Pub Date : 2025-12-01Epub Date: 2025-10-28DOI: 10.1016/j.padiff.2025.101311
Alaa A. Sharhan , Adnan K. Farhood , Alaa H. Al-Muslimawi
This research looks at the flow of inelastic fluids in an axisymmetric 4:1:4 contraction-expansion with a sharp corner. The finite element approach is used to simulate the flow of inelastic fluid numerically. The continuity equation and the conversation equation of momentum equation are used in combination with the power law model. This study presents the extent of the influence of many factors, including the Reynolds number (Re) and the power law index , on the solution behavior. Our focus in this work is specifically on how these parameters effect the component of the solution and the convergence rate. The values of pressure and velocity were on of the interests of our research paper, as was the extent to which these are effected by the power law index and the Reynolds number. The influence of index of power law model on viscosity was also one of the subjects of the investigation.
{"title":"Numerical study for inelastic fluid flow in a contraction-expansion axisymmetric channel by using the finite element method","authors":"Alaa A. Sharhan , Adnan K. Farhood , Alaa H. Al-Muslimawi","doi":"10.1016/j.padiff.2025.101311","DOIUrl":"10.1016/j.padiff.2025.101311","url":null,"abstract":"<div><div>This research looks at the flow of inelastic fluids in an axisymmetric 4:1:4 contraction-expansion with a sharp corner. The finite element approach is used to simulate the flow of inelastic fluid numerically. The continuity equation and the conversation equation of momentum equation are used in combination with the power law model. This study presents the extent of the influence of many factors, including the Reynolds number (Re) and the power law index <span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, on the solution behavior. Our focus in this work is specifically on how these parameters effect the component of the solution and the convergence rate. The values of pressure and velocity were on of the interests of our research paper, as was the extent to which these are effected by the power law index and the Reynolds number. The influence of index <span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> of power law model on viscosity was also one of the subjects of the investigation.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101311"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466072","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}
Pub Date : 2025-12-01Epub Date: 2025-10-06DOI: 10.1016/j.padiff.2025.101307
K. Rajyalakshmi, G. Ravi Kiran, N. Lavanya
This study provides an analytical examination of the hemodynamic characteristics of two-layered blood flow in a diverging narrow channel featuring multiple symmetrical stenoses, porous wall effects, and slip boundary conditions. The central region, characterized by a concentration of RBCs, is modeled as a Jeffrey fluid, whereas the peripheral region is considered Newtonian. Under the assumption of mild stenosis and incompressible, completely developed laminar movement, the governing equations are precisely formulated and solved through direct integration. Closed-form expressions for velocity, mean hematocrit, core hematocrit and effective viscosity have been obtained. Parametric analysis indicates that velocity escalates with the Jeffrey parameter and slip, whereas effective viscosity diminishes with elevated Jeffrey parameter and Darcy number values, but augments with slip and stenosis height. The core and mean hematocrit diminish with most parameter variations, yet increase with the Jeffrey parameter. These findings improve comprehension of pathological conditions such as arterial occlusions and illustrate microcirculatory effects, including the Fåhraeus–Lindqvist phenomenon. The integrated modeling framework enhances physiological relevance and facilitates biomedical applications in the diagnosis and treatment of vascular diseases.
