Pub Date : 2025-09-01Epub Date: 2025-08-24DOI: 10.1016/j.padiff.2025.101295
M. Nurul Islam , M. Al-Amin , M. Ali Akbar
The space-time fractional Chen-Lee-Liu (CLL) model is a significant optical fiber model utilized to analyze the performance of communication systems in optical fibers. It studies numerous features that may have impacts on the data transmission rates and signal excellence in optical fibers networks, nonlinearity, and noise. By developing this model, the engineers and researchers can optimize the design and performance in optical fiber communication systems. The optical solitons pulses of the CLL model are the fundamental construction block of soliton transmission technology, the telecommunication sector, and data transfer of optical fiber. In this study, we establish the significant soliton solutions which can be functional in optics of the stated model through the beta derivative employing the generalized exponential rational function technique (GERFT) which are not been investigated in the recent literature. The numerical simulations of the establishing solitons illustrates the bell-shaped, periodic, and some other soliton-like feature sand the examined shapes show the structure and influence of the fractional parameters. The results of this study exhibits that the implemented technique is efficient, reliable, and capable of establishing solutions to other complex nonlinear models in optical fiber communication systems.
{"title":"Mathematical analysis of novel soliton solutions of the space-time fractional Chen-Lee-Liu model in optical fibers communication systems","authors":"M. Nurul Islam , M. Al-Amin , M. Ali Akbar","doi":"10.1016/j.padiff.2025.101295","DOIUrl":"10.1016/j.padiff.2025.101295","url":null,"abstract":"<div><div>The space-time fractional Chen-Lee-Liu (CLL) model is a significant optical fiber model utilized to analyze the performance of communication systems in optical fibers. It studies numerous features that may have impacts on the data transmission rates and signal excellence in optical fibers networks, nonlinearity, and noise. By developing this model, the engineers and researchers can optimize the design and performance in optical fiber communication systems. The optical solitons pulses of the CLL model are the fundamental construction block of soliton transmission technology, the telecommunication sector, and data transfer of optical fiber. In this study, we establish the significant soliton solutions which can be functional in optics of the stated model through the beta derivative employing the generalized exponential rational function technique (GERFT) which are not been investigated in the recent literature. The numerical simulations of the establishing solitons illustrates the bell-shaped, periodic, and some other soliton-like feature sand the examined shapes show the structure and influence of the fractional parameters. The results of this study exhibits that the implemented technique is efficient, reliable, and capable of establishing solutions to other complex nonlinear models in optical fiber communication systems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101295"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996471","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 : 2025-09-01Epub Date: 2025-06-30DOI: 10.1016/j.padiff.2025.101253
Mohammad Alaroud
Both linear and nonlinear differential, partial equations of fractional order can be solved efficiently using the residual power series method (RPSM). Nevertheless, the process requires the residual function's (n − 1)ϱ fractional derivative(FD). We all know that figuring out the FD of a function can be difficult. A straightforward and effective analytical technique known as the Laplace transform-residual power series method (LT-RPSM) is used in this study to provide the approximate and exact solutions to nonlinear fractional partial differential equations(NFPDEs) under Caputo fractional differentiation including the nonlinear Fokker-Planck, nonlinear gas dynamics and nonlinear Klein-Gordon equations. The computations needed to find the coefficients of an expansion series are modest because the proposed method just requires the concept of an infinite limit. Three nonlinear fractional physical problems are successfully solved by the used investigation, which provides closed- form solutions and exact solutions in ordinary case, also a thorough graphical and numerical comparisons of the findings discovered. These outcomes are compared with existing solutions in the literature, especially in the meaning of absolute errors against the Laplace Adomin decompostion method LADM in light of different FD operators. Strong agreement between the results of the used method and several series solution techniques. Consequently, LT-RPSM can be considered a very successful technique and the most effective analytical algorithm to deal with numerous NFPDEs emerging in physics and engineering.
