Ruifeng Xie, Jian Zhang, Jing Niu, Wen Li, Guangming Yao
In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.
{"title":"A reproducing Kernel method for solving singularly perturbed delay parabolic Partial differential equations","authors":"Ruifeng Xie, Jian Zhang, Jing Niu, Wen Li, Guangming Yao","doi":"10.3846/mma.2023.16852","DOIUrl":"https://doi.org/10.3846/mma.2023.16852","url":null,"abstract":"In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69999906","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 the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.
{"title":"CONVERGENCE AND STABILITY OF GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS OF ADVANCED TYPE","authors":"Yongtang Chen, Qi Wang","doi":"10.3846/mma.2023.16677","DOIUrl":"https://doi.org/10.3846/mma.2023.16677","url":null,"abstract":"This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135403589","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}
Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune it according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise a direct relation between the shape parameter of RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable RBFs lie in the -class which means that their curvature at the point xj is proportional to, where cj is the corresponding spatially variable shape parameter at xj. Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then, the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs with shape parameters depending on the function values and approximate curvature values of the function to be approximated. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods.
{"title":"Curvature based characterization of radial Basis Functions: Application to interpolation","authors":"Mohammad Heidari, Maryam Mohammadi, S. Marchi","doi":"10.3846/mma.2023.16897","DOIUrl":"https://doi.org/10.3846/mma.2023.16897","url":null,"abstract":"Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune it according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise a direct relation between the shape parameter of RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable RBFs lie in the -class which means that their curvature at the point xj is proportional to, where cj is the corresponding spatially variable shape parameter at xj. Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then, the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs with shape parameters depending on the function values and approximate curvature values of the function to be approximated. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69999975","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}
M. Sapagovas, Kristina Pupalaigė, R. Čiupaila, T. Meškauskas
The difference eigenvalue problem approximating the one-dimensional differential equation with the variable weight coefficients in an integral conditions is considered. The cases without negative eigenvalue in the spectrum of difference eigenvalue problem were analyzed. Analysis of the conditions of stability of difference schemes for parabolic equations was carried out according to the theoretical results and results of the numerical experiment.
{"title":"On the spectrum Structure for One difference eigenvalue Problem with nonlocal boundary conditions","authors":"M. Sapagovas, Kristina Pupalaigė, R. Čiupaila, T. Meškauskas","doi":"10.3846/mma.2023.17503","DOIUrl":"https://doi.org/10.3846/mma.2023.17503","url":null,"abstract":"The difference eigenvalue problem approximating the one-dimensional differential equation with the variable weight coefficients in an integral conditions is considered. The cases without negative eigenvalue in the spectrum of difference eigenvalue problem were analyzed. Analysis of the conditions of stability of difference schemes for parabolic equations was carried out according to the theoretical results and results of the numerical experiment.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70000141","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 propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.
{"title":"A numerical method for 3D Time-dependent Maxwell's equations in axisymmetric singular Domains with Arbitrary Data","authors":"Franck Assous, I. Raichik","doi":"10.3846/mma.2023.17553","DOIUrl":"https://doi.org/10.3846/mma.2023.17553","url":null,"abstract":"In this article, we propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70000202","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 investigates the existence result of entropy solution for some nonlinear degenerate parabolic problem in weighted Sobolov space with Dirichlet type boundary conditions and L1 data.
