and as a consequence, their solitary waves in the presence of perturbation terms are formally derived using the Kudryashov method. The effect of Kerr and parabolic laws on the dynamics of solitary waves is examined in detail. The outcomes of the current paper present suitable ways to control the width and amplitude of solitary waves. The authors believe that the results reported in the current study will contribute significantly to studies related to Ginzburg–Landau equations with distinct laws.
{"title":"Ginzburg–Landau equations involving different effects and their solitary waves","authors":"K. Hosseini , F. Alizadeh , S. Kheybari , E. Hinçal , B. Kaymakamzade , M.S. Osman","doi":"10.1016/j.padiff.2024.100987","DOIUrl":"10.1016/j.padiff.2024.100987","url":null,"abstract":"<div><div>Ginzburg–Landau (GL) equations describe a wide range of phenomena involving superconductivity, superfluidity, etc. In the present paper, Ginzburg–Landau equations involving distinct laws are considered, and as a consequence, their solitary waves in the presence of perturbation terms are formally derived using the Kudryashov method. The effect of Kerr and parabolic laws on the dynamics of solitary waves is examined in detail. The outcomes of the current paper present suitable ways to control the width and amplitude of solitary waves. The authors believe that the results reported in the current study will contribute significantly to studies related to Ginzburg–Landau equations with distinct laws.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100987"},"PeriodicalIF":0.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.padiff.2024.100994
Subhajit Panda , B. Nayak , Rupa Baithalu , S.R. Mishra
In biomedical engineering, the behavior of gyrotactic microorganisms with non-Newtonian fluids such as tangent hyperbolic fluids improve the design of targeted drug delivery systems. In this system control over microorganism movement is essential. The present study deals with the synergistic influence of gyrotactic microorganisms and bimolecular reactions on the bidirectional flow of tangent hyperbolic fluids under Nield boundary conditions. Further, the flow characteristic of the non-Newtonian fluid is enhanced by incorporating the impact of thermal radiation, heat sources, Brownian motion, and thermophoresis. The presentation of these phenomena is vital for an extensive range of applications, including industrial processes, biomedical engineering, and environmental management. The analysis employs advanced mathematical modeling which needs suitable transformation rules to get the non-dimensional form and further numerical simulation is presented with the assistance of the “shooting-based fourth-order Runge–Kutta technique”. The results are depicted for the several contributing factors via the built- in-house function bvp4c in “MATLAB”. The authentication of the study with the prior research is a benchmark to precede further research in this direction. However, the outstanding results are; the fluid velocity is controlled by increasing non-Newtonian Weissenberg number whereas the velocity slip shows dual characteristics on the axial velocity distribution. Further, the motile microorganism profile is controlled by the enhanced bioconvection Lewis number.
{"title":"Synergistic influence of gyrotactic microorganisms and bimolecular reaction on bidirectional tangent hyperbolic fluid with Nield boundary conditions: A biomathematical model","authors":"Subhajit Panda , B. Nayak , Rupa Baithalu , S.R. Mishra","doi":"10.1016/j.padiff.2024.100994","DOIUrl":"10.1016/j.padiff.2024.100994","url":null,"abstract":"<div><div>In biomedical engineering, the behavior of gyrotactic microorganisms with non-Newtonian fluids such as tangent hyperbolic fluids improve the design of targeted drug delivery systems. In this system control over microorganism movement is essential. The present study deals with the synergistic influence of gyrotactic microorganisms and bimolecular reactions on the bidirectional flow of tangent hyperbolic fluids under Nield boundary conditions. Further, the flow characteristic of the non-Newtonian fluid is enhanced by incorporating the impact of thermal radiation, heat sources, Brownian motion, and thermophoresis. The presentation of these phenomena is vital for an extensive range of applications, including industrial processes, biomedical engineering, and environmental management. The analysis employs advanced mathematical modeling which needs suitable transformation rules to get the non-dimensional form and further numerical simulation is presented with the assistance of the “shooting-based fourth-order Runge–Kutta technique”. The results are depicted for the several contributing factors via the built- in-house function bvp4c in “MATLAB”. The authentication of the study with the prior research is a benchmark to precede further research in this direction. However, the outstanding results are; the fluid velocity is controlled by increasing non-Newtonian Weissenberg number whereas the velocity slip shows dual characteristics on the axial velocity distribution. Further, the motile microorganism profile is controlled by the enhanced bioconvection Lewis number.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100994"},"PeriodicalIF":0.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-10DOI: 10.1016/j.padiff.2024.100993
R.K. Sahoo , S.R. Mishra , Subhajit Panda