{"title":"A comparative analysis using the Laplace transform approach for some nonlinear fractional physical problems","authors":"Mohammad Alaroud","doi":"10.1016/j.padiff.2025.101253","DOIUrl":"10.1016/j.padiff.2025.101253","url":null,"abstract":"<div><div>Both linear and nonlinear differential, partial equations of fractional order can be solved efficiently using the residual power series method (RPSM). Nevertheless, the process requires the residual function's (<em>n</em> − 1)ϱ fractional derivative(FD). We all know that figuring out the FD of a function can be difficult. A straightforward and effective analytical technique known as the Laplace transform-residual power series method (LT-RPSM) is used in this study to provide the approximate and exact solutions to nonlinear fractional partial differential equations(NFPDEs) under Caputo fractional differentiation including the nonlinear Fokker-Planck, nonlinear gas dynamics and nonlinear Klein-Gordon equations. The computations needed to find the coefficients of an expansion series are modest because the proposed method just requires the concept of an infinite limit. Three nonlinear fractional physical problems are successfully solved by the used investigation, which provides closed- form solutions and exact solutions in ordinary case, also a thorough graphical and numerical comparisons of the findings discovered. These outcomes are compared with existing solutions in the literature, especially in the meaning of absolute errors against the Laplace Adomin decompostion method LADM in light of different FD operators. Strong agreement between the results of the used method and several series solution techniques. Consequently, LT-RPSM can be considered a very successful technique and the most effective analytical algorithm to deal with numerous NFPDEs emerging in physics and engineering.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101253"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144548424","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 : 2025-09-01Epub Date: 2025-07-29DOI: 10.1016/j.padiff.2025.101261
Abderrahim El Ayboudi , Radoine Belkanoufi , Abdelkarim Hajjaj
This paper investigates the indirect boundary observability properties of one-dimensional strongly coupled wave equations in an approximated setting. Classical numerical discretization methods, such as finite differences and finite elements, typically fail to maintain uniform observability inequalities when applied to wave systems. This failure is primarily attributed to the emergence of high-frequency numerical solutions. The present work demonstrates a different approach through the implementation of these discretization schemes on a carefully designed non-uniform mesh. This study successfully establishes uniform observability inequalities for the coupled system. This methodology effectively recovers the system’s total energy through boundary observations, overcoming the well-documented limitations of traditional numerical approaches in wave equation systems.
{"title":"Uniform indirect boundary observability for a spatial discretization of strongly coupled wave equations","authors":"Abderrahim El Ayboudi , Radoine Belkanoufi , Abdelkarim Hajjaj","doi":"10.1016/j.padiff.2025.101261","DOIUrl":"10.1016/j.padiff.2025.101261","url":null,"abstract":"<div><div>This paper investigates the indirect boundary observability properties of one-dimensional strongly coupled wave equations in an approximated setting. Classical numerical discretization methods, such as finite differences and finite elements, typically fail to maintain uniform observability inequalities when applied to wave systems. This failure is primarily attributed to the emergence of high-frequency numerical solutions. The present work demonstrates a different approach through the implementation of these discretization schemes on a carefully designed non-uniform mesh. This study successfully establishes uniform observability inequalities for the coupled system. This methodology effectively recovers the system’s total energy through boundary observations, overcoming the well-documented limitations of traditional numerical approaches in wave equation systems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101261"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772096","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 : 2025-09-01Epub Date: 2025-08-22DOI: 10.1016/j.padiff.2025.101291
Pramod S , Sujatha N , Sreekala C. K , Hanumagowda B. N , Kiran S , Jagadish V. Tawade , Manish Gupta , Barno Abdullaeva , M. Ijaz Khan
This research paper comprehensively investigates magnetohydrodynamic free convection in a ferrofluid-filled rectangular cavity. The researchers designed a rectangular cavity where the left vertical wall maintains a warmer temperature than the right, while the horizontal walls (top and bottom) are adiabatic. A uniform magnetic field is imposed horizontally along the positive x-axis. The main objective is to analyse the impacts of various parameters, such as Hartmann number (0 ≤ Ha ≤ 60), Rayleigh number (103 ≤ Ra ≤ 106), and volume fraction (0 ≤ ϕ ≤ 0.04), on the heat transfer characteristics and fluid flow behavior within the enclosure. The governing equations are rigorously solved using the Galerkin finite element method. Quality plots like streamlines and isotherms and quantity plots like average Nusselt number (Nua) are presented to elucidate the underlying physics. The findings indicate that increasing Rayleigh numbers increases the convective flow, whereas increasing Hartmann numbers decreases the convective flow, promoting conduction as the primary mode of heat transfer. It is also notable that the inclusion of a magnetic field significantly alters the flow and temperature distributions, leading to a notable reduction in average Nusselt number. Furthermore, the incorporation of nanoparticles is found to intensify the heat transfer rates, with higher volume fractions yielding greater thermal performance. These findings offer significant implications for advancing thermal management, material processing techniques, and magnetohydrodynamic power generation, thereby providing innovative heat transfer solutions across diverse engineering applications.
