In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.
{"title":"A Composite collocation method based on the fractional Chelyshkov wavelets for Distributed-order fractional Mobile-immobile advection-dispersion equation","authors":"H. Marasi, M. Derakhshan","doi":"10.3846/mma.2022.15311","DOIUrl":"https://doi.org/10.3846/mma.2022.15311","url":null,"abstract":"In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73000305","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 two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details.
{"title":"Nonlinear Elliptic equation with nonlocal integral boundary condition depending on two parameters","authors":"Kristina Pupalaigė, M. Sapagovas, R. Čiupaila","doi":"10.3846/mma.2022.16209","DOIUrl":"https://doi.org/10.3846/mma.2022.16209","url":null,"abstract":"In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75348078","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 proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.
{"title":"Virtual element Approximations for two species Model of the advection-diffusion-reaction in Poroelastic Media","authors":"Nitesh Verma, Sarvesh Kumar","doi":"10.3846/mma.2022.15542","DOIUrl":"https://doi.org/10.3846/mma.2022.15542","url":null,"abstract":"This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81747816","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 present paper, a new family of multi-layers (deep) neural network (NN) operators is introduced. Density results have been established in the space of continuous functions on [−1,1], with respect to the uniform norm. First, the case of the operators with two-layers is considered in detail, then the definition and the corresponding density results have been extended to the general case of multi-layers operators. All the above definitions allow us to prove approximation results by a constructive approach, in the sense that, for any given f all the weights, the thresholds, and the coefficients of the deep NN operators can be explicitly determined. Finally, examples of activation functions have been provided, together with graphical examples. The main motivation of this work resides in the aim to provide the corresponding multi-layers version of the well-known (shallow) NN operators, according to what is done in the applications with the construction of deep neural models.
{"title":"Density Results by Deep Neural Network operators with Integer weights","authors":"D. Costarelli","doi":"10.3846/mma.2022.15974","DOIUrl":"https://doi.org/10.3846/mma.2022.15974","url":null,"abstract":"In the present paper, a new family of multi-layers (deep) neural network (NN) operators is introduced. Density results have been established in the space of continuous functions on [−1,1], with respect to the uniform norm. First, the case of the operators with two-layers is considered in detail, then the definition and the corresponding density results have been extended to the general case of multi-layers operators. All the above definitions allow us to prove approximation results by a constructive approach, in the sense that, for any given f all the weights, the thresholds, and the coefficients of the deep NN operators can be explicitly determined. Finally, examples of activation functions have been provided, together with graphical examples. The main motivation of this work resides in the aim to provide the corresponding multi-layers version of the well-known (shallow) NN operators, according to what is done in the applications with the construction of deep neural models.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88794658","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}
A new variational approach for a boundary value problem in mathematical physics is proposed. By considering two-field Lagrange multipliers, we deliver a variational formulation consisting of a mixed variational problem which is equivalent with a saddle point problem. Thus, the unique solvability of the weak formulation we propose is governed by the saddle point theory. Alternative variational formulations and some of their connections are also discussed.
{"title":"Weak solutions via two-field Lagrange Multipliers for boundary Value Problems in Mathematical Physics","authors":"Mariana Chivu Cojocaru, A. Matei","doi":"10.3846/mma.2022.15827","DOIUrl":"https://doi.org/10.3846/mma.2022.15827","url":null,"abstract":"A new variational approach for a boundary value problem in mathematical physics is proposed. By considering two-field Lagrange multipliers, we deliver a variational formulation consisting of a mixed variational problem which is equivalent with a saddle point problem. Thus, the unique solvability of the weak formulation we propose is governed by the saddle point theory. Alternative variational formulations and some of their connections are also discussed.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74899070","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, we prove the existence and uniqueness results of weak solutions to a class of nonlinear fractional parabolic p(.)-Laplacian problem with variable order. The main tool used here is the Rothe’s method combined with the theory of variable-order fractional Sobolev spaces with variable exponent.
{"title":"Weak solution for nonlinear fractional P(.)-Laplacian Problem with variable order via Rothe's Time-discretization method","authors":"Abdelali Sabri","doi":"10.3846/mma.2022.15740","DOIUrl":"https://doi.org/10.3846/mma.2022.15740","url":null,"abstract":"In this paper, we prove the existence and uniqueness results of weak solutions to a class of nonlinear fractional parabolic p(.)-Laplacian problem with variable order. The main tool used here is the Rothe’s method combined with the theory of variable-order fractional Sobolev spaces with variable exponent.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83477273","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. M. Shoheib, S. Shahrooi, M. Shishehsaz, M. Hamzehei
A new procedure in the field of Bézier base extended isogeometric method (XIGA) has been introduced to analyze the effect of welding residual stress and thermal load on crack propagation rate and fatigue life. This new procedure is based on the constitutive thermoelastic plastic equation. The main parts of this procedure are using the B´ezier base XIGA method to calculate the redistribution of welding residual stress due to crack growth and to compute the value of stress intensity factor (SIF) in the welding residual stress field. For this purpose, the grid points of Bézier elements (with C0-continuity) around the crack line and the crack tip are identified by the level set representation. Then, discontinuous enrichment functions are added to the isogeometric analysis approximation. Thus, this method does not require the re-meshing process. The results show that there is a good agreement between the results of proposed numerical method and the Hole-Drilling Strain-Gage method. The interaction integral method has been used to extract SIF. The effects of welding residual stress and thermal load on the SIF are considered using the superposition method. Also, the Walker equation has been modified to calculate the fatigue life caused by thermal loading and welding residual stress. The results display a good agreement between the proposed method and the finite element method. Due to the advantages of the Bézier based XIGA method, which eliminates parametric space and allows the precise addition of enrichment functions to the basis functions of cracked elements (crack line or crack tip), the obtained results are highly accurate that shows this method is effective for analyzing discontinuous problems.