{"title":"Hemodynamic analysis of Jeffrey blood flow with two-layered model through a multiple stenoses in a diverging narrow channel with a porous layer under slip conditions","authors":"K. Rajyalakshmi, G. Ravi Kiran, N. Lavanya","doi":"10.1016/j.padiff.2025.101307","DOIUrl":"10.1016/j.padiff.2025.101307","url":null,"abstract":"<div><div>This study provides an analytical examination of the hemodynamic characteristics of two-layered blood flow in a diverging narrow channel featuring multiple symmetrical stenoses, porous wall effects, and slip boundary conditions. The central region, characterized by a concentration of RBCs, is modeled as a Jeffrey fluid, whereas the peripheral region is considered Newtonian. Under the assumption of mild stenosis and incompressible, completely developed laminar movement, the governing equations are precisely formulated and solved through direct integration. Closed-form expressions for velocity, mean hematocrit, core hematocrit and effective viscosity have been obtained. Parametric analysis indicates that velocity escalates with the Jeffrey parameter and slip, whereas effective viscosity diminishes with elevated Jeffrey parameter and Darcy number values, but augments with slip and stenosis height. The core and mean hematocrit diminish with most parameter variations, yet increase with the Jeffrey parameter. These findings improve comprehension of pathological conditions such as arterial occlusions and illustrate microcirculatory effects, including the Fåhraeus–Lindqvist phenomenon. The integrated modeling framework enhances physiological relevance and facilitates biomedical applications in the diagnosis and treatment of vascular diseases.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101307"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268619","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}
Pub Date : 2025-12-01Epub Date: 2025-09-08DOI: 10.1016/j.padiff.2025.101283
Yeşim Sağlam Özkan , Esra Ünal Yılmaz
This study investigates the -dimensional generalized Hietarinta equation, which models the propagation of waves on water surfaces in the presence of gravity and surface tension. Solitary wave solutions are obtained using the method and the -expansion method, and are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. Two- and three-dimensional plots illustrate various wave structures, such as dark, kinked, and singular kinked waves, highlighting their dynamic behaviors under different parameter settings. Hamiltonian functions and bifurcation theory are employed to analyze phase portraits and nonlinear wave dynamics, including chaotic behavior. Numerical simulations has been conducted using Mathematica and Maple confirm the theoretical findings. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed techniques are effective, computationally efficient and reliable. In this context, considering previous studies, the findings of this research contribute to the existing literature.
{"title":"Exploring solitary wave structures and bifurcation dynamics in the (2+1)-dimensional generalized Hietarinta equation","authors":"Yeşim Sağlam Özkan , Esra Ünal Yılmaz","doi":"10.1016/j.padiff.2025.101283","DOIUrl":"10.1016/j.padiff.2025.101283","url":null,"abstract":"<div><div>This study investigates the <span><math><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional generalized Hietarinta equation, which models the propagation of waves on water surfaces in the presence of gravity and surface tension. Solitary wave solutions are obtained using the <span><math><mrow><mi>e</mi><mi>x</mi><mi>p</mi><mrow><mo>(</mo><mo>−</mo><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> method and the <span><math><mi>F</mi></math></span>-expansion method, and are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. Two- and three-dimensional plots illustrate various wave structures, such as dark, kinked, and singular kinked waves, highlighting their dynamic behaviors under different parameter settings. Hamiltonian functions and bifurcation theory are employed to analyze phase portraits and nonlinear wave dynamics, including chaotic behavior. Numerical simulations has been conducted using Mathematica and Maple confirm the theoretical findings. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed techniques are effective, computationally efficient and reliable. In this context, considering previous studies, the findings of this research contribute to the existing literature.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101283"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145044691","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}
Pub Date : 2025-12-01Epub Date: 2025-09-23DOI: 10.1016/j.padiff.2025.101303
Makhdoom Ali , Muhammad Bilal Riaz , Nauman Ahmed , Muhammad Zafarullah Baber , Ali Akgül
In this work, we investigates the conformable time-fractional (3+1)-dimensional p-type model for the analytical solutions. The underlying model is explained the material characteristics and spontaneous processes in solid-state physics, such as magnetism and conventional particle physics. To obtain the analytical solutions, we used the novel Kumar–Malik method and the new extended direct algebraic method. We derived the analytical solutions through the application of the conformal fractional derivative and the fractional wave transformation. We successfully obtain several solutions in the form of rational, hyperbolic, mixed trigonometric, mixed hyperbolic, exponential, Jacobi elliptic, and trigonometric functions by using these methods. The found solutions include various solitary wave solutions as well as bright, dark, and w-shaped soliton solutions. With the use of Mathematica 13.0, the analytical soliton solutions are further shown in 3D, contour and 2D representations, assisting in the understanding of these complex wave phenomena.