{"title":"Magnetically driven free convection of nanofluids in rectangular cavities: A FEM approach","authors":"Pramod S , Sujatha N , Sreekala C. K , Hanumagowda B. N , Kiran S , Jagadish V. Tawade , Manish Gupta , Barno Abdullaeva , M. Ijaz Khan","doi":"10.1016/j.padiff.2025.101291","DOIUrl":"10.1016/j.padiff.2025.101291","url":null,"abstract":"<div><div>This research paper comprehensively investigates magnetohydrodynamic free convection in a ferrofluid-filled rectangular cavity. The researchers designed a rectangular cavity where the left vertical wall maintains a warmer temperature than the right, while the horizontal walls (top and bottom) are adiabatic. A uniform magnetic field is imposed horizontally along the positive <em>x-</em>axis. The main objective is to analyse the impacts of various parameters, such as Hartmann number (0 ≤ <em>Ha</em> ≤ 60), Rayleigh number (10<sup>3</sup> ≤ <em>Ra</em> ≤ 10<sup>6</sup>), and volume fraction (0 ≤ ϕ ≤ 0.04), on the heat transfer characteristics and fluid flow behavior within the enclosure. The governing equations are rigorously solved using the Galerkin finite element method. Quality plots like streamlines and isotherms and quantity plots like average Nusselt number (<em>Nu<sub>a</sub></em>) are presented to elucidate the underlying physics. The findings indicate that increasing Rayleigh numbers increases the convective flow, whereas increasing Hartmann numbers decreases the convective flow, promoting conduction as the primary mode of heat transfer. It is also notable that the inclusion of a magnetic field significantly alters the flow and temperature distributions, leading to a notable reduction in average Nusselt number. Furthermore, the incorporation of nanoparticles is found to intensify the heat transfer rates, with higher volume fractions yielding greater thermal performance. These findings offer significant implications for advancing thermal management, material processing techniques, and magnetohydrodynamic power generation, thereby providing innovative heat transfer solutions across diverse engineering applications.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101291"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902467","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 : 2025-09-01Epub Date: 2025-06-18DOI: 10.1016/j.padiff.2025.101242
Muhammad Kazim, Mubashir Abbas, Safder Hussain, Munawwar Ali Abbas
In this paper, we investigate a fractional model of an incompressible and unstable MHD viscous fluid with heat transfer pass across a porous medium. To quantify this, we used a vertical plate with a fluid connected to it. When an angled magnetic field is supplied, the plate moves in its own plane. The required nonlinear partial differential equations are used to convert the governing equations into a non-dimensional form. To find the solution of the simplified nonlinear partial differential equations, the Constant Proportional Caputo fractional derivatives are utilized. The Laplace transform techniques are used to simplify the non-dimensional governing equations of the model and the boundary conditions we discovered explicit formulations for each field. The resultant equation is solved for momentum and energy, and the solutions are given as series. The performance of velocity and temperature values are graphically plotted using MATHCAD software. In numerical simulation, the Local Skin fraction and local Nusselt number are considered and evaluated additionally. It has been concluded that the fluid’s temperature and velocity decreases by increasing the value of fractional parameter. It has also been found that the velocity and temperature increase with increasing values of.
{"title":"Fractional modelling of heat transfer through porous media for incompressible MHD fluid flow with laplace transform approach","authors":"Muhammad Kazim, Mubashir Abbas, Safder Hussain, Munawwar Ali Abbas","doi":"10.1016/j.padiff.2025.101242","DOIUrl":"10.1016/j.padiff.2025.101242","url":null,"abstract":"<div><div>In this paper, we investigate a fractional model of an incompressible and unstable MHD viscous fluid with heat transfer pass across a porous medium. To quantify this, we used a vertical plate with a fluid connected to it. When an angled magnetic field is supplied, the plate moves in its own plane. The required nonlinear partial differential equations are used to convert the governing equations into a non-dimensional form. To find the solution of the simplified nonlinear partial differential equations, the Constant Proportional Caputo fractional derivatives are utilized. The Laplace transform techniques are used to simplify the non-dimensional governing equations of the model and the boundary conditions we discovered explicit formulations for each field. The resultant equation is solved for momentum and energy, and the solutions are given as series. The performance of velocity and temperature values are graphically plotted using MATHCAD software. In numerical simulation, the Local Skin fraction and local Nusselt number are considered and evaluated additionally. It has been concluded that the fluid’s temperature and velocity decreases by increasing the value of fractional parameter. It has also been found that the velocity and temperature increase with increasing values of<span><math><msub><mi>Q</mi><mn>0</mn></msub></math></span>.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101242"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144472180","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 : 2025-09-01Epub Date: 2025-08-29DOI: 10.1016/j.padiff.2025.101276
V. Sowbakiya , R. Nirmalkumar , K. Loganathan , C. Selvamani
In this paper, we study the existence, uniqueness, and stability analysis of non-linear implicit neutral fractional differential equations involving the Atangana–Baleanu derivative in the Caputo sense. The Banach contraction principle theorem is employed to establish the existence and uniqueness of solutions, while Krasnoselskii’s fixed-point theorem is utilized to further analyze the existence of solutions. Stability analysis is also examined, including results for Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias stability. Finally, an example is presented to illustrate the existence and uniqueness of solutions, along with a discussion on their stability.
