This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.
{"title":"Asymptotic stability of the nonlocal diffusion equation with nonlocal delay","authors":"Yiming Tang, Xin Wu, Rong Yuan, Zhaohai Ma","doi":"10.1002/mma.10385","DOIUrl":"https://doi.org/10.1002/mma.10385","url":null,"abstract":"This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934962","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 focuses on the existence of normalized solutions for the Chern–Simons–Schrödinger system with mixed dispersion and critical exponential growth. These solutions correspond to critical points of the underlying energy functional under the ‐norm constraint, namely, . Under certain mild assumptions, we establish the existence of nontrivial solutions by developing new mathematical strategies and analytical techniques for the given system. These results extend and improve the results in the existing literature.
{"title":"Normalized solutions for Chern–Simons–Schrödinger system with mixed dispersion and critical exponential growth","authors":"Chenlu Wei, Lixi Wen","doi":"10.1002/mma.10383","DOIUrl":"https://doi.org/10.1002/mma.10383","url":null,"abstract":"This paper focuses on the existence of normalized solutions for the Chern–Simons–Schrödinger system with mixed dispersion and critical exponential growth. These solutions correspond to critical points of the underlying energy functional under the ‐norm constraint, namely, . Under certain mild assumptions, we establish the existence of nontrivial solutions by developing new mathematical strategies and analytical techniques for the given system. These results extend and improve the results in the existing literature.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141934961","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 deals with compatibility of space‐time kernels with (either) full, spatially dynamical, or space‐time compact support. We deal with the dilemma of statistical accuracy versus computational scalability, which are in a notorious trade‐off. Apparently, models with full support ensure maximal information but are computationally expensive, while compactly supported models achieve computational scalability at the expense of loss of information. Hence, an inspection of whether these models might be compatible is necessary. The criterion we use for such an inspection is based on equivalence of Gaussian measures. We provide sufficient conditions for space‐time compatibility. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics. Some results of independent interest relate about the space‐time spectrum associated with the classes of kernels proposed in the paper.
{"title":"Compatibility of space‐time kernels with full, dynamical, or compact support","authors":"Tarik Faouzi, Reinhard Furrer, Emilio Porcu","doi":"10.1002/mma.10379","DOIUrl":"https://doi.org/10.1002/mma.10379","url":null,"abstract":"This paper deals with compatibility of space‐time kernels with (either) full, spatially dynamical, or space‐time compact support. We deal with the dilemma of statistical accuracy <jats:styled-content>versus</jats:styled-content> computational scalability, which are in a notorious trade‐off. Apparently, models with full support ensure maximal information but are computationally expensive, while compactly supported models achieve computational scalability at the expense of loss of information. Hence, an inspection of whether these models might be compatible is necessary. The criterion we use for such an inspection is based on equivalence of Gaussian measures. We provide sufficient conditions for space‐time compatibility. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics. Some results of independent interest relate about the space‐time spectrum associated with the classes of kernels proposed in the paper.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882111","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 explores the pure‐cubic nonlinear Schrödinger equation (PC‐NLSE) with different nonlinearities. According to qualitative analysis, we get the dynamic systems and show that solitons and periodic solutions exist. The corresponding traveling wave solutions of these equations are constructed to demonstrate the correctness of qualitative analysis, and some solutions are initially given. In particular, a special kind of soliton solution, the Gaussian soliton, is constructed, which is rarely identified in non‐logarithmic equation. Next, the solitons stability and modulation instability (MI) of PC‐NLSE with two types of nonlinearity are discussed. Finally, by adding perturbed terms to the dynamic systems, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are the chaotic behaviors in PC‐NLSE. To the best of our knowledge, the Gaussian solitons, stability analysis and chaotic behaviors we obtained are first presented, which improves the study and proposes a new direction for the future researches on PC‐NLSE.
{"title":"Chaotic behaviors and stability analysis of pure‐cubic nonlinear Schrödinger equation with full nonlinearity","authors":"Yaxi Li, Yue Kai","doi":"10.1002/mma.10374","DOIUrl":"https://doi.org/10.1002/mma.10374","url":null,"abstract":"This paper explores the pure‐cubic nonlinear Schrödinger equation (PC‐NLSE) with different nonlinearities. According to qualitative analysis, we get the dynamic systems and show that solitons and periodic solutions exist. The corresponding traveling wave solutions of these equations are constructed to demonstrate the correctness of qualitative analysis, and some solutions are initially given. In particular, a special kind of soliton solution, the Gaussian soliton, is constructed, which is rarely identified in non‐logarithmic equation. Next, the solitons stability and modulation instability (MI) of PC‐NLSE with two types of nonlinearity are discussed. Finally, by adding perturbed terms to the dynamic systems, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are the chaotic behaviors in PC‐NLSE. To the best of our knowledge, the Gaussian solitons, stability analysis and chaotic behaviors we obtained are first presented, which improves the study and proposes a new direction for the future researches on PC‐NLSE.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882202","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 aim of this study is to present the evolution of COVID‐19 pandemic in Turkey. For this, the SIR (Susceptible, Infected, Removed) model with the fractional order derivative is employed. By applying the collocation method via the Pell–Lucas polynomials (PLPs) to this model, the approximate solutions of model with fractional order derivative are obtained. Hence, the comments are made about the susceptible population, the infected population, and the recovered population. For the method, firstly, PLPs are expressed in matrix form for a selected number of . With the help of this matrix relationship, the matrix forms of each term in the SIR model with the fractional order derivative are constituted. For implementation and visualization, we utilize MATLAB. Moreover, the outcomes for the Runge–Kutta method (RKM) are obtained using MATLAB, and these results are compared with the results obtained with the Pell–Lucas collocation method (PLCM). From all simulations, it is concluded that the presented method is effective and reliable.