The recent industrial needs for the production process depending upon heat transfer properties of the fluids. However, the utility of the nanofluid in comparison to the conventional fluid is widely used because of its advanced coolant efficiency. In particular cooling of electronic devices, drug delivery systems, operation theatre, etc. the use of nanofluid shows its influential characteristics. As a result, the contemporary study aims to inspect the heat transmission effects of alloy nanoparticles via the base fluid methanol is presented through a diverging channel. Particularly, the aluminium alloy of AA7075 containing base metal Aluminium (Al) about 87.1–91.4 %, Zinc (Zn) up to 5.1–6.1 %, Magnesium (Mg) about 2.1–2.9 %, Copper (Cu) within the range of 1.2–2.0 %, Chromium (Cr) amounts 0.18–0.28 %, Silicon (Si) usually <0.4 %, Iron (Fe) <0.5 %, Manganese (Mn) up to 0.3 %, and Titanium (Ti) usually <0.2 %. However, the flow through a permeable medium, the interaction of Darcy dissipation energies the flow phenomena. An appropriate similarity transform rule is employed for the transformation of the basic equations and solved analytically via the differential transform method (DTM). Further, a comparative analysis with previously establish outputs is presented to ensure the accuracy of the adopted methodology. The impact of characterizing factors on the flow profiles are presented graphically and the important outcomes are; the velocity profile shows its dual characteristic for the variation of alloy nanoparticles whereas the fluid temperature enhances significantly. Further, heat transport feature enhances for the augmentation in the Eckert number which is exhibited for the inclusion of dissipative heat.
{"title":"Significant properties of AA7075-methanol nanofluid flow through diverging channel with porous material: Differential transform method","authors":"R.K. Sahoo , S.R. Mishra , Subhajit Panda","doi":"10.1016/j.padiff.2024.100993","DOIUrl":"10.1016/j.padiff.2024.100993","url":null,"abstract":"<div><div>The recent industrial needs for the production process depending upon heat transfer properties of the fluids. However, the utility of the nanofluid in comparison to the conventional fluid is widely used because of its advanced coolant efficiency. In particular cooling of electronic devices, drug delivery systems, operation theatre, etc. the use of nanofluid shows its influential characteristics. As a result, the contemporary study aims to inspect the heat transmission effects of alloy nanoparticles via the base fluid methanol is presented through a diverging channel. Particularly, the aluminium alloy of AA7075 containing base metal Aluminium (Al) about 87.1–91.4 %, Zinc (Zn) up to 5.1–6.1 %, Magnesium (Mg) about 2.1–2.9 %, Copper (Cu) within the range of 1.2–2.0 %, Chromium (Cr) amounts 0.18–0.28 %, Silicon (Si) usually <0.4 %, Iron (Fe) <0.5 %, Manganese (Mn) up to 0.3 %, and Titanium (Ti) usually <0.2 %. However, the flow through a permeable medium, the interaction of Darcy dissipation energies the flow phenomena. An appropriate similarity transform rule is employed for the transformation of the basic equations and solved analytically via the differential transform method (DTM). Further, a comparative analysis with previously establish outputs is presented to ensure the accuracy of the adopted methodology. The impact of characterizing factors on the flow profiles are presented graphically and the important outcomes are; the velocity profile shows its dual characteristic for the variation of alloy nanoparticles whereas the fluid temperature enhances significantly. Further, heat transport feature enhances for the augmentation in the Eckert number which is exhibited for the inclusion of dissipative heat.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100993"},"PeriodicalIF":0.0,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-10DOI: 10.1016/j.padiff.2024.100980
Rami AlAhmad , Mohammad Al-Khaleel
In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.
{"title":"Analytical solutions for autonomous differential equations with weighted derivatives","authors":"Rami AlAhmad , Mohammad Al-Khaleel","doi":"10.1016/j.padiff.2024.100980","DOIUrl":"10.1016/j.padiff.2024.100980","url":null,"abstract":"<div><div>In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100980"},"PeriodicalIF":0.0,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-09DOI: 10.1016/j.padiff.2024.100983
Muhammad Moneeb Tariq , Muhammad Aziz-ur-Rehman , Muhammad Bilal Riaz
This paper focuses on obtaining exact solutions for the nonlinear Van der Waals gas system using the modified Khater method. Renowned as one of the latest and most precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The governing equation is transformed into an ordinary differential equation through a well-suited wave transformation. This analytical simplification makes it possible to use the provided methods to derive trigonometric, rational, and hyperbolic solutions. To illuminate the physical behavior of the model, graphical plots of selected solutions are presented. By selecting appropriate values for arbitrary factors, this visual representation enhances comprehension of the dynamical system. Furthermore, the system undergoes a certain transformation to become a planar dynamical system, and the bifurcation analysis is examined. Additionally, the sensitivity analysis of the dynamical system is conducted using the Runge–Kutta method to confirm that slight alterations in the initial conditions have minimal impact on the stability of the solution. The presence of chaotic dynamics in the Van der Waals gas system is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase profiles are used to illustrate these chaotic behaviors.