{"title":"Bézier Base Extended isogeometric numerical method for thermo elastic-plastic Analysis of crack Propagation in cracked plate under welding residual stress and thermal Load","authors":"M. M. Shoheib, S. Shahrooi, M. Shishehsaz, M. Hamzehei","doi":"10.3846/mma.2022.15503","DOIUrl":"https://doi.org/10.3846/mma.2022.15503","url":null,"abstract":"A new procedure in the field of Bézier base extended isogeometric method (XIGA) has been introduced to analyze the effect of welding residual stress and thermal load on crack propagation rate and fatigue life. This new procedure is based on the constitutive thermoelastic plastic equation. The main parts of this procedure are using the B´ezier base XIGA method to calculate the redistribution of welding residual stress due to crack growth and to compute the value of stress intensity factor (SIF) in the welding residual stress field. For this purpose, the grid points of Bézier elements (with C0-continuity) around the crack line and the crack tip are identified by the level set representation. Then, discontinuous enrichment functions are added to the isogeometric analysis approximation. Thus, this method does not require the re-meshing process. The results show that there is a good agreement between the results of proposed numerical method and the Hole-Drilling Strain-Gage method. The interaction integral method has been used to extract SIF. The effects of welding residual stress and thermal load on the SIF are considered using the superposition method. Also, the Walker equation has been modified to calculate the fatigue life caused by thermal loading and welding residual stress. The results display a good agreement between the proposed method and the finite element method. Due to the advantages of the Bézier based XIGA method, which eliminates parametric space and allows the precise addition of enrichment functions to the basis functions of cracked elements (crack line or crack tip), the obtained results are highly accurate that shows this method is effective for analyzing discontinuous problems.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72992799","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 a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then, we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function b[k](·) that depends analytically on a real parameter k and vanishes for k = 0 and we multiply the Dirichlet-like part of the Robin condition by b[k](·). We show that the displacement solution can be written in terms of power series of k that converge for k in a whole neighborhood of 0. For our analysis we use the Functional Analytic Approach.
{"title":"A Degenerating Robin-Type Traction Problem in a periodic Domain","authors":"M. D. Riva, Gennady Mishuris, P. Musolino","doi":"10.3846/mma.2023.17681","DOIUrl":"https://doi.org/10.3846/mma.2023.17681","url":null,"abstract":"We consider a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then, we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function b[k](·) that depends analytically on a real parameter k and vanishes for k = 0 and we multiply the Dirichlet-like part of the Robin condition by b[k](·). We show that the displacement solution can be written in terms of power series of k that converge for k in a whole neighborhood of 0. For our analysis we use the Functional Analytic Approach.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70000226","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 current article presents a degenerating diffusion-precipitation model including vanishing porosity and focuses primarily on uniqueness results. This is accomplished by assuming sufficient conditions under which the uniqueness of weak solutions can be established. Moreover, a proof of existence based on a compactness argument yields rather regular solutions, satisfying these unique conditions. The results show that every strong solution is unique, though a slightly different condition is additionally required in three dimensions. The analysis presents particular challenges due to the nonlinear structure of the underlying problem and the necessity to work with appropriate weights and manage possible degeneration.
{"title":"Uniqueness of Degenerating solutions to a diffusion-precipitation Model for Clogging porous Media","authors":"R. Schulz","doi":"10.3846/mma.2022.15132","DOIUrl":"https://doi.org/10.3846/mma.2022.15132","url":null,"abstract":"The current article presents a degenerating diffusion-precipitation model including vanishing porosity and focuses primarily on uniqueness results. This is accomplished by assuming sufficient conditions under which the uniqueness of weak solutions can be established. Moreover, a proof of existence based on a compactness argument yields rather regular solutions, satisfying these unique conditions. The results show that every strong solution is unique, though a slightly different condition is additionally required in three dimensions. The analysis presents particular challenges due to the nonlinear structure of the underlying problem and the necessity to work with appropriate weights and manage possible degeneration.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85930115","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 a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.
{"title":"Stability Analysis and numerical simