In this paper, we consider the continuity of solutions for non-autonomous stochastic reaction-diffusion equation driven by additive noise over a Wiener probability space. It is proved that the solutions are strongly continuous in $ H^1( {mathbb{R}}^N)cap L^p( {mathbb{R}}^N) $ with respect to the $ L^2 $-initial data and the samples in the double limit sense. As applications of the results on the continuity we obtain that the pullback random attractor for this equation is measurable, compact and attracting in the topology of the space $ H^1( {mathbb{R}}^N)cap L^p( {mathbb{R}}^N) $ under a weak assumption on the forcing term and the noise coefficient. More precisely, the continuity of solutions in the initial data implies the asymptotic compactness of system and therefore the attraction of attractor, and the continuity in the samples indicates its measurability. The main technique employed here is the difference estimate method, by which an appropriate multiplier is carefully selected.
{"title":"CONTINUITY OF SOLUTIONS IN <inline-formula><tex-math id=\"M1\">$ H^1( {mathbb{R}}^N)cap L^{p}( {mathbb{R}}^N) $</tex-math></inline-formula> FOR STOCHASTIC REACTION-DIFFUSION EQUATIONS AND ITS APPLICATIONS TO PULLBACK ATTRACTOR","authors":"Wenqiang Zhao, Zhi Li","doi":"10.11948/20230009","DOIUrl":"https://doi.org/10.11948/20230009","url":null,"abstract":"In this paper, we consider the continuity of solutions for non-autonomous stochastic reaction-diffusion equation driven by additive noise over a Wiener probability space. It is proved that the solutions are strongly continuous in $ H^1( {mathbb{R}}^N)cap L^p( {mathbb{R}}^N) $ with respect to the $ L^2 $-initial data and the samples in the double limit sense. As applications of the results on the continuity we obtain that the pullback random attractor for this equation is measurable, compact and attracting in the topology of the space $ H^1( {mathbb{R}}^N)cap L^p( {mathbb{R}}^N) $ under a weak assumption on the forcing term and the noise coefficient. More precisely, the continuity of solutions in the initial data implies the asymptotic compactness of system and therefore the attraction of attractor, and the continuity in the samples indicates its measurability. The main technique employed here is the difference estimate method, by which an appropriate multiplier is carefully selected.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, an insect-parasite-host model with Ricker’s type reproduction of triatomines and the standard incidence rate of the interaction between insects and hosts is formulated to study the transmission dynamics of Chagas disease. Two thresholds of the ecological basic reproduction number of triatomines and the epidemiological basic reproduction number of Chagas disease are derived, which determine the dynamics of this model. As a result, the existence of equilibria and the local/global stabilities of the equilibrium are accordingly obtained. Moreover, backward bifurcation, forward bifurcation and saddle-node bifurcation are also shown analytically and numerically. Biologically speaking, Chagas disease may undergo outbreak if the number of bites of per triatomine bug per unit time or the transmission probability from infected bugs to susceptible competent hosts per bite increase.
{"title":"TRANSMISSION DYNAMICS OF A CHAGAS DISEASE MODEL WITH STANDARD INCIDENCE INFECTION","authors":"Fanwei Meng, Lin Chen, Xianchao Zhang, Yancong Xu","doi":"10.11948/20230071","DOIUrl":"https://doi.org/10.11948/20230071","url":null,"abstract":"In this paper, an insect-parasite-host model with Ricker’s type reproduction of triatomines and the standard incidence rate of the interaction between insects and hosts is formulated to study the transmission dynamics of Chagas disease. Two thresholds of the ecological basic reproduction number of triatomines and the epidemiological basic reproduction number of Chagas disease are derived, which determine the dynamics of this model. As a result, the existence of equilibria and the local/global stabilities of the equilibrium are accordingly obtained. Moreover, backward bifurcation, forward bifurcation and saddle-node bifurcation are also shown analytically and numerically. Biologically speaking, Chagas disease may undergo outbreak if the number of bites of per triatomine bug per unit time or the transmission probability from infected bugs to susceptible competent hosts per bite increase.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we use the dynamical system method to investigate the wave solutions of the KdV-nKdV equation. We prove Wazwaz’s proposal that the KdV-nKdV equation has continuous periodic wave solutions and give their exact expressions by elliptic integral theory. We confirm that the KdV-nKdV equation has no classical solitary wave solution although it can be regarded as a fusion of the KdV equation with classical solitary wave and the nKdV equation. In addition, we obtain some novel traveling wave solutions of it including trapezoidal wave, inverted ‘N’ wave, and blow-up wave solutions.
{"title":"TRAVELING WAVES OF THE KDV-NKDV EQUATION","authors":"Xueqiong Yi, Yuqian Zhou, Qian Liu","doi":"10.11948/20230100","DOIUrl":"https://doi.org/10.11948/20230100","url":null,"abstract":"In this paper, we use the dynamical system method to investigate the wave solutions of the KdV-nKdV equation. We prove Wazwaz’s proposal that the KdV-nKdV equation has continuous periodic wave solutions and give their exact expressions by elliptic integral theory. We confirm that the KdV-nKdV equation has no classical solitary wave solution although it can be regarded as a fusion of the KdV equation with classical solitary wave and the nKdV equation. In addition, we obtain some novel traveling wave solutions of it including trapezoidal wave, inverted ‘N’ wave, and blow-up wave solutions.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"146 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, the new waveforms of two nonlinear evolution models are investigated by an analytical method, namely the sigmoid function method. The considered nonlinear complex models for this are the full nonlinearity form of the Fokas-Lenells equation and the paraxial wave equation, which play an important role in the field of fiber optics by balancing the nonlinearity with the dispersion terms. Under different numeric values of the free terms, the obtained results represent varieties of wave shapes, specifically anti-kink, dark, bright, singular soliton, anti-peakon, kink, two-lump propagation during breather periodic form, single lump, two lump solutions, periodic peakon, and periodic wave solutions, which have not been obtained in the previous studies. These dynamical characteristics are discussed in detail with the help of a pictorial presentation of the derived solutions. These resultants of both the considered nonlinear equations can be useful in both fiber optics as well as in other optics-related fields.
{"title":"COMPLEX NONLINEAR EVOLUTION EQUATIONS IN THE CONTEXT OF OPTICAL FIBERS: NEW WAVE-FORM ANALYSIS","authors":"A. Tripathy, S. Sahoo, S. Saha Ray, M. A. Abdou","doi":"10.11948/20230080","DOIUrl":"https://doi.org/10.11948/20230080","url":null,"abstract":"In this study, the new waveforms of two nonlinear evolution models are investigated by an analytical method, namely the sigmoid function method. The considered nonlinear complex models for this are the full nonlinearity form of the Fokas-Lenells equation and the paraxial wave equation, which play an important role in the field of fiber optics by balancing the nonlinearity with the dispersion terms. Under different numeric values of the free terms, the obtained results represent varieties of wave shapes, specifically anti-kink, dark, bright, singular soliton, anti-peakon, kink, two-lump propagation during breather periodic form, single lump, two lump solutions, periodic peakon, and periodic wave solutions, which have not been obtained in the previous studies. These dynamical characteristics are discussed in detail with the help of a pictorial presentation of the derived solutions. These resultants of both the considered nonlinear equations can be useful in both fiber optics as well as in other optics-related fields.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The current study is concerned with the existence and uniqueness of the solution to the Langevin equation of two separate fractional orders. With the infinite-point boundary condition, the boundary value problem is studied. The Banach contraction principle, Leray-nonlinear Schauder's alternative, and Leray-Schauder degree theorems are all implemented. A numerical example is presented to demonstrate the accuracy of our results. In addition, as an application of our results, the mean and variance of a fractional harmonic oscillator with the undamped angular frequency of the oscillator under the effect of a random force described as Gaussian colored noise are calculated.
{"title":"FRACTIONAL LANGEVIN EQUATIONS WITH INFINITE-POINT BOUNDARY CONDITION: APPLICATION TO FRACTIONAL HARMONIC OSCILLATOR","authors":"Lamya Almaghamsi, Ahmed Salem","doi":"10.11948/20230124","DOIUrl":"https://doi.org/10.11948/20230124","url":null,"abstract":"The current study is concerned with the existence and uniqueness of the solution to the Langevin equation of two separate fractional orders. With the infinite-point boundary condition, the boundary value problem is studied. The Banach contraction principle, Leray-nonlinear Schauder's alternative, and Leray-Schauder degree theorems are all implemented. A numerical example is presented to demonstrate the accuracy of our results. In addition, as an application of our results, the mean and variance of a fractional harmonic oscillator with the undamped angular frequency of the oscillator under the effect of a random force described as Gaussian colored noise are calculated.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The monitoring and controlling of systemic risk have increasingly become the focus of attention in the financial field. It is important and difficult to accurately forecast systemic financial risk. In this paper, we propose a spatio-temporal partial differential equation model to describe the systemic risk of China's Banking Industry based on network, clustering, and real date of 24 China's A-share listed banks. The model considers the combined influence of local risk and transboundary contagion effects, and the prediction relative accuracy is up to 95%. Simulation results confirm that strict joint control measures, the timeliness of central bank intervention, and differences in bank strategies are efficient for reducing systemic risk. To our knowledge, this is the first paper to apply a PDE model to forecast systemic financial risk.