{"title":"Study on existence and stability analysis for implicit neutral fractional differential equations of ABC derivative","authors":"V. Sowbakiya , R. Nirmalkumar , K. Loganathan , C. Selvamani","doi":"10.1016/j.padiff.2025.101276","DOIUrl":"10.1016/j.padiff.2025.101276","url":null,"abstract":"<div><div>In this paper, we study the existence, uniqueness, and stability analysis of non-linear implicit neutral fractional differential equations involving the Atangana–Baleanu derivative in the Caputo sense. The Banach contraction principle theorem is employed to establish the existence and uniqueness of solutions, while Krasnoselskii’s fixed-point theorem is utilized to further analyze the existence of solutions. Stability analysis is also examined, including results for Ulam–Hyers, generalized Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias stability. Finally, an example is presented to illustrate the existence and uniqueness of solutions, along with a discussion on their stability.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101276"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144920091","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 : 2025-09-01Epub Date: 2025-07-04DOI: 10.1016/j.padiff.2025.101251
Paul M. Matao , Jumanne Mng’ang’a , B. Prabhakar Reddy
This study investigates the consequence of thermal radiation on the fractional magnetohydrodynamic (MHD) Couette flow of a Jeffrey fluid in a vertical channel, incorporating the influences of activation energy and Joule heating. The mathematical model is derived using appropriate governing equations that account for the non-Newtonian behavior of the Jeffrey fluid, combined with the impacts of thermal radiation, magnetic field, and activation energy mechanisms. The classical mathematical framework has been transformed into a system of fractal fractional-order derivatives using the Caputo–Fabrizio derivative operator. To solve these systems, the finite difference technique was employed. The behavior of fluid flow fields in response to several significant parameters was analyzed and represented graphically. It is ascertained that velocity distribution upsurges as Hall current parameter rises, while a more substantial effect from the Jeffrey fluid parameter results in a decrease in the velocity field. Additionally, thermal field profiles exhibited higher values in response to increased thermal radiation and Joule heating parameters, whereas the temperature distribution showed a decline with improving in Hall current parameter values. The concentration field improved with higher activation energy parameter values, in contrast to the opposite trend observed with temperature difference and chemical reaction parameters. Furthermore, it is remarked that fractal fractional-order derivatives operator produced a more pronounced boundary layer compared to both fractional and classical models. It is ascertained that the Nusselt number showing a 15.7% improvement in thermal efficiency as thermal radiation varied from 2 to 4. These findings are important for applications in geothermal energy extraction, and biomedical engineering.