{"title":"Existence of Entropy solution for a nonlinear parabolic Problem in Weighted Sobolev Space via Optimization method","authors":"Lhoucine Hmidouch, A. Jamea, Mohamed Laghdir","doi":"10.3846/mma.2023.17010","DOIUrl":"https://doi.org/10.3846/mma.2023.17010","url":null,"abstract":"This paper investigates the existence result of entropy solution for some nonlinear degenerate parabolic problem in weighted Sobolov space with Dirichlet type boundary conditions and L1 data.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70000031","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}
V. Zavialov, Oleksii Lobok, T. Mysiura, Nataliia Popova, V. Chornyi, Taras Pohorilyi
An iterative algorithm for identifying unknown parameters of a mathematical model based on the Bayesian approach is proposed, which makes it possible to determine the most probable maximum informative estimates of these parameters. The example of the mathematical model of mass transfer dynamics shows the algorithm for finding the most probable and most informative estimate of the vector of unknown parameters, and also an analysis of the sequence of the corresponding steps is given. The results of computational experiments showed a significant dependence of the results of the calculations on the choice of the initial approximation point and slowing down the rate of convergence of the iterative process (and even its divergence) with an unsuccessful choice of the initial approximation. The validity of the obtained results is provided by analytical conclusions, the results of computational experiments, and statistical modeling. The results of computational experiments make it possible to assert that the proposed algorithm has a sufficiently high convergence for a given degree of accuracy and makes it possible to derive not only estimates of point values of mathematical model parameters based on a posteriori analysis, but also confidence intervals of these estimates. At the same time, it should be noted that the results of calculations depend significantly on the choice of the initial approximation point and the slowing of the convergence rate of the iterative process with an unsuccessful choice of the initial approximation. Analytical studies and results of calculations confirm the effectiveness of the proposed identification algorithm, which makes it possible, with the help of active, purposeful experiments, to build more accurate mathematical models. In accordance with the algorithm, a program was developed in the MatLab mathematics package and computational experiments were performed.
{"title":"Identification of unknown parameters of the Dynamic Model of mass Transfer","authors":"V. Zavialov, Oleksii Lobok, T. Mysiura, Nataliia Popova, V. Chornyi, Taras Pohorilyi","doi":"10.3846/mma.2023.16403","DOIUrl":"https://doi.org/10.3846/mma.2023.16403","url":null,"abstract":"An iterative algorithm for identifying unknown parameters of a mathematical model based on the Bayesian approach is proposed, which makes it possible to determine the most probable maximum informative estimates of these parameters. The example of the mathematical model of mass transfer dynamics shows the algorithm for finding the most probable and most informative estimate of the vector of unknown parameters, and also an analysis of the sequence of the corresponding steps is given. The results of computational experiments showed a significant dependence of the results of the calculations on the choice of the initial approximation point and slowing down the rate of convergence of the iterative process (and even its divergence) with an unsuccessful choice of the initial approximation. The validity of the obtained results is provided by analytical conclusions, the results of computational experiments, and statistical modeling. The results of computational experiments make it possible to assert that the proposed algorithm has a sufficiently high convergence for a given degree of accuracy and makes it possible to derive not only estimates of point values of mathematical model parameters based on a posteriori analysis, but also confidence intervals of these estimates. At the same time, it should be noted that the results of calculations depend significantly on the choice of the initial approximation point and the slowing of the convergence rate of the iterative process with an unsuccessful choice of the initial approximation. Analytical studies and results of calculations confirm the effectiveness of the proposed identification algorithm, which makes it possible, with the help of active, purposeful experiments, to build more accurate mathematical models. In accordance with the algorithm, a program was developed in the MatLab mathematics package and computational experiments were performed.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69999349","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, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.
{"title":"Eigenvalues of Sturm-Liouville Problems with eigenparameter dependent boundary and Interface conditions","authors":"Jiajia Zheng, Kun Li, Zhaowen Zheng","doi":"10.3846/mma.2023.17094","DOIUrl":"https://doi.org/10.3846/mma.2023.17094","url":null,"abstract":"In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70000087","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 the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.
{"title":"Barycentric rational interpolation method of the Helmholtz equation with Irregular Domain","authors":"Miaomiao Yang, Wentao Ma, Y. Ge","doi":"10.3846/mma.2023.16408","DOIUrl":"https://doi.org/10.3846/mma.2023.16408","url":null,"abstract":"In the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77899302","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 main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.
{"title":"Numerical simulation of fractional Power diffusion Biosensors","authors":"Ignas Dapšys, R. Čiegis","doi":"10.3846/mma.2023.17583","DOIUrl":"https://doi.org/10.3846/mma.2023.17583","url":null,"abstract":"The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90214344","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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