{"title":"FORECASTING SYSTEMIC RISK OF CHINA'S BANKING INDUSTRY BY PARTIAL DIFFERENTIAL EQUATIONS MODEL AND COMPLEX NETWORK","authors":"Xiaofeng Yan, Haiyan Wang, Yulian An","doi":"10.11948/20230306","DOIUrl":"https://doi.org/10.11948/20230306","url":null,"abstract":"The monitoring and controlling of systemic risk have increasingly become the focus of attention in the financial field. It is important and difficult to accurately forecast systemic financial risk. In this paper, we propose a spatio-temporal partial differential equation model to describe the systemic risk of China's Banking Industry based on network, clustering, and real date of 24 China's A-share listed banks. The model considers the combined influence of local risk and transboundary contagion effects, and the prediction relative accuracy is up to 95%. Simulation results confirm that strict joint control measures, the timeliness of central bank intervention, and differences in bank strategies are efficient for reducing systemic risk. To our knowledge, this is the first paper to apply a PDE model to forecast systemic financial risk.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 a diffusive autocatalysis model with saturation under Neumann boundary conditions. Firstly, some stability and Turing instability results are obtained. Then by the maximum principle, H$ ddot{o} $lder inequality and Poincar$ acute{e} $ inequality, a priori estimates and some basic characterizations of non-constant positive solutions are given. Moreover, some non-existence results are presented for three different situations. In particular, we find that the model does not have any non-constant positive solution when the parameter which represents the saturation rate is large enough. In addition, we use the theories of Leray-Schauder degree and bifurcation to get the existence of non-constant positive solutions, respectively. The steady-state bifurcations at both simple and double eigenvalues are intensively studied and we establish some specific condition to determine the bifurcation direction. Finally, a few of numerical simulations are provided to illustrate theoretical results.
{"title":"THE NON-EXISTENCE AND EXISTENCE OF NON-CONSTANT POSITIVE SOLUTIONS FOR A DIFFUSIVE AUTOCATALYSIS MODEL WITH SATURATION","authors":"Gaihui Guo, Feiyan Guo, Bingfang Li, Lixin Yang","doi":"10.11948/20230002","DOIUrl":"https://doi.org/10.11948/20230002","url":null,"abstract":"This paper deals with a diffusive autocatalysis model with saturation under Neumann boundary conditions. Firstly, some stability and Turing instability results are obtained. Then by the maximum principle, H$ ddot{o} $lder inequality and Poincar$ acute{e} $ inequality, a priori estimates and some basic characterizations of non-constant positive solutions are given. Moreover, some non-existence results are presented for three different situations. In particular, we find that the model does not have any non-constant positive solution when the parameter which represents the saturation rate is large enough. In addition, we use the theories of Leray-Schauder degree and bifurcation to get the existence of non-constant positive solutions, respectively. The steady-state bifurcations at both simple and double eigenvalues are intensively studied and we establish some specific condition to determine the bifurcation direction. Finally, a few of numerical simulations are provided to illustrate theoretical results.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we studied the existence and unique solution of the Volterra-Fredholm integral equation of the second kind (V-FIESK). The general singular kernel is considered to be in position with the Fredholm integral term. Singular kernel will tend to a logarithmic function under exceptional conditions and new discussions. The Volterra-Fredholm integral equation with the logarithmic form will be solved using Legendre polynomials, where the kernel of Volterra integral term is a positive continuous function in time. A system of infinite linear algebraic equations is obtained by solving the problem in series, where the convergence of this system is discussed. Finally, The error is calculated using Maple software after the numerical results have been acquired.
