We consider the convex combinations , of a pair of sequences of real numbers and such that , converging to , and study the location of the limit inside the intervals , for every or for sufficiently large . We also investigate the same problem for the case of two corresponding sequences converging to . Among other results, we prove some, a bit, unexpected ones. Namely, for each , we determine the exact index at which the sequence changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.
{"title":"Convex combinations of some convergent sequences","authors":"Stevo Stević","doi":"10.1002/mma.10463","DOIUrl":"https://doi.org/10.1002/mma.10463","url":null,"abstract":"We consider the convex combinations , of a pair of sequences of real numbers and such that , converging to , and study the location of the limit inside the intervals , for every or for sufficiently large . We also investigate the same problem for the case of two corresponding sequences converging to . Among other results, we prove some, a bit, unexpected ones. Namely, for each , we determine the exact index at which the sequence changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220044","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}
This paper is concerned with the class of quasilinear Schrödinger equations: which models the self‐channeling of a high‐power ultra short laser in matter provided , where is a parameter, and . Under some appropriate assumptions on , we establish the nonexistence of solutions for the above problem.
{"title":"Nonexistence of solutions to quasilinear Schrödinger equations with a parameter","authors":"Hongwang Yu, Yunfeng Wei, Caisheng Chen, Qiang Chen","doi":"10.1002/mma.10452","DOIUrl":"https://doi.org/10.1002/mma.10452","url":null,"abstract":"This paper is concerned with the class of quasilinear Schrödinger equations: <jats:disp-formula> </jats:disp-formula>which models the self‐channeling of a high‐power ultra short laser in matter provided , where is a parameter, and . Under some appropriate assumptions on , we establish the nonexistence of solutions for the above problem.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220049","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}
This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations.
{"title":"Spectral methods utilizing generalized Bernstein‐like basis functions for time‐fractional advection–diffusion equations","authors":"Shahad Adil Taher Algazaa, Jamshid Saeidian","doi":"10.1002/mma.10390","DOIUrl":"https://doi.org/10.1002/mma.10390","url":null,"abstract":"This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220045","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}
We study the Cauchy problem for the three‐dimensional isentropic compressible Navier–Stokes/Allen–Cahn system, which models the phase transitions in two‐component patterns interacting with a compressible fluid. Under the assumption that the initial perturbation is small and decays spatially, we establish the global existence and the pointwise behavior of strong solutions to this nonconserved system. To deal with the source terms involving the phase variable, we employ the Green's function and space‐time weighted estimates. The analysis shows that the phase variable mainly contains the diffusion wave with exponential decaying amplitude over time, and consequently the density and momentum of the compressible fluid adhere to a generalized Huygens principle.
{"title":"Wave propagation for the isentropic compressible Navier–Stokes/Allen–Cahn system","authors":"Yazhou Chen, Houzhi Tang, Yue Zhang","doi":"10.1002/mma.10458","DOIUrl":"https://doi.org/10.1002/mma.10458","url":null,"abstract":"We study the Cauchy problem for the three‐dimensional isentropic compressible Navier–Stokes/Allen–Cahn system, which models the phase transitions in two‐component patterns interacting with a compressible fluid. Under the assumption that the initial perturbation is small and decays spatially, we establish the global existence and the pointwise behavior of strong solutions to this nonconserved system. To deal with the source terms involving the phase variable, we employ the Green's function and space‐time weighted estimates. The analysis shows that the phase variable mainly contains the diffusion wave with exponential decaying amplitude over time, and consequently the density and momentum of the compressible fluid adhere to a generalized Huygens principle.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220046","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}
We consider the compressible Navier–Stokes equations with viscosities in bounded domains when the initial data are spherically symmetric, which covers the Saint‐Venant model for the motion of shallow water. First, based on the exploitation of the one‐dimensional feature of symmetric solutions, we prove the global existence of weak solutions with initial vacuum, where the upper bound of the density is obtained. Then, with more conditions imposed on the nonvacuum initial data, we obtain the global weak solution which is a strong one away from the symmetry center. The analysis allows for the possibility that a vacuum state emerges at the symmetry center; in particular, we give the uniform bound of the radius of the vacuum domain.