{"title":"An analytical investigation of the Van Der Waals gas system: Dynamics insights into bifurcation, optical pattern along with sensitivity and chaotic analysis","authors":"Muhammad Moneeb Tariq , Muhammad Aziz-ur-Rehman , Muhammad Bilal Riaz","doi":"10.1016/j.padiff.2024.100983","DOIUrl":"10.1016/j.padiff.2024.100983","url":null,"abstract":"<div><div>This paper focuses on obtaining exact solutions for the nonlinear Van der Waals gas system using the modified Khater method. Renowned as one of the latest and most precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The governing equation is transformed into an ordinary differential equation through a well-suited wave transformation. This analytical simplification makes it possible to use the provided methods to derive trigonometric, rational, and hyperbolic solutions. To illuminate the physical behavior of the model, graphical plots of selected solutions are presented. By selecting appropriate values for arbitrary factors, this visual representation enhances comprehension of the dynamical system. Furthermore, the system undergoes a certain transformation to become a planar dynamical system, and the bifurcation analysis is examined. Additionally, the sensitivity analysis of the dynamical system is conducted using the Runge–Kutta method to confirm that slight alterations in the initial conditions have minimal impact on the stability of the solution. The presence of chaotic dynamics in the Van der Waals gas system is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase profiles are used to illustrate these chaotic behaviors.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100983"},"PeriodicalIF":0.0,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-09DOI: 10.1016/j.padiff.2024.100985
Md. Nur Alam, Md. Azizur Rahman
Time-space fractional nonlinear problems (T-SFNLPs) play a crucial role in the study of nonlinear wave propagation. Time-space nonlinearity is prevalent across various fields of applied science, nonlinear dynamics, mathematical physics, and engineering, including biosciences, neurosciences, plasma physics, geochemistry, and fluid mechanics. In this context, we examine the time-space fractional soliton neuron model (TSFSNM), which holds significant importance in neuroscience. This model explains how action potentials are initiated and propagated by axons, based on a thermodynamic theory of nerve pulse transmission. The signals passing through the cell membrane (CM) are proposed to take the form of solitary sound pulses, which can be represented as solitons. To investigate these soliton solutions, nonlinear fractional differential equations (NLFDEs) are transformed into corresponding partial differential equations (PDEs) using a fractional complex transform (FCT). The Kudryashov method is then applied to determine the wave profiles for the TSFSNM equation. We present 3D, 2D, contour, and density plots of the TSFSNM equation, and further analyze how fractional and time-space parameters influence these wave profiles through additional graphical representations. Kink, singular kink and different types of soliton solutions are successfully recovered through the Kudryashov method. The outcomes of various studies show that the applied method is highly efficient and well-suited for tackling problems in applied sciences and mathematical physics. Graphical representations, coupled with numerical data, reinforce the validity and accuracy of the technique. The proposed method is a convenient and powerful tool for handling the solution of nonlinear equations, making it particularly effective in exploring complex wave phenomena in diverse scientific fields.
{"title":"Study of the parametric effect of the wave profiles of the time-space fractional soliton neuron model equation arising in the topic of neuroscience","authors":"Md. Nur Alam, Md. Azizur Rahman","doi":"10.1016/j.padiff.2024.100985","DOIUrl":"10.1016/j.padiff.2024.100985","url":null,"abstract":"<div><div>Time-space fractional nonlinear problems (T-SFNLPs) play a crucial role in the study of nonlinear wave propagation. Time-space nonlinearity is prevalent across various fields of applied science, nonlinear dynamics, mathematical physics, and engineering, including biosciences, neurosciences, plasma physics, geochemistry, and fluid mechanics. In this context, we examine the time-space fractional soliton neuron model (TSFSNM), which holds significant importance in neuroscience. This model explains how action potentials are initiated and propagated by axons, based on a thermodynamic theory of nerve pulse transmission. The signals passing through the cell membrane (CM) are proposed to take the form of solitary sound pulses, which can be represented as solitons. To investigate these soliton solutions, nonlinear fractional differential equations (NLFDEs) are transformed into corresponding partial differential equations (PDEs) using a fractional complex transform (FCT). The Kudryashov method is then applied to determine the wave profiles for the TSFSNM equation. We present 3D, 2D, contour, and density plots of the TSFSNM equation, and further analyze how fractional and time-space parameters influence these wave profiles through additional graphical representations. Kink, singular kink and different types of soliton solutions are successfully recovered through the Kudryashov method. The outcomes of various studies show that the applied method is highly efficient and well-suited for tackling problems in applied sciences and mathematical physics. Graphical representations, coupled with numerical data, reinforce the validity and accuracy of the technique. The proposed method is a convenient and powerful tool for handling the solution of nonlinear equations, making it particularly effective in exploring complex wave phenomena in diverse scientific fields.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100985"},"PeriodicalIF":0.0,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-09DOI: 10.1016/j.padiff.2024.100977
Kwassi Anani , Mensah Folly-Gbetoula
In this study, we consider Burgers’ equation with fixed Dirichlet boundary conditions on generic bounded intervals. By employing the Hopf–Cole transformation and a recently established exact operational solution for linear reaction–diffusion equations, an exact solution in the time domain is derived through inverse Laplace transforms. In the event that analytic inverses do in fact exist, they can be obtained in closed form through the use of Mellin transforms. Nevertheless, highly efficient algorithms are available, and numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. Two illustrative tests demonstrate that the results align closely with those of classical exact solutions. In comparison to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent a novel alternative, offering new insights and perspectives. The exact solution via the inverse Laplace transform is shown to be more computationally efficient, providing a reference point for numerical and semi-analytical methods.
{"title":"Exact solution to a class of problems for the Burgers’ equation on bounded intervals","authors":"Kwassi Anani , Mensah Folly-Gbetoula","doi":"10.1016/j.padiff.2024.100977","DOIUrl":"10.1016/j.padiff.2024.100977","url":null,"abstract":"<div><div>In this study, we consider Burgers’ equation with fixed Dirichlet boundary conditions on generic bounded intervals. By employing the Hopf–Cole transformation and a recently established exact operational solution for linear reaction–diffusion equations, an exact solution in the time domain is derived through inverse Laplace transforms. In the event that analytic inverses do in fact exist, they can be obtained in closed form through the use of Mellin transforms. Nevertheless, highly efficient algorithms are available, and numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. Two illustrative tests demonstrate that the results align closely with those of classical exact solutions. In comparison to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent a novel alternative, offering new insights and perspectives. The exact solution via the inverse Laplace transform is shown to be more computationally efficient, providing a reference point for numerical and semi-analytical methods.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100977"},"PeriodicalIF":0.0,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-06DOI: 10.1016/j.padiff.2024.100779
Shams Ul Arifeen , Ihteram Ali , Imtiaz Ahmad , Sadaf Shaheen
This study presents two numerical methods focused on Quintic B-spline (QBS) and Galerkin finite element method (GFEM) for solving time-fractional Kawahara equations. The QBS is utilized as both the basis and test function in the FEM approach. We apply Caputo formula with quadrature rule for evaluation of temporal fractional part. The QBS and GFEM formulation are used to approximate the space functions and their derivatives. Furthermore, a four-point Gauss Legendre quadrature is employed to evaluate the source term in the GFEM. The efficiency and accuracy of the proposed scheme are evaluated using the and norms. Additionally, Fourier stability analysis is conducted, and it is revealed that the method exhibits unconditional stability. The results, presented in the form of tables and graphs to demonstrate the effectiveness of the scheme.
{"title":"Computational study of time-fractional non-linear Kawahara equations using Quintic B-spline and Galerkin’s method","authors":"Shams Ul Arifeen , Ihteram Ali , Imtiaz Ahmad , Sadaf Shaheen","doi":"10.1016/j.padiff.2024.100779","DOIUrl":"10.1016/j.padiff.2024.100779","url":null,"abstract":"<div><div>This study presents two numerical methods focused on Quintic B-spline (QBS) and Galerkin finite element method (GFEM) for solving time-fractional Kawahara equations. The QBS is utilized as both the basis and test function in the FEM approach. We apply Caputo formula with quadrature rule for evaluation of temporal fractional part. The QBS and GFEM formulation are used to approximate the space functions and their derivatives. Furthermore, a four-point Gauss Legendre quadrature is employed to evaluate the source term in the GFEM. The efficiency and accuracy of the proposed scheme are evaluated using the <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> norms. Additionally, Fourier stability analysis is conducted, and it is revealed that the method exhibits unconditional stability. The results, presented in the form of tables and graphs to demonstrate the effectiveness of the scheme.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"12 ","pages":"Article 100779"},"PeriodicalIF":0.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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