{"title":"ANALYTICAL AND NUMERICAL DISCUSSION FOR THE PHASE-LAG VOLTERRA-FREDHOLM INTEGRAL EQUATION WITH SINGULAR KERNEL","authors":"Mohammed Abdel-Aty, Mohammed Abdou","doi":"10.11948/20220547","DOIUrl":"https://doi.org/10.11948/20220547","url":null,"abstract":"In this paper, we studied the existence and unique solution of the Volterra-Fredholm integral equation of the second kind (V-FIESK). The general singular kernel is considered to be in position with the Fredholm integral term. Singular kernel will tend to a logarithmic function under exceptional conditions and new discussions. The Volterra-Fredholm integral equation with the logarithmic form will be solved using Legendre polynomials, where the kernel of Volterra integral term is a positive continuous function in time. A system of infinite linear algebraic equations is obtained by solving the problem in series, where the convergence of this system is discussed. Finally, The error is calculated using Maple software after the numerical results have been acquired.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Le Thi Mai Thanh, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long
In this paper, a class of fourth-order viscoelastic wave equations with damping terms is studied. First, the local existence and uniqueness of weak solutions for the proposed problem are proved by the linear approximation and the Faedo-Galerkin method. Next, a special case of the original problem is considered. Then, under some suitablely sufficient conditions on the relaxation functions and by using contrary arguments, we show that the corresponding problem in this case does not admit any global solutions. Ultimately, we prove the finite-time blow up of solutions in case of negative initial energy.
{"title":"FINITE-TIME BLOW UP OF SOLUTIONS FOR A FOURTH-ORDER VISCOELASTIC WAVE EQUATION WITH DAMPING TERMS","authors":"Le Thi Mai Thanh, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long","doi":"10.11948/20230162","DOIUrl":"https://doi.org/10.11948/20230162","url":null,"abstract":"In this paper, a class of fourth-order viscoelastic wave equations with damping terms is studied. First, the local existence and uniqueness of weak solutions for the proposed problem are proved by the linear approximation and the Faedo-Galerkin method. Next, a special case of the original problem is considered. Then, under some suitablely sufficient conditions on the relaxation functions and by using contrary arguments, we show that the corresponding problem in this case does not admit any global solutions. Ultimately, we prove the finite-time blow up of solutions in case of negative initial energy.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, the (2+1)-dimensional variable coefficients dispersive long wave equations (vcDLWs) are studied by the Lie symmetry analysis method. The infinitesimal generators and geometric vector fields are given. Optimal system of the (2+1)-dimensional vcDLWs are analyzed by Olver's method. Based on the optimal system, the (2+1)-dimensional vcDLW equations are reduced to (1+1)-dimensional equations. A number of new exact solutions of vcDLW equations are derived. Some kink solutions and 2-soliton solutions are obtained by using $left( {1/G'} right)$-expansion method and $left( {G'/G} right)$-expansion method. Many different types of exact solutions can be obtained by changing the coefficient functions. By exploring the evolution of the solutions with function of the coefficients and time $t$, the dynamic behaviors of the solutions are analysed. At last, the conservation laws of the (2+1)-dimensional vcDLWs are derived based on the nonlinear self-adjointness.
{"title":"THE LIE SYMMETRY ANALYSIS, OPTIMAL SYSTEM, EXACT SOLUTIONS AND CONSERVATION LAWS OF THE (2+1)-DIMENSIONAL VARIABLE COEFFICIENTS DISPERSIVE LONG WAVE EQUATIONS","authors":"Meng Jin, Jiajia Yang, Jinzhou Liu, Xiangpeng Xin","doi":"10.11948/20230147","DOIUrl":"https://doi.org/10.11948/20230147","url":null,"abstract":"In this article, the (2+1)-dimensional variable coefficients dispersive long wave equations (vcDLWs) are studied by the Lie symmetry analysis method. The infinitesimal generators and geometric vector fields are given. Optimal system of the (2+1)-dimensional vcDLWs are analyzed by Olver's method. Based on the optimal system, the (2+1)-dimensional vcDLW equations are reduced to (1+1)-dimensional equations. A number of new exact solutions of vcDLW equations are derived. Some kink solutions and 2-soliton solutions are obtained by using $left( {1/G'} right)$-expansion method and $left( {G'/G} right)$-expansion method. Many different types of exact solutions can be obtained by changing the coefficient functions. By exploring the evolution of the solutions with function of the coefficients and time $t$, the dynamic behaviors of the solutions are analysed. At last, the conservation laws of the (2+1)-dimensional vcDLWs are derived based on the nonlinear self-adjointness.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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