{"title":"Dynamical wave structures for time-fractional (3+1)-dimensional p-type model via two improved techniques","authors":"Makhdoom Ali , Muhammad Bilal Riaz , Nauman Ahmed , Muhammad Zafarullah Baber , Ali Akgül","doi":"10.1016/j.padiff.2025.101303","DOIUrl":"10.1016/j.padiff.2025.101303","url":null,"abstract":"<div><div>In this work, we investigates the conformable time-fractional (3+1)-dimensional p-type model for the analytical solutions. The underlying model is explained the material characteristics and spontaneous processes in solid-state physics, such as magnetism and conventional particle physics. To obtain the analytical solutions, we used the novel Kumar–Malik method and the new extended direct algebraic method. We derived the analytical solutions through the application of the conformal fractional derivative and the fractional wave transformation. We successfully obtain several solutions in the form of rational, hyperbolic, mixed trigonometric, mixed hyperbolic, exponential, Jacobi elliptic, and trigonometric functions by using these methods. The found solutions include various solitary wave solutions as well as bright, dark, and w-shaped soliton solutions. With the use of Mathematica 13.0, the analytical soliton solutions are further shown in 3D, contour and 2D representations, assisting in the understanding of these complex wave phenomena.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101303"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221967","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}
Pub Date : 2025-12-01Epub Date: 2025-09-10DOI: 10.1016/j.padiff.2025.101301
E. Momoniat , C. Harley
The equations of motion for the pseudo-planar motions of a classical linearized elastic solid and an incompressible linearized elastic solid undergoing non-uniform rotation about a vertical axis are derived. The pseudo-planar motions for both a classical linearized and an incompressible linearized elastic solid are determined numerically. For a classical linearized elastic solid, the non-uniform rotation is time-dependent and is specified. We derive a wave equation that models the non-uniform rotation for an incompressible linearized elastic solid. A pressure Poisson equation is derived and depends on the time derivative of the non-uniform rotation. The locus of the equations of motion coupled with the pseudo-planar motions of a cylindrical solid are plotted and the results are discussed. We show that the pseudo-planar motions of a classical linearized elastic solid with zero rotation are translations of the pseudo-planes about the locus. The pseudo-plane motions for classical and incompressible linearized elastic solids undergo translations and rotations about the locus. The motions are bound and stable when the pressure is symmetric. Unsymmetric pressure, which is just the mechanical pressure, results in a distortion of the pseudo-planar curves.
{"title":"Pseudo-planar deformations of a linearized elastic solid","authors":"E. Momoniat , C. Harley","doi":"10.1016/j.padiff.2025.101301","DOIUrl":"10.1016/j.padiff.2025.101301","url":null,"abstract":"<div><div>The equations of motion for the pseudo-planar motions of a classical linearized elastic solid and an incompressible linearized elastic solid undergoing non-uniform rotation about a vertical axis are derived. The pseudo-planar motions for both a classical linearized and an incompressible linearized elastic solid are determined numerically. For a classical linearized elastic solid, the non-uniform rotation is time-dependent and is specified. We derive a wave equation that models the non-uniform rotation for an incompressible linearized elastic solid. A pressure Poisson equation is derived and depends on the time derivative of the non-uniform rotation. The locus of the equations of motion coupled with the pseudo-planar motions of a cylindrical solid are plotted and the results are discussed. We show that the pseudo-planar motions of a classical linearized elastic solid with zero rotation are translations of the pseudo-planes about the locus. The pseudo-plane motions for classical and incompressible linearized elastic solids undergo translations and rotations about the locus. The motions are bound and stable when the pressure is symmetric. Unsymmetric pressure, which is just the mechanical pressure, results in a distortion of the pseudo-planar curves.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101301"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107802","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}
Pub Date : 2025-12-01Epub Date: 2025-10-03DOI: 10.1016/j.padiff.2025.101304
Kosuke Kita , Kei Matsushima , Tomoyuki Oka
This paper is concerned with configurations of two-material thermal conductors that minimize the Dirichlet energy for steady-state diffusion equations with nonlinear boundary conditions described mainly by maximal monotone operators. To find such configurations, a homogenization theorem will be proved and applied to an existence theorem for minimizers of a relaxation problem whose minimum value is equivalent to an original design problem. As a typical example of nonlinear boundary conditions, thermal radiation boundary conditions will be the focus, and then the sensitivity of the Dirichlet energy will be derived, which is used to estimate the minimum value. Since optimal configurations of the relaxation problem involve the so-called grayscale domains that do not make sense in general, a perimeter constraint problem via the positive part of the level set function will be introduced as an approximation problem to avoid such domains, and moreover, the existence theorem for minimizers of the perimeter constraint problem will be proved. In particular, it will also be proved that the limit of minimizers for the approximation problem becomes that of the relaxation problem in a specific case, and then candidates for minimizers of the approximation problem will be constructed by employing a nonlinear diffusion-based level set method. In this paper, it will be shown that optimized configurations deeply depend on force terms as a characteristic of nonlinear problems and will also be applied to real physical problems.
