Pub Date : 2025-09-01Epub Date: 2025-07-28DOI: 10.1016/j.padiff.2025.101257
Thokozani Blessing Shiba, Khadijo Rashid Adem
This study examines the Camassa–Holm–Kadomtsev–Petviashvili equation with power law nonlinearity in (3+1)-D. The highlighted equation appears in mathematical physics, particularly in the study of nonlinear optics, plasma, integrable systems, and soliton theory, among other areas. The integration of the underlying equation is done using Lie symmetry analysis. To get more precise answers, the ansatz approach is applied. Traveling wave solutions are then obtained. The multiplier approach will be used to obtain conservation laws for the underlying equation.
{"title":"On the exact explicit solutions and conservation laws of the generalized (3+1)-D Camassa–Holm–Kadomtsev–Petviashvili equation with power law nonlinearity","authors":"Thokozani Blessing Shiba, Khadijo Rashid Adem","doi":"10.1016/j.padiff.2025.101257","DOIUrl":"10.1016/j.padiff.2025.101257","url":null,"abstract":"<div><div>This study examines the Camassa–Holm–Kadomtsev–Petviashvili equation with power law nonlinearity in (3+1)-D. The highlighted equation appears in mathematical physics, particularly in the study of nonlinear optics, plasma, integrable systems, and soliton theory, among other areas. The integration of the underlying equation is done using Lie symmetry analysis. To get more precise answers, the ansatz approach is applied. Traveling wave solutions are then obtained. The multiplier approach will be used to obtain conservation laws for the underlying equation.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101257"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779770","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: 2025-07-02DOI: 10.1016/j.padiff.2025.101223
Mehwish Saleem , Arshed Ali , Fazal-i-Haq , Hassan Khan
Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions.
{"title":"Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods","authors":"Mehwish Saleem , Arshed Ali , Fazal-i-Haq , Hassan Khan","doi":"10.1016/j.padiff.2025.101223","DOIUrl":"10.1016/j.padiff.2025.101223","url":null,"abstract":"<div><div>Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101223"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144672280","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: 2025-06-06DOI: 10.1016/j.padiff.2025.101233
M.C. Sebogodi , A.R. Adem , B. Muatjetjeja
A nonlinear evolution equation of the type is a generalized (2+1) dimensional equation in which and are real parameters. Together with an ansatz process and the multiple-exp function method, it will be examined from the perspective of Lie symmetry analysis. Lastly, conservation laws and their significant implications will be illustrated. To produce solitonic solutions, the multiple exp-function algorithm, a generalization of Hirota’s perturbation will be used.
{"title":"A generalized (2+1)-dimensional generalized KdV equation with a plethora of analytical solutions and conservation laws","authors":"M.C. Sebogodi , A.R. Adem , B. Muatjetjeja","doi":"10.1016/j.padiff.2025.101233","DOIUrl":"10.1016/j.padiff.2025.101233","url":null,"abstract":"<div><div>A nonlinear evolution equation of the type <span><span><span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>α</mi><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>−</mo><mi>∞</mi></mrow><mrow><mi>x</mi></mrow></msubsup><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>y</mi></mrow></msub><mi>d</mi><mi>x</mi><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><mi>β</mi><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>+</mo><mi>β</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>−</mo><mi>∞</mi></mrow><mrow><mi>x</mi></mrow></msubsup><msub><mrow><mi>u</mi></mrow><mrow><mi>y</mi></mrow></msub><mi>d</mi><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math></span></span></span>is a generalized (2+1) dimensional equation in which <span><math><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></math></span> are real parameters. Together with an ansatz process and the multiple-exp function method, it will be examined from the perspective of Lie symmetry analysis. Lastly, conservation laws and their significant implications will be illustrated. To produce solitonic solutions, the multiple exp-function algorithm, a generalization of Hirota’s perturbation will be used.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101233"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279950","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: 2025-07-07DOI: 10.1016/j.padiff.2024.100811
Ziad Khan , Fawad Hussain , Ikhtesham Ullah , Tariq Rahim , Madad Khan , Rashid Jan , Ibrahim Mekawy , Asma Alharbi
The complex intuitionistic fuzzy set is an extension of the intuitionistic fuzzy set where the membership and non-membership functions are expressed by a complex numbers. Ring theory is a well-known field of abstract algebra that is used in a broad area of present study in mathematics and computer science. The study of ideals is important in numerous ways in ring theory. Keeping in view the importance of complex intuitionistic fuzzy sets and ring theory, in this paper, we define the notion of complex intuitionistic fuzzy ideals in a classical ring and investigate its various algebraic properties. We obtain that the intersection of any two complex intuitionistic fuzzy ideals of a classical ring is again a complex intuitionistic fuzzy ideal of . We also define the notion of a complex intuitionistic fuzzy level set. Furthermore, we define the concept of complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal of a classical ring and prove that the set of all complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal forms a ring under certain binary operations. Finally, we prove a complex intuitionistic fuzzy version of the fundamental theorem of a ring homomorphism.