{"title":"Thermal radiation effect on fractional MHD Couette Flow of Jeffrey fluid in a vertical channel with activation energy and Joule Heating","authors":"Paul M. Matao , Jumanne Mng’ang’a , B. Prabhakar Reddy","doi":"10.1016/j.padiff.2025.101251","DOIUrl":"10.1016/j.padiff.2025.101251","url":null,"abstract":"<div><div>This study investigates the consequence of thermal radiation on the fractional magnetohydrodynamic (MHD) Couette flow of a Jeffrey fluid in a vertical channel, incorporating the influences of activation energy and Joule heating. The mathematical model is derived using appropriate governing equations that account for the non-Newtonian behavior of the Jeffrey fluid, combined with the impacts of thermal radiation, magnetic field, and activation energy mechanisms. The classical mathematical framework has been transformed into a system of fractal fractional-order derivatives using the Caputo–Fabrizio derivative operator. To solve these systems, the finite difference technique was employed. The behavior of fluid flow fields in response to several significant parameters was analyzed and represented graphically. It is ascertained that velocity distribution upsurges as Hall current parameter rises, while a more substantial effect from the Jeffrey fluid parameter results in a decrease in the velocity field. Additionally, thermal field profiles exhibited higher values in response to increased thermal radiation and Joule heating parameters, whereas the temperature distribution showed a decline with improving in Hall current parameter values. The concentration field improved with higher activation energy parameter values, in contrast to the opposite trend observed with temperature difference and chemical reaction parameters. Furthermore, it is remarked that fractal fractional-order derivatives operator produced a more pronounced boundary layer compared to both fractional and classical models. It is ascertained that the Nusselt number showing a 15.7% improvement in thermal efficiency as thermal radiation varied from 2 to 4. These findings are important for applications in geothermal energy extraction, and biomedical engineering.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101251"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557626","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 : 2025-09-01Epub Date: 2025-06-26DOI: 10.1016/j.padiff.2025.101244
Ayelet Goldstein, Ofer Eyal, Jorge Berger
This work examines the behavior of fields near corners under various boundary conditions (BCs), focusing on singularities arising from fully constrained and relaxed BCs. We analyze this behavior across diverse physical systems governed by similar equations, including electromagnetism, superconductivity, and two-phase fluid flow. The corner geometry presents a challenge due to potentially diverging field solutions as the corner is approached (r 0). This motivates the investigation of relaxed BCs, which regularize the field by introducing a characteristic length (Ls) that relates the field’s value to its normal derivative at the boundary.
We explore both single-medium (single-phase) and double-medium (two-phase) systems. While prior research has addressed relaxed BCs in specific contexts, their application to corners, particularly in diverse physical systems, remains under-explored. We develop a series solution method to analyze the field behavior near the corner under different BCs. Concrete examples illustrate the theoretical framework, examining both fully constrained and relaxed scenarios. The implications of this work extend to fields such as fluid mechanics, electromagnetism, and heat transfer.
{"title":"Detailed analysis under fully constrained and relaxed boundary conditions of linear fields in the vicinity of a corner","authors":"Ayelet Goldstein, Ofer Eyal, Jorge Berger","doi":"10.1016/j.padiff.2025.101244","DOIUrl":"10.1016/j.padiff.2025.101244","url":null,"abstract":"<div><div>This work examines the behavior of fields near corners under various boundary conditions (BCs), focusing on singularities arising from fully constrained and relaxed BCs. We analyze this behavior across diverse physical systems governed by similar equations, including electromagnetism, superconductivity, and two-phase fluid flow. The corner geometry presents a challenge due to potentially diverging field solutions as the corner is approached (r<span><math><mo>→</mo></math></span> 0). This motivates the investigation of relaxed BCs, which regularize the field by introducing a characteristic length (Ls) that relates the field’s value to its normal derivative at the boundary.</div><div>We explore both single-medium (single-phase) and double-medium (two-phase) systems. While prior research has addressed relaxed BCs in specific contexts, their application to corners, particularly in diverse physical systems, remains under-explored. We develop a series solution method to analyze the field behavior near the corner under different BCs. Concrete examples illustrate the theoretical framework, examining both fully constrained and relaxed scenarios. The implications of this work extend to fields such as fluid mechanics, electromagnetism, and heat transfer.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101244"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144557625","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 : 2025-09-01Epub Date: 2025-07-26DOI: 10.1016/j.padiff.2025.101264
Ali Ahadi , Seyed Mostafa Mousavi , Amir Mohammad Alinia , Hossein Khademi
Partial differential equations (PDEs), particularly those involving fractional derivatives, have garnered considerable attention due to their ability to model complex systems with memory and hereditary properties. This paper focuses on the generalized Caputo fractional equation and presents a comparative analysis of three powerful solution techniques: the Homotopy Perturbation Method (HPM), the Variational Iteration Method (VIM), and the Akbari-Ganji Method (AGM). These methods are applied to fractional differential equations (FDEs) to derive approximate solutions. The accuracy and effectiveness of the methods are demonstrated through detailed comparisons with exact solutions and previous works in the field.