{"title":"Optimal design problem with thermal radiation","authors":"Kosuke Kita , Kei Matsushima , Tomoyuki Oka","doi":"10.1016/j.padiff.2025.101304","DOIUrl":"10.1016/j.padiff.2025.101304","url":null,"abstract":"<div><div>This paper is concerned with configurations of two-material thermal conductors that minimize the Dirichlet energy for steady-state diffusion equations with nonlinear boundary conditions described mainly by maximal monotone operators. To find such configurations, a homogenization theorem will be proved and applied to an existence theorem for minimizers of a relaxation problem whose minimum value is equivalent to an original design problem. As a typical example of nonlinear boundary conditions, thermal radiation boundary conditions will be the focus, and then the sensitivity of the Dirichlet energy will be derived, which is used to estimate the minimum value. Since optimal configurations of the relaxation problem involve the so-called grayscale domains that do not make sense in general, a perimeter constraint problem via the positive part of the level set function will be introduced as an approximation problem to avoid such domains, and moreover, the existence theorem for minimizers of the perimeter constraint problem will be proved. In particular, it will also be proved that the limit of minimizers for the approximation problem becomes that of the relaxation problem in a specific case, and then candidates for minimizers of the approximation problem will be constructed by employing a nonlinear diffusion-based level set method. In this paper, it will be shown that optimized configurations deeply depend on force terms as a characteristic of nonlinear problems and will also be applied to real physical problems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101304"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362460","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}
Pub Date : 2025-12-01Epub Date: 2025-09-13DOI: 10.1016/j.padiff.2025.101275
Amna H.A. Ibrahim, Hermane Mambili Mamboundou
This paper presents a mathematical model that comprehensively analyzes the dynamics of Hepatitis C Virus (HCV) infection. The model based on a system of nonlinear differential equations captures the interactions between liver cells (hepatocytes), the Hepatitis C virus, immune cells, and cytokines dynamics. We establish the well-posedness of the model within a biologically feasible region. Using the next-generation method, we calculate the basic reproduction number, , a threshold parameter that determines whether the infection will spread or die. A sensitivity analysis is also performed to identify the parameters that most significantly influence this number. We derive the conditions for the stability of disease-free and endemic equilibrium. The model is then used to investigate the system’s behavior under various scenarios: a weak immune response, the absence of T helper cell support, and a strong immune response. Our simulations show that the lack of interleukin-2 (IL-2) significantly affects the activation of cytotoxic T lymphocyte (CTLs). These results underscore the importance of including T helper cells, Interferon (IFN-) and IL-2 for an accurate representation of the dynamics of hepatitis C virus infection. Ultimately, this study deepens our understanding of the dynamics of HCV infection and simplifies how specific immune components shape the course of the disease.