{"title":"Some novel properties of complex intuitionistic fuzzy ideals in classical ring","authors":"Ziad Khan , Fawad Hussain , Ikhtesham Ullah , Tariq Rahim , Madad Khan , Rashid Jan , Ibrahim Mekawy , Asma Alharbi","doi":"10.1016/j.padiff.2024.100811","DOIUrl":"10.1016/j.padiff.2024.100811","url":null,"abstract":"<div><div>The complex intuitionistic fuzzy set is an extension of the intuitionistic fuzzy set where the membership and non-membership functions are expressed by a complex numbers. Ring theory is a well-known field of abstract algebra that is used in a broad area of present study in mathematics and computer science. The study of ideals is important in numerous ways in ring theory. Keeping in view the importance of complex intuitionistic fuzzy sets and ring theory, in this paper, we define the notion of complex intuitionistic fuzzy ideals in a classical ring <span><math><mi>R</mi></math></span> and investigate its various algebraic properties. We obtain that the intersection of any two complex intuitionistic fuzzy ideals of a classical ring <span><math><mi>R</mi></math></span> is again a complex intuitionistic fuzzy ideal of <span><math><mi>R</mi></math></span>. We also define the notion of a complex intuitionistic fuzzy level set. Furthermore, we define the concept of complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal of a classical ring and prove that the set of all complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal forms a ring under certain binary operations. Finally, we prove a complex intuitionistic fuzzy version of the fundamental theorem of a ring homomorphism.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 100811"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144842010","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: 2025-01-28DOI: 10.1016/j.padiff.2025.101097
Ihteram Ali , Nadia Kamal , Abdulrahman Obaid Alshammari , Rahman Ullah , Imtiaz Ahmad
The diffusion equation with variable coefficients is widely applied in heat transfer to model the distribution of temperature in materials with spatially varying thermal properties, allowing for accurate simulation of heat conduction in heterogeneous media, accounting for changes in material composition and thermal conductivity. In this work, an effective method for numerically solving the one-dimensional diffusion equation with variable coefficients is presented. This approach utilizes a hybrid method that combines Lucas and Fibonacci polynomials with finite differences to convert the problem into a time-discrete form using the forward finite difference approach. Then, the derivative of the function is estimated using Fibonacci polynomials. The proposed method is applied to both linear and nonlinear one-dimensional problems, and its efficacy is verified by comparing the obtained results with those of alternative methods. The accuracy and effectiveness of the proposed method are demonstrated through comparison with the exact solution and other methods found in the literature.
