Pub Date : 2026-10-01Epub Date: 2026-02-13DOI: 10.1016/j.cam.2026.117452
Sachin Saini, Uaday Singh
In this paper, a new family of nonlinear neural network operators with a single hidden layer has been introduced for a class of continuous functions and Lebesgue integrable functions on a compact interval. We used specific activation functions constructed using a class of sigmoidal functions. The convergence of these operators is examined, and we establish both pointwise and uniform approximation theorems for the functions in the class of continuous functions , . We also extend the study for the class of Lebesgue integrable functions for p ≥ 1. Investigation of these nonlinear operators provides valuable insights into their generalization ability and practical applications in signal processing. We also discuss some examples, quantitative estimates, and graphical representations to highlight the applications.
{"title":"Approximation capabilities of certain nonlinear neural network operators","authors":"Sachin Saini, Uaday Singh","doi":"10.1016/j.cam.2026.117452","DOIUrl":"10.1016/j.cam.2026.117452","url":null,"abstract":"<div><div>In this paper, a new family of nonlinear neural network operators with a single hidden layer has been introduced for a class of continuous functions and Lebesgue integrable functions on a compact interval. We used specific activation functions constructed using a class of sigmoidal functions. The convergence of these operators is examined, and we establish both pointwise and uniform approximation theorems for the functions in the class of continuous functions <span><math><mrow><mi>C</mi><mo>(</mo><mi>I</mi><mo>)</mo></mrow></math></span>, <span><math><mrow><mi>I</mi><mo>=</mo><mo>[</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow></math></span>. We also extend the study for the class of Lebesgue integrable functions <span><math><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></mrow></math></span> for <em>p</em> ≥ 1. Investigation of these nonlinear operators provides valuable insights into their generalization ability and practical applications in signal processing. We also discuss some examples, quantitative estimates, and graphical representations to highlight the applications.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117452"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-11DOI: 10.1016/j.cam.2026.117403
Victorita Dolean , Mark Fry , Matthias Langer
Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the most effective preconditioners, yet their theoretical guarantees often lag behind practical performance. In this work, we introduce and analyse the Δk-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the Δ-GenEO coarse space. Our results sharpen the k-explicit conditions for GMRES convergence, reducing the restrictions on the subdomain size and eigenvalue threshold. This narrows the long-standing gap between pessimistic theory and empirical evidence, and reveals why GenEO spaces based on SPD (symmetric positive definite) eigenvalue problems remain surprisingly effective despite their apparent limitations. Numerical experiments confirm the theory, demonstrating scalability, robustness to heterogeneity for low to moderate frequencies (while experiencing limitations in the high frequency cases), and significantly milder coarse-space growth than conservative estimates predict.
{"title":"Can symmetric positive definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?","authors":"Victorita Dolean , Mark Fry , Matthias Langer","doi":"10.1016/j.cam.2026.117403","DOIUrl":"10.1016/j.cam.2026.117403","url":null,"abstract":"<div><div>Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the most effective preconditioners, yet their theoretical guarantees often lag behind practical performance. In this work, we introduce and analyse the Δ<sub><em>k</em></sub>-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the Δ-GenEO coarse space. Our results sharpen the <em>k</em>-explicit conditions for GMRES convergence, reducing the restrictions on the subdomain size and eigenvalue threshold. This narrows the long-standing gap between pessimistic theory and empirical evidence, and reveals why GenEO spaces based on SPD (symmetric positive definite) eigenvalue problems remain surprisingly effective despite their apparent limitations. Numerical experiments confirm the theory, demonstrating scalability, robustness to heterogeneity for low to moderate frequencies (while experiencing limitations in the high frequency cases), and significantly milder coarse-space growth than conservative estimates predict.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117403"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-29DOI: 10.1016/j.nonrwa.2026.104610
Petar Ćirković , Jelena V. Manojlović