The study highlights the strengths of each technique in handling non-linear and fractional-order problems, providing reliable results with minimal error. Specifically, the HPM and VIM show remarkable convergence properties, while the AGM proves efficient in solving both linear and non-linear equations. These methods are validated by comparing the results with known solutions, which shows that these techniques work for a wide range of FDEs. The present study underscores the applicability of these approaches in several scientific and technological domains, hence promoting more advancements in the numerical examination of fractional systems.
{"title":"Analytical simulation of the nonlinear Caputo fractional equations","authors":"Ali Ahadi , Seyed Mostafa Mousavi , Amir Mohammad Alinia , Hossein Khademi","doi":"10.1016/j.padiff.2025.101264","DOIUrl":"10.1016/j.padiff.2025.101264","url":null,"abstract":"<div><div>Partial differential equations (PDEs), particularly those involving fractional derivatives, have garnered considerable attention due to their ability to model complex systems with memory and hereditary properties. This paper focuses on the generalized Caputo fractional equation and presents a comparative analysis of three powerful solution techniques: the Homotopy Perturbation Method (HPM), the Variational Iteration Method (VIM), and the Akbari-Ganji Method (AGM). These methods are applied to fractional differential equations (FDEs) to derive approximate solutions. The accuracy and effectiveness of the methods are demonstrated through detailed comparisons with exact solutions and previous works in the field.</div><div>The study highlights the strengths of each technique in handling non-linear and fractional-order problems, providing reliable results with minimal error. Specifically, the HPM and VIM show remarkable convergence properties, while the AGM proves efficient in solving both linear and non-linear equations. These methods are validated by comparing the results with known solutions, which shows that these techniques work for a wide range of FDEs. The present study underscores the applicability of these approaches in several scientific and technological domains, hence promoting more advancements in the numerical examination of fractional systems.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101264"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the geometric properties of Smarandache ruled surfaces generated by integral binormal curves in the Euclidean 3-space . Specifically, we study four types of Smarandache ruled surfaces: the , , , and surfaces, each defined by different combinations of the tangent, normal, and binormal vectors of the integral curves. For each type of surface, we derive the parametric representations and compute the fundamental geometric properties, including the striction lines, distribution parameters, and the first and second fundamental forms. Additionally, we provide explicit expressions for the Gaussian and mean curvatures, which characterize the local shape of the surfaces. We also analyze the geodesic curvature, normal curvature, and geodesic torsion associated with the base curves on these surfaces. Furthermore, we establish necessary and sufficient conditions for these surfaces to be developable or minimal. The paper concludes with detailed conditions under which the base curves can be classified as geodesic or asymptotic lines on the surfaces. The results are supported by rigorous proofs and illustrative examples, offering a comprehensive understanding of the geometric behavior of these Smarandache ruled surfaces.
{"title":"Geometric properties of Smarandache ruled surfaces generated by integral binormal curves in Euclidean 3-space","authors":"Ayman Elsharkawy , Hanene Hamdani , Clemente Cesarano , Noha Elsharkawy","doi":"10.1016/j.padiff.2025.101298","DOIUrl":"10.1016/j.padiff.2025.101298","url":null,"abstract":"<div><div>This paper investigates the geometric properties of Smarandache ruled surfaces generated by integral binormal curves in the Euclidean 3-space <span><math><msup><mrow><mi>E</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Specifically, we study four types of Smarandache ruled surfaces: the <span><math><mrow><mi>t</mi><mi>n</mi></mrow></math></span>, <span><math><mrow><mi>t</mi><mi>b</mi></mrow></math></span>, <span><math><mrow><mi>n</mi><mi>b</mi></mrow></math></span>, and <span><math><mrow><mi>t</mi><mi>n</mi><mi>b</mi></mrow></math></span> surfaces, each defined by different combinations of the tangent, normal, and binormal vectors of the integral curves. For each type of surface, we derive the parametric representations and compute the fundamental geometric properties, including the striction lines, distribution parameters, and the first and second fundamental forms. Additionally, we provide explicit expressions for the Gaussian and mean curvatures, which characterize the local shape of the surfaces. We also analyze the geodesic curvature, normal curvature, and geodesic torsion associated with the base curves on these surfaces. Furthermore, we establish necessary and sufficient conditions for these surfaces to be developable or minimal. The paper concludes with detailed conditions under which the base curves can be classified as geodesic or asymptotic lines on the surfaces. The results are supported by rigorous proofs and illustrative examples, offering a comprehensive understanding of the geometric behavior of these Smarandache ruled surfaces.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101298"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144931558","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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