{"title":"Exploring spectral collocation methods for diffusion models with variable coefficients in heat transfer studies","authors":"Ihteram Ali , Nadia Kamal , Abdulrahman Obaid Alshammari , Rahman Ullah , Imtiaz Ahmad","doi":"10.1016/j.padiff.2025.101097","DOIUrl":"10.1016/j.padiff.2025.101097","url":null,"abstract":"<div><div>The diffusion equation with variable coefficients is widely applied in heat transfer to model the distribution of temperature in materials with spatially varying thermal properties, allowing for accurate simulation of heat conduction in heterogeneous media, accounting for changes in material composition and thermal conductivity. In this work, an effective method for numerically solving the one-dimensional diffusion equation with variable coefficients is presented. This approach utilizes a hybrid method that combines Lucas and Fibonacci polynomials with finite differences to convert the problem into a time-discrete form using the forward finite difference approach. Then, the derivative of the function is estimated using Fibonacci polynomials. The proposed method is applied to both linear and nonlinear one-dimensional problems, and its efficacy is verified by comparing the obtained results with those of alternative methods. The accuracy and effectiveness of the proposed method are demonstrated through comparison with the exact solution and other methods found in the literature.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101097"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518751","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: 2025-07-03DOI: 10.1016/j.padiff.2025.101256
Nazia Parvin , Hasibun Naher , M. Ali Akbar
In this study, we investigate the soliton solutions of the (2 + 1)-dimensional time-fractional nonlinear generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation using the extended sinh-Gordon expansion approach. This equation is useful in modeling the hydro-magnetic waves in cold plasma, acoustic waves in harmonic crystals, shallow water waves, and acoustic gravity waves. By utilizing the suggested approach, we derive some rich structured soliton solutions, including bell-shaped soliton, anti-bell-shaped soliton, anti-peakon, periodic soliton and singular solitons of the model. These solutions are expressed in hyperbolic and trigonometric forms, and their dynamical behaviors are illustrated through 3D and 2D plots for various values of the fractional parameter and other physical parameters. The impact of the time-fractional derivative on the introduced model is examined using the beta derivative framework, which provides a more general and flexible way to enhance the accuracy of the solutions. The stability of the model is also examined through the linear stability theory, confirming that all analytical findings are stable. The results unambiguously demonstrate that the extended sinh-Gordon expansion approach is compatible, reliable, and efficient for investigating various nonlinear evolution equations in fields of applied science and engineering.
{"title":"Analytical new soliton solutions and stability analysis of the (2 + 1)-dimensional time-fractional nonlinear GZKBBM equation","authors":"Nazia Parvin , Hasibun Naher , M. Ali Akbar","doi":"10.1016/j.padiff.2025.101256","DOIUrl":"10.1016/j.padiff.2025.101256","url":null,"abstract":"<div><div>In this study, we investigate the soliton solutions of the (2 + 1)-dimensional time-fractional nonlinear generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation using the extended sinh-Gordon expansion approach. This equation is useful in modeling the hydro-magnetic waves in cold plasma, acoustic waves in harmonic crystals, shallow water waves, and acoustic gravity waves. By utilizing the suggested approach, we derive some rich structured soliton solutions, including bell-shaped soliton, anti-bell-shaped soliton, anti-peakon, periodic soliton and singular solitons of the model. These solutions are expressed in hyperbolic and trigonometric forms, and their dynamical behaviors are illustrated through 3D and 2D plots for various values of the fractional parameter <span><math><mrow><mi>β</mi><mspace></mspace></mrow></math></span>and other physical parameters. The impact of the time-fractional derivative on the introduced model is examined using the beta derivative framework, which provides a more general and flexible way to enhance the accuracy of the solutions. The stability of the model is also examined through the linear stability theory, confirming that all analytical findings are stable. The results unambiguously demonstrate that the extended sinh-Gordon expansion approach is compatible, reliable, and efficient for investigating various nonlinear evolution equations in fields of applied science and engineering.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101256"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571611","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: 2025-06-15DOI: 10.1016/j.padiff.2025.101245
Md. Mamunur Roshid, Sayma Akter, Bithi Akter
The current research investigates the M-fractional modified complex Ginzburg-Landau equation, a crucial nonlinear model for characterizing the behavior and evolution of optical solitary waves in dynamic fiber optics. Examining wave propagation in nonlinear dispersive media is essential since it promotes progress in data transmission for communication systems and allows for generating ultrafast optical pulses. the M-fractional derivative for the MCGL model is applied for the first time, which is more meaningful. The equation is converted into an ordinary differential equation via wave transformation, enabling the use of a unified technique to get many soliton solutions. By applying the unified method, we obtain more solutions than other methods, such as the function transformation technique.23 The solutions are expressed as tanh, sec, tan, sech functions and their combinations. For the special values of free parameters, we have periodic waves, kinky-periodic waves, periodic lump waves, periodic waves with lump waves, interactions of anti-kink and periodic waves, double periodic waves, and multi-kink waves. This work's innovative component is applying this approach to derive various soliton structures, analyzed using 2D, 3D, and contour representations. Additionally, the influence of fractional parameter presents with 3D plots for γ = 0.1, 0.4, 0.8. we also compare the fractional effect with the classical form in 2D plots. The results highlight the efficacy of this approach in examining soliton solutions in diverse nonlinear models, hence enhancing the comprehension of wave dynamics in mediums with differing stability.