{"title":"Numerical solutions and simulations of the fractional COVID‐19 model via Pell–Lucas collocation algorithm","authors":"Gamze Yıldırım, Şuayip Yüzbaşı","doi":"10.1002/mma.10284","DOIUrl":"https://doi.org/10.1002/mma.10284","url":null,"abstract":"The aim of this study is to present the evolution of COVID‐19 pandemic in Turkey. For this, the SIR (Susceptible, Infected, Removed) model with the fractional order derivative is employed. By applying the collocation method via the Pell–Lucas polynomials (PLPs) to this model, the approximate solutions of model with fractional order derivative are obtained. Hence, the comments are made about the susceptible population, the infected population, and the recovered population. For the method, firstly, PLPs are expressed in matrix form for a selected number of . With the help of this matrix relationship, the matrix forms of each term in the SIR model with the fractional order derivative are constituted. For implementation and visualization, we utilize MATLAB. Moreover, the outcomes for the Runge–Kutta method (RKM) are obtained using MATLAB, and these results are compared with the results obtained with the Pell–Lucas collocation method (PLCM). From all simulations, it is concluded that the presented method is effective and reliable.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882199","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}
S. Sivasankar, K. Nadhaprasadh, M. Sathish Kumar, Shrideh Al‐Omari, R. Udhayakumar
In this study, we examine whether mild solutions to a fractional stochastic evolution system with a fractional Caputo derivative on an infinite interval exist and are attractive. We use semigroup theory, fractional calculus, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. There are several suggested sufficient requirements for the existence of mild solutions to the stated problem. Examples that highlight the key findings are provided.
{"title":"New study on Cauchy problems of fractional stochastic evolution systems on an infinite interval","authors":"S. Sivasankar, K. Nadhaprasadh, M. Sathish Kumar, Shrideh Al‐Omari, R. Udhayakumar","doi":"10.1002/mma.10365","DOIUrl":"https://doi.org/10.1002/mma.10365","url":null,"abstract":"In this study, we examine whether mild solutions to a fractional stochastic evolution system with a fractional Caputo derivative on an infinite interval exist and are attractive. We use semigroup theory, fractional calculus, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. There are several suggested sufficient requirements for the existence of mild solutions to the stated problem. Examples that highlight the key findings are provided.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882102","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 article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.
{"title":"ψ$$ psi $$‐Bernstein–Kantorovich operators","authors":"Hüseyin Aktuğlu, Mustafa Kara, Erdem Baytunç","doi":"10.1002/mma.10375","DOIUrl":"https://doi.org/10.1002/mma.10375","url":null,"abstract":"In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869844","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 article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.
{"title":"Global asymptotical stability and Hopf bifurcation for a three‐species Lotka‐Volterra food web model","authors":"Zhan‐Ping Ma, Jin‐Zuo Han","doi":"10.1002/mma.10376","DOIUrl":"https://doi.org/10.1002/mma.10376","url":null,"abstract":"In this article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869977","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, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.
{"title":"Global solution for semilinear σ$$ sigma $$‐evolution models with memory term","authors":"Ting Xie, Han Yang","doi":"10.1002/mma.10373","DOIUrl":"https://doi.org/10.1002/mma.10373","url":null,"abstract":"In this paper, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869975","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 energy conservation problem for the reduced‐gravity two‐and‐a‐half layer model remains a challenging open problem, with several recent contributions addressing various aspects of it. In this paper, we establish sufficient conditions on the regularity of weak solutions that guarantee energy conservation even in the presence of a vacuum. Our theorem corresponds to an improvement of some recent results on this problem and contains some well‐known results as a particular case.
{"title":"Energy equality of weak solutions of the 2D reduced‐gravity two‐and‐a‐half layer system allowing vacuum","authors":"Xiang Ji, Fan Wu, Yanping Gao","doi":"10.1002/mma.10380","DOIUrl":"https://doi.org/10.1002/mma.10380","url":null,"abstract":"The energy conservation problem for the reduced‐gravity two‐and‐a‐half layer model remains a challenging open problem, with several recent contributions addressing various aspects of it. In this paper, we establish sufficient conditions on the regularity of weak solutions that guarantee energy conservation even in the presence of a vacuum. Our theorem corresponds to an improvement of some recent results on this problem and contains some well‐known results as a particular case.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869974","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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