{"title":"Mathematical model of immune response to Hepatitis C virus (HCV) disease","authors":"Amna H.A. Ibrahim, Hermane Mambili Mamboundou","doi":"10.1016/j.padiff.2025.101275","DOIUrl":"10.1016/j.padiff.2025.101275","url":null,"abstract":"<div><div>This paper presents a mathematical model that comprehensively analyzes the dynamics of Hepatitis C Virus (HCV) infection. The model based on a system of nonlinear differential equations captures the interactions between liver cells (hepatocytes), the Hepatitis C virus, immune cells, and cytokines dynamics. We establish the well-posedness of the model within a biologically feasible region. Using the next-generation method, we calculate the basic reproduction number, <span><math><msub><mrow><mi>ℜ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, a threshold parameter that determines whether the infection will spread or die. A sensitivity analysis is also performed to identify the parameters that most significantly influence this number. We derive the conditions for the stability of disease-free and endemic equilibrium. The model is then used to investigate the system’s behavior under various scenarios: a weak immune response, the absence of T helper cell support, and a strong immune response. Our simulations show that the lack of interleukin-2 (IL-2) significantly affects the activation of cytotoxic T lymphocyte (CTLs). These results underscore the importance of including T helper cells, Interferon<span><math><mrow><mo>−</mo><mi>γ</mi></mrow></math></span> (IFN-<span><math><mi>γ</mi></math></span>) and IL-2 for an accurate representation of the dynamics of hepatitis C virus infection. Ultimately, this study deepens our understanding of the dynamics of HCV infection and simplifies how specific immune components shape the course of the disease.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101275"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145108167","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 research investigates diverse wave behaviors of innovative higher dimensional Boussinesq model (BM) based on Hirota bi-linear technique. From this, firstly we derive lump and multiple soliton solutions. The study explores various dynamic behaviors, including interactions involving one up to four solitons. Additionally, the study analyzes breather waves, twofold periodic wave, periodic line lump wave, and the interactions among bell solitons. Other interactions analyze include lump wave with periodic wave, 1-stripe soliton and 2-stripe solitons. Many of these dynamic properties not yet explored in previous research. The trajectories of these solutions are visualized using Maple software, providing deeper insights into the model's dynamical behavior.
{"title":"Intrinsic dynamics of lumps and multi-soliton solutions to the higher dimensional Boussinesq model","authors":"M. Belal Hossen , Md. Towhiduzzaman , Mst. Shekha Khatun , Harun-Or- Roshid , Md. Amanat Ullah","doi":"10.1016/j.padiff.2025.101320","DOIUrl":"10.1016/j.padiff.2025.101320","url":null,"abstract":"<div><div>This research investigates diverse wave behaviors of innovative higher dimensional Boussinesq model (BM) based on Hirota bi-linear technique. From this, firstly we derive lump and multiple soliton solutions. The study explores various dynamic behaviors, including interactions involving one up to four solitons. Additionally, the study analyzes breather waves, twofold periodic wave, periodic line lump wave, and the interactions among bell solitons. Other interactions analyze include lump wave with periodic wave, 1-stripe soliton and 2-stripe solitons. Many of these dynamic properties not yet explored in previous research. The trajectories of these solutions are visualized using Maple software, providing deeper insights into the model's dynamical behavior.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101320"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466071","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}
Pub Date : 2025-12-01Epub Date: 2025-09-10DOI: 10.1016/j.padiff.2025.101293
Sonal Jain , Kolade M. Owolabi , Edson Pindza , Eben Mare
In this study, a novel implicit numerical approach is introduced by combining finite-difference techniques with innovative L1 schemes. This method is designed to solve time-fractional reaction–diffusion systems occurring in one and two dimensions. Specifically, the focus is on ecological systems with mixed boundary conditions, which are commonly found in biological and chemical processes. This research focuses on the spatiotemporal behavior of a predator–prey model with a Holling III functional response, taking into account the presence of prey refuges. This study revealed that this model does not exhibit a Turing pattern, which is typically associated with diffusion-driven instability. Consequently, this investigation explored alternative non-Turing patterns using extensive numerical simulations. In scenarios involving two-dimensional subdiffusion, the study observed a variety of spatiotemporal dynamics within the diffusive prey–predator model. When prey refuge availability was low, the system displayed a circular pattern that gradually expanded over time to encompass the entire spatial domain. As the availability of refugees decreased, the system transitioned from a spiral to a chaotic pattern. Furthermore, the research revealed that, as the ratio of predator-to-prey diffusion rates increased, the system exhibited a subdiffusive spiral pattern, which then transformed into a spot-like pattern. Eventually, these spots merged to form stripe-like patterns as the ratio increased. This investigation highlights the rich and intricate dynamics that can emerge in fractional predator–prey interactions when considering both spatial and temporal factors. To further confirm the complexity of the dynamical behaviors, Lyapunov exponents were estimated numerically.