The present paper discusses the dynamics and optimal harvesting of an intraguild predation three-level food web model incorporating nonlinear Michaelis-Menten type harvesting on the intermediate predator and proportional harvesting on the intraguild predator. The positivity and boundedness of solutions, as well as the existence and stability of equilibria, are established, and unconditional survival of the prey species is observed. The effect of harvesting is studied through a detailed bifurcation analysis, revealing rich dynamical behaviors and threshold harvesting levels that prevent predator extinction. The existence of saddle-node, transcritical, pitchfork, and Hopf bifurcations is shown. The qualitative dynamics are discussed through two-parameter bifurcation diagram. Parameter regions of extinction and coexistence are identified. At higher harvesting rates, Bogdanov-Takens and generalized Hopf bifurcations reveal parametric regions in which either both predator species will eventually be driven to extinction or all three species may coexist, depending on the initial values. At lower harvesting rates, Zero-Hopf and generalized Hopf bifurcations reveal parametric regions in which either intermediate predator eventually goes extinct or all three species may coexist, depending on the initial population densities. It is shown that the system can exhibit multistability and sensitivity to initial conditions, with bistability between coexistence attractors and predator-free attractors. From an economic perspective, an optimal harvesting policy is derived, maximizing the total economic return from harvesting while preventing overharvesting and ensuring ecological sustainability. A numerical example shows that both economic benefits and ecological balance can be achieved by controlling both predators harvesting rates.
{"title":"Bifurcation analysis and optimal harvesting of an intraguild predation three-level food web model with harvesting on top two levels","authors":"Petar Ćirković , Jelena V. Manojlović","doi":"10.1016/j.nonrwa.2026.104610","DOIUrl":"10.1016/j.nonrwa.2026.104610","url":null,"abstract":"<div><div>The present paper discusses the dynamics and optimal harvesting of an intraguild predation three-level food web model incorporating nonlinear Michaelis-Menten type harvesting on the intermediate predator and proportional harvesting on the intraguild predator. The positivity and boundedness of solutions, as well as the existence and stability of equilibria, are established, and unconditional survival of the prey species is observed. The effect of harvesting is studied through a detailed bifurcation analysis, revealing rich dynamical behaviors and threshold harvesting levels that prevent predator extinction. The existence of saddle-node, transcritical, pitchfork, and Hopf bifurcations is shown. The qualitative dynamics are discussed through two-parameter bifurcation diagram. Parameter regions of extinction and coexistence are identified. At higher harvesting rates, Bogdanov-Takens and generalized Hopf bifurcations reveal parametric regions in which either both predator species will eventually be driven to extinction or all three species may coexist, depending on the initial values. At lower harvesting rates, Zero-Hopf and generalized Hopf bifurcations reveal parametric regions in which either intermediate predator eventually goes extinct or all three species may coexist, depending on the initial population densities. It is shown that the system can exhibit multistability and sensitivity to initial conditions, with bistability between coexistence attractors and predator-free attractors. From an economic perspective, an optimal harvesting policy is derived, maximizing the total economic return from harvesting while preventing overharvesting and ensuring ecological sustainability. A numerical example shows that both economic benefits and ecological balance can be achieved by controlling both predators harvesting rates.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104610"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-20DOI: 10.1016/j.cam.2026.117464
Yize Wang , Li Yuan
In this paper we apply the transported probability density function (PDF) method to model the Burgers turbulence governed by the one-dimensional Burgers equation with a stochastic external force and random initial data. The PDF modeling method can provide the mean and high order moments of the random solution variable which are useful for the characterization of Burgers turbulence. Firstly, an exact one-point PDF transport equation is derived from the Burgers equation with a stochastic external force, and then the conditional advection and diffusion terms are modeled for closure by using two respective models. Secondly, three numerical methods are employed to solve the modeled PDF transport equation: the mesh-based Lagrangian particle Monte Carlo (MC) method, the original quadrature method of moments (QMOM), and the linear QMOM (LQMOM) as modified in this work. The numerical tests verify the viability of using the PDF modeling method to study Burgers turbulence and show that the MC method, LQMOM and QMOM can yield results comparable to the DNS solution, and the LQMOM is more efficient than the QMOM and MC method. However, the MC method has less dissipation for turbulence in high wavenumber regimes.
{"title":"A transported PDF modeling method applied to one-dimensional stochastic Burgers equation","authors":"Yize Wang , Li Yuan","doi":"10.1016/j.cam.2026.117464","DOIUrl":"10.1016/j.cam.2026.117464","url":null,"abstract":"<div><div>In this paper we apply the transported probability density function (PDF) method to model the Burgers turbulence governed by the one-dimensional Burgers equation with a stochastic external force and random initial data. The PDF modeling method can provide the mean and high order moments of the random solution variable which are useful for the characterization of Burgers turbulence. Firstly, an exact one-point PDF transport equation is derived from the Burgers equation with a stochastic external force, and then the conditional advection and diffusion terms are modeled for closure by using two respective models. Secondly, three numerical methods are employed to solve the modeled PDF transport equation: the mesh-based Lagrangian particle Monte Carlo (MC) method, the original quadrature method of moments (QMOM), and the linear QMOM (LQMOM) as modified in this work. The numerical tests verify the viability of using the PDF modeling method to study Burgers turbulence and show that the MC method, LQMOM and QMOM can yield results comparable to the DNS solution, and the LQMOM is more efficient than the QMOM and MC method. However, the MC method has less dissipation for turbulence in high wavenumber regimes.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117464"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-24DOI: 10.1016/j.cam.2026.117518
Li Zhou , Wensheng Jia
Under the assumption that the payoffs of players are fuzzy-valued vector functions, we prove the existence and stability of fuzzy equilibria for multi-objective leader-follower games. To begin with, based on the order relation of fuzzy vectors, we define the notions of continuity and properly quasi-concavity of fuzzy-valued vector functions. We then propose the concept of fuzzy equilibria for fuzzy multi-objective games and prove their existence by utilizing Fan-Glicksberg fixed point theorem. Subsequently, the existence results of fuzzy equilibria for fuzzy multi-objective single-leader-multi-follower games and fuzzy multi-objective multi-leader-follower games are respectively shown. Finally, by constructing a problem space of fuzzy multi-objective multi-leader-follower games, we show that most of the games are essential in the sense of Baire category. Several examples are provided to illustrate and support the developed results.
{"title":"Existence and continuity theorems of equilibrium solutions of multi-objective leader-follower games with fuzzy payoffs","authors":"Li Zhou , Wensheng Jia","doi":"10.1016/j.cam.2026.117518","DOIUrl":"10.1016/j.cam.2026.117518","url":null,"abstract":"<div><div>Under the assumption that the payoffs of players are fuzzy-valued vector functions, we prove the existence and stability of fuzzy equilibria for multi-objective leader-follower games. To begin with, based on the order relation of fuzzy vectors, we define the notions of continuity and properly quasi-concavity of fuzzy-valued vector functions. We then propose the concept of fuzzy equilibria for fuzzy multi-objective games and prove their existence by utilizing Fan-Glicksberg fixed point theorem. Subsequently, the existence results of fuzzy equilibria for fuzzy multi-objective single-leader-multi-follower games and fuzzy multi-objective multi-leader-follower games are respectively shown. Finally, by constructing a problem space of fuzzy multi-objective multi-leader-follower games, we show that most of the games are essential in the sense of Baire category. Several examples are provided to illustrate and support the developed results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117518"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-24DOI: 10.1016/j.cam.2026.117537
Kabiru Ahmed , Mohammed Yusuf Waziri , Abubakar Sani Halilu , Salisu Murtala
In this paper, a modified version of the method by Dai and Kou for unconstrained optimization is presented for solving constrained nonlinear system of equations with applications to blurry image recovery. To develop the scheme, an effective choice for the Dai-Kou parameter τk, which has been declared a subject of research, was obtained by conducting singular value analysis and minimizing upper bound of the condition number of the search direction matrix of the classical Dai-Kou scheme. The backtracking linesearch presented by Li and Li is used to obtain step-size in the algorithm. The scheme satisfies the vital condition for analyzing global convergence. It is also suitable for solving some large-scale nonsmooth problems since it avoids computing the Jacobian. Mild conditions are employed to prove the schemes’ global convergence, while numerical experiments with related methods are carried out to illustrate its effectiveness for solving constrained nonlinear system of equations. The method was also tested with two effective schemes in the literature to illustrate its effectiveness in image de-blurring problems.