{"title":"Global solution of spherically symmetric compressible Navier–Stokes equations with bounded density and density‐dependent viscosity","authors":"Xueyao Zhang","doi":"10.1002/mma.10433","DOIUrl":"https://doi.org/10.1002/mma.10433","url":null,"abstract":"We consider the compressible Navier–Stokes equations with viscosities in bounded domains when the initial data are spherically symmetric, which covers the Saint‐Venant model for the motion of shallow water. First, based on the exploitation of the one‐dimensional feature of symmetric solutions, we prove the global existence of weak solutions with initial vacuum, where the upper bound of the density is obtained. Then, with more conditions imposed on the nonvacuum initial data, we obtain the global weak solution which is a strong one away from the symmetry center. The analysis allows for the possibility that a vacuum state emerges at the symmetry center; in particular, we give the uniform bound of the radius of the vacuum domain.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220054","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}
In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems.
{"title":"Spectral properties for discontinuous Dirac system with eigenparameter‐dependent boundary condition","authors":"Jiajia Zheng, Kun Li, Zhaowen Zheng","doi":"10.1002/mma.10364","DOIUrl":"https://doi.org/10.1002/mma.10364","url":null,"abstract":"In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220048","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}
Ghaus Ur Rahman, Dildar Ahmad, José Francisco Gómez‐Aguilar, Ravi P. Agarwal, Amjad Ali
In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.
{"title":"Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations","authors":"Ghaus Ur Rahman, Dildar Ahmad, José Francisco Gómez‐Aguilar, Ravi P. Agarwal, Amjad Ali","doi":"10.1002/mma.10392","DOIUrl":"https://doi.org/10.1002/mma.10392","url":null,"abstract":"In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220047","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}
The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.
{"title":"The analysis of traveling wave structures and chaos of the cubic–quartic perturbed Biswas–Milovic equation with Kudryashov's nonlinear form and two generalized nonlocal laws","authors":"Shuang Li, Xing‐Hua Du","doi":"10.1002/mma.10462","DOIUrl":"https://doi.org/10.1002/mma.10462","url":null,"abstract":"The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220053","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}
Periodic peaked waves of the Novikov equation have been proved to be ‐orbital stable. Utilizing the method of characteristics, we establish that the periodic peakons of the Novikov equation are linearly unstable under perturbations. Moreover, it is proved that the small initial perturbations of the above periodic peakons can lead to the blow‐up phenomenon in finite time in the nonlinear evolution of the Novikov equation.
{"title":"Instability of H1$$ {H}^1 $$‐stable periodic peakons for the Novikov equation","authors":"Gezi Chong, Ying Fu, Hao Wang","doi":"10.1002/mma.10436","DOIUrl":"https://doi.org/10.1002/mma.10436","url":null,"abstract":"Periodic peaked waves of the Novikov equation have been proved to be ‐orbital stable. Utilizing the method of characteristics, we establish that the periodic peakons of the Novikov equation are linearly unstable under perturbations. Moreover, it is proved that the small initial perturbations of the above periodic peakons can lead to the blow‐up phenomenon in finite time in the nonlinear evolution of the Novikov equation.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220050","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}
Muhammad Toseef, Abdul Mateen, Muhammad Aamir Ali, Zhiyue Zhang
In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities.
{"title":"A family of quadrature formulas with their error bounds for convex functions and applications","authors":"Muhammad Toseef, Abdul Mateen, Muhammad Aamir Ali, Zhiyue Zhang","doi":"10.1002/mma.10460","DOIUrl":"https://doi.org/10.1002/mma.10460","url":null,"abstract":"In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220052","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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