{"title":"Dynamic complexity in fractional multispecies ecological systems: A Caputo derivative approach","authors":"Sonal Jain , Kolade M. Owolabi , Edson Pindza , Eben Mare","doi":"10.1016/j.padiff.2025.101293","DOIUrl":"10.1016/j.padiff.2025.101293","url":null,"abstract":"<div><div>In this study, a novel implicit numerical approach is introduced by combining finite-difference techniques with innovative L1 schemes. This method is designed to solve time-fractional reaction–diffusion systems occurring in one and two dimensions. Specifically, the focus is on ecological systems with mixed boundary conditions, which are commonly found in biological and chemical processes. This research focuses on the spatiotemporal behavior of a predator–prey model with a Holling III functional response, taking into account the presence of prey refuges. This study revealed that this model does not exhibit a Turing pattern, which is typically associated with diffusion-driven instability. Consequently, this investigation explored alternative non-Turing patterns using extensive numerical simulations. In scenarios involving two-dimensional subdiffusion, the study observed a variety of spatiotemporal dynamics within the diffusive prey–predator model. When prey refuge availability was low, the system displayed a circular pattern that gradually expanded over time to encompass the entire spatial domain. As the availability of refugees decreased, the system transitioned from a spiral to a chaotic pattern. Furthermore, the research revealed that, as the ratio of predator-to-prey diffusion rates increased, the system exhibited a subdiffusive spiral pattern, which then transformed into a spot-like pattern. Eventually, these spots merged to form stripe-like patterns as the ratio increased. This investigation highlights the rich and intricate dynamics that can emerge in fractional predator–prey interactions when considering both spatial and temporal factors. To further confirm the complexity of the dynamical behaviors, Lyapunov exponents were estimated numerically.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101293"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048529","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}
Pub Date : 2025-12-01Epub Date: 2025-10-19DOI: 10.1016/j.padiff.2025.101314
M.A. El-Shorbagy , Sonia Akram , Mati Ur Rahman , Hossam A. Nabwey
This study focuses on the Rosenau–Hyman equation, which is a fundamental model in nonlinear wave dynamics, and investigates it through the lens of nonclassical symmetry analysis. The approach employs symbolic computation to derive determining equations and uncover new invariant formulations, from which several explicit exact solutions are constructed. To further understand the system’s behavior, dynamical tools such as bifurcation analysis, sensitivity tests, Lyapunov exponents, and phase portraits are applied, highlighting the presence of stability transitions, multistability, and chaotic regimes. In addition, travelling wave solutions are obtained using the enhanced modified extended tanh function method (eMETFM), providing complementary wave structures. The findings deepen our understanding of nonlinear dispersive wave propagation and soliton interactions, with particular relevance to shallow water dynamics. More broadly, the developed solutions and their graphical interpretations contribute valuable insights for theoretical studies and applied research in fluid dynamics and wave modeling.
{"title":"Invariant formulation of nonclassical symmetries and explicit solutions of Rosenau-Hyman equation along with bifurcation analysis","authors":"M.A. El-Shorbagy , Sonia Akram , Mati Ur Rahman , Hossam A. Nabwey","doi":"10.1016/j.padiff.2025.101314","DOIUrl":"10.1016/j.padiff.2025.101314","url":null,"abstract":"<div><div>This study focuses on the Rosenau–Hyman equation, which is a fundamental model in nonlinear wave dynamics, and investigates it through the lens of nonclassical symmetry analysis. The approach employs symbolic computation to derive determining equations and uncover new invariant formulations, from which several explicit exact solutions are constructed. To further understand the system’s behavior, dynamical tools such as bifurcation analysis, sensitivity tests, Lyapunov exponents, and phase portraits are applied, highlighting the presence of stability transitions, multistability, and chaotic regimes. In addition, travelling wave solutions are obtained using the enhanced modified extended tanh function method (eMETFM), providing complementary wave structures. The findings deepen our understanding of nonlinear dispersive wave propagation and soliton interactions, with particular relevance to shallow water dynamics. More broadly, the developed solutions and their graphical interpretations contribute valuable insights for theoretical studies and applied research in fluid dynamics and wave modeling.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"16 ","pages":"Article 101314"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466073","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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