{"title":"Image recovery via modified Dai-Kou method for constrained system of nonlinear equations","authors":"Kabiru Ahmed , Mohammed Yusuf Waziri , Abubakar Sani Halilu , Salisu Murtala","doi":"10.1016/j.cam.2026.117537","DOIUrl":"10.1016/j.cam.2026.117537","url":null,"abstract":"<div><div>In this paper, a modified version of the method by Dai and Kou for unconstrained optimization is presented for solving constrained nonlinear system of equations with applications to blurry image recovery. To develop the scheme, an effective choice for the Dai-Kou parameter <em>τ<sub>k</sub></em>, which has been declared a subject of research, was obtained by conducting singular value analysis and minimizing upper bound of the condition number of the search direction matrix of the classical Dai-Kou scheme. The backtracking linesearch presented by Li and Li is used to obtain step-size in the algorithm. The scheme satisfies the vital condition for analyzing global convergence. It is also suitable for solving some large-scale nonsmooth problems since it avoids computing the Jacobian. Mild conditions are employed to prove the schemes’ global convergence, while numerical experiments with related methods are carried out to illustrate its effectiveness for solving constrained nonlinear system of equations. The method was also tested with two effective schemes in the literature to illustrate its effectiveness in image de-blurring problems.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117537"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-02-24DOI: 10.1016/j.cam.2026.117513
Zijun Hao , Qiong Wu, Chunmin Zhao
In this paper, a smooth-like lower order penalty approach is proposed for solving second-order cone mixed complementarity problems (SOCMCPs), which is closely related to second-order cone programming. Four kinds of special smoothing functions are considered in this approach. In light of this approach, the SOCMCP is approximated by asymptotic smooth-like lower order penalty equations (SLOPEs) with penalty parameter and smoothing parameter. Under a mild assumption, the main result shows that the solution sequence of the asymptotic SLOPEs converges exponentially to the solution of the SOCMCP while the penalty parameter tends to positive infinity and the smoothing parameter monotonically decreases to zero. A corresponding algorithm is constructed and the numerical experimental results are reported to illustrate the feasibility of the proposed algorithm. The performance profile of four specific smoothing functions is presented, and the numerical performance comparison of the proposed approach with other three algorithms is also investigated.
{"title":"Smooth-like lower order penalty approach for solving second-order cone mixed complementarity problems","authors":"Zijun Hao , Qiong Wu, Chunmin Zhao","doi":"10.1016/j.cam.2026.117513","DOIUrl":"10.1016/j.cam.2026.117513","url":null,"abstract":"<div><div>In this paper, a smooth-like lower order penalty approach is proposed for solving second-order cone mixed complementarity problems (SOCMCPs), which is closely related to second-order cone programming. Four kinds of special smoothing functions are considered in this approach. In light of this approach, the SOCMCP is approximated by asymptotic smooth-like lower order penalty equations (SLOPEs) with penalty parameter and smoothing parameter. Under a mild assumption, the main result shows that the solution sequence of the asymptotic SLOPEs converges exponentially to the solution of the SOCMCP while the penalty parameter tends to positive infinity and the smoothing parameter monotonically decreases to zero. A corresponding algorithm is constructed and the numerical experimental results are reported to illustrate the feasibility of the proposed algorithm. The performance profile of four specific smoothing functions is presented, and the numerical performance comparison of the proposed approach with other three algorithms is also investigated.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117513"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-21DOI: 10.1016/j.nonrwa.2026.104602
Meina Sun
The Riemann solutions for the simplified liquid-gas two-phase modified Chaplygin flow model are obtained constructively by virtue of the equality of velocity and pressure across the second characteristic field. Then, we are mainly concerned with the transition of Riemann solutions for this model when the equation of state varies from the modified Chaplygin flow to the Chaplygin flow by letting the perturbed parameter drop to zero. The formation of delta shock Riemann solution for the Chaplygin flow model is explored carefully by sending the limit in the Riemann solution made up of first-shock wave, second-contact discontinuity and third-shock wave for the modified Chaplygin flow model. In addition, the formation of the association of three contact discontinuities for the Chaplygin flow model is also carried out by taking the limit in all the four different structural Riemann solutions for the modified Chaplygin flow model.