{"title":"Optical soliton solutions of M-fractional modified complex Ginzburg-Landau equation using unified method: A comparative study","authors":"Md. Mamunur Roshid, Sayma Akter, Bithi Akter","doi":"10.1016/j.padiff.2025.101245","DOIUrl":"10.1016/j.padiff.2025.101245","url":null,"abstract":"<div><div>The current research investigates the M-fractional modified complex Ginzburg-Landau equation, a crucial nonlinear model for characterizing the behavior and evolution of optical solitary waves in dynamic fiber optics. Examining wave propagation in nonlinear dispersive media is essential since it promotes progress in data transmission for communication systems and allows for generating ultrafast optical pulses. the M-fractional derivative for the MCGL model is applied for the first time, which is more meaningful. The equation is converted into an ordinary differential equation via wave transformation, enabling the use of a unified technique to get many soliton solutions. By applying the unified method, we obtain more solutions than other methods, such as the function transformation technique.<sup>23</sup> The solutions are expressed as <em>tanh</em>, <em>sec</em>, <em>tan</em>, <em>sech</em> functions and their combinations. For the special values of free parameters, we have periodic waves, kinky-periodic waves, periodic lump waves, periodic waves with lump waves, interactions of anti-kink and periodic waves, double periodic waves, and multi-kink waves. This work's innovative component is applying this approach to derive various soliton structures, analyzed using 2D, 3D, and contour representations. Additionally, the influence of fractional parameter presents with 3D plots for γ = 0.1, 0.4, 0.8. we also compare the fractional effect with the classical form in 2D plots. The results highlight the efficacy of this approach in examining soliton solutions in diverse nonlinear models, hence enhancing the comprehension of wave dynamics in mediums with differing stability.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101245"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144614844","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: 2025-06-27DOI: 10.1016/j.padiff.2025.101243
Sachit Kumar , Varun Joshi , Mamta Kapoor
In this work, we use the Caputo fractional calculus to methodically examine the Coupled Jaulent–Miodek (CJM) fractional equation and the fractional Whitham–Broer–Kaup (WBK) system. The Sumudu residual power series approach and the Sumudu iteration transform method are used to analyze the nonlinear fractional differential equation systems, providing a comprehensive analytical analysis. The Sumudu iteration transform approach is used to achieve the fractional WBK system’s dynamics, as well as the Sumudu power series residual approach is utilized to investigate the CJM equation’s behavior for fractions. We thoroughly examine their interactions using known solutions, using both symbolic calculations and numerical simulations. This leads to the identification of new solutions and the clarification of the way in which certain systems of fractions behave in terms of the operator of Caputo. The outcomes demonstrate the efficacy of the strategies used to decipher the intricate dynamics of fractional nonlinear systems by demonstrating a strong convergence agreement between analytical and numerical solutions.
{"title":"Analysis of fractional viewpoints on the Jaulent–Miodek and Whitham–Broer–Kaup coupled equations","authors":"Sachit Kumar , Varun Joshi , Mamta Kapoor","doi":"10.1016/j.padiff.2025.101243","DOIUrl":"10.1016/j.padiff.2025.101243","url":null,"abstract":"<div><div>In this work, we use the Caputo fractional calculus to methodically examine the Coupled Jaulent–Miodek (CJM) fractional equation and the fractional Whitham–Broer–Kaup (WBK) system. The Sumudu residual power series approach and the Sumudu iteration transform method are used to analyze the nonlinear fractional differential equation systems, providing a comprehensive analytical analysis. The Sumudu iteration transform approach is used to achieve the fractional WBK system’s dynamics, as well as the Sumudu power series residual approach is utilized to investigate the CJM equation’s behavior for fractions. We thoroughly examine their interactions using known solutions, using both symbolic calculations and numerical simulations. This leads to the identification of new solutions and the clarification of the way in which certain systems of fractions behave in terms of the operator of Caputo. The outcomes demonstrate the efficacy of the strategies used to decipher the intricate dynamics of fractional nonlinear systems by demonstrating a strong convergence agreement between analytical and numerical solutions.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101243"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144562992","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: 2025-08-23DOI: 10.1016/j.padiff.2025.101285
Hamida Parvin , Md. Nur Alam , Md. Abdullah Bin Masud , Md. Jakir Hossen
This research discovers traveling wave solutions (TWSs) of the van der Waals normal form for fluidized granular matter using the modified S-expansion (MS-E) method. The model captures key behaviors such as phase transitions, clustering, and shock structures in granular flows. Applying a traveling wave transformation reduces the governing equation to a nonlinear ordinary differential equation (NODE), enabling the construction of TWSs relevant to geophysical and industrial applications. The MS-E technique is implemented to systematically derive TWSs—such as kink, bright, and dark solitons—that model density waves, shock fronts, and clustering in granular media. Comprehensive 2D, 3D, and contour plots are presented to validate and visualize the results, offering insights into wave behavior and soliton stability. This work highlights the MS-E method as a powerful tool for solving nonlinear integral and fractional partial differential equations (NLIFPDEs), with broad applications in granular physics, fluid mechanics, plasma waves, and nonlinear optics. This experiment offers a novel procedure to explore additional compound nonlinear wave phenomena by integrating the MS-E method, opening novel opportunities for additional expansions in soliton-driven knowledge. This method offers a promising pathway for future researchers to explore closed-form traveling wave solutions of other NLIFPDEs.