{"title":"The transition of Riemann solutions for a simplified liquid-gas two-phase modified Chaplygin flow model","authors":"Meina Sun","doi":"10.1016/j.nonrwa.2026.104602","DOIUrl":"10.1016/j.nonrwa.2026.104602","url":null,"abstract":"<div><div>The Riemann solutions for the simplified liquid-gas two-phase modified Chaplygin flow model are obtained constructively by virtue of the equality of velocity and pressure across the second characteristic field. Then, we are mainly concerned with the transition of Riemann solutions for this model when the equation of state varies from the modified Chaplygin flow to the Chaplygin flow by letting the perturbed parameter drop to zero. The formation of delta shock Riemann solution for the Chaplygin flow model is explored carefully by sending the limit in the Riemann solution made up of first-shock wave, second-contact discontinuity and third-shock wave for the modified Chaplygin flow model. In addition, the formation of the association of three contact discontinuities for the Chaplygin flow model is also carried out by taking the limit in all the four different structural Riemann solutions for the modified Chaplygin flow model.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104602"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146037631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-20DOI: 10.1016/j.nonrwa.2026.104601
Lennon Ó Náraigh , Khang Ee Pang , Richard J. Smith
The Geometric Thin-Film Equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time – these are known as ‘particle solutions’. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are 1/2-Hölder continuous in time and are uniquely determined by the initial conditions.
{"title":"Convergence analysis of the geometric thin-film equation","authors":"Lennon Ó Náraigh , Khang Ee Pang , Richard J. Smith","doi":"10.1016/j.nonrwa.2026.104601","DOIUrl":"10.1016/j.nonrwa.2026.104601","url":null,"abstract":"<div><div>The Geometric Thin-Film Equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time – these are known as ‘particle solutions’. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are 1/2-Hölder continuous in time and are uniquely determined by the initial conditions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104601"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146037630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-31DOI: 10.1016/j.nonrwa.2026.104607
Jaume Llibre , Angela C.T. Sánchez , Durval J. Tonon
A hybrid dynamical system is one whose behavior is governed by both continuous and discrete dynamics; that is, it exhibits both flows and jumps. The field of hybrid dynamical systems is relatively recent and encompasses a broad range of phenomena, and is often used to model various natural processes. In this paper, we investigate the maximum number of limit cycles that can arise in certain classes of discontinuous piecewise differential systems. These systems consist of two hybrid rigid subsystems separated by a straight line, where each rigid subsystem is composed of a linear center perturbed by a homogeneous polynomial of degree 2, 3, 4, 5 or 6. For these classes of piecewise systems, we address the extended 16th Hilbert problem.
{"title":"Limit cycles of discontinuous piecewise hybrid rigid systems separated by a straight line","authors":"Jaume Llibre , Angela C.T. Sánchez , Durval J. Tonon","doi":"10.1016/j.nonrwa.2026.104607","DOIUrl":"10.1016/j.nonrwa.2026.104607","url":null,"abstract":"<div><div>A hybrid dynamical system is one whose behavior is governed by both continuous and discrete dynamics; that is, it exhibits both flows and jumps. The field of hybrid dynamical systems is relatively recent and encompasses a broad range of phenomena, and is often used to model various natural processes. In this paper, we investigate the maximum number of limit cycles that can arise in certain classes of discontinuous piecewise differential systems. These systems consist of two hybrid rigid subsystems separated by a straight line, where each rigid subsystem is composed of a linear center perturbed by a homogeneous polynomial of degree 2, 3, 4, 5 or 6. For these classes of piecewise systems, we address the extended 16th Hilbert problem.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104607"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146188909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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