{"title":"Investigating traveling wave structures in the van der Waals normal form for fluidized granular matter through the modified S-expansion method","authors":"Hamida Parvin , Md. Nur Alam , Md. Abdullah Bin Masud , Md. Jakir Hossen","doi":"10.1016/j.padiff.2025.101285","DOIUrl":"10.1016/j.padiff.2025.101285","url":null,"abstract":"<div><div>This research discovers traveling wave solutions (TWSs) of the van der Waals normal form for fluidized granular matter using the modified S-expansion (MS-E) method. The model captures key behaviors such as phase transitions, clustering, and shock structures in granular flows. Applying a traveling wave transformation reduces the governing equation to a nonlinear ordinary differential equation (NODE), enabling the construction of TWSs relevant to geophysical and industrial applications. The MS-E technique is implemented to systematically derive TWSs—such as kink, bright, and dark solitons—that model density waves, shock fronts, and clustering in granular media. Comprehensive 2D, 3D, and contour plots are presented to validate and visualize the results, offering insights into wave behavior and soliton stability. This work highlights the MS-E method as a powerful tool for solving nonlinear integral and fractional partial differential equations (NLIFPDEs), with broad applications in granular physics, fluid mechanics, plasma waves, and nonlinear optics. This experiment offers a novel procedure to explore additional compound nonlinear wave phenomena by integrating the MS-E method, opening novel opportunities for additional expansions in soliton-driven knowledge. This method offers a promising pathway for future researchers to explore closed-form traveling wave solutions of other NLIFPDEs.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"15 ","pages":"Article 101285"},"PeriodicalIF":0.0,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018631","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: 2025-07-25DOI: 10.1016/j.padiff.2025.101263
P. Kumar , AR. Ajaykumar , F. Almeida , S. Saranya , Qasem Al-Mdallal
Statistical and numerical approach is provided in the current article for Casson-Carreau nanofluid transient flow over continuously elongated sheet of curved feature. The flow is subjected under the various generation, Joule heating, non-linear thermal radiation, activation energy, second order slip, and convective peripheral conditions. Identifying the parameters that optimize the heat transfer rate and using those parameters applying the appropriate statistical tool to optimize the heat transfer rate are the two motives behind this study. A regression analysis is executed on the entropy generated; it has analyzed statistically using response surface methodology. For the issue under consideration, a Runge-Kutta-Fehlberg 4–5th order scheme has been implemented. Here, the study shows that although the Darcy number and first order slip decelerates velocity, the second order slip improves the velocity regime. Additionally, the study has showed that the activation energy parameter leverages the same, while chemical reaction parameter has negative effect on mass dispersion. With an increase in Brinkmann number, entropy production likewise rises, and fluid friction irreversibilities become more prevalent. As unsteadiness and activation energy parameters increase, Sherwood number declines. The visual representation of isotherms and streamlines is presented to display the flow and temperature pattern as a summary of the study. For the experimental setup by RSM, the better correlation coefficient is 99.93 % attained. The Pareto-chart specifies 2.2 to be the vital point for the statistical experimental design considered. For all the levels of heat source parameter and Eckert number, Radiation parameter exhibits positive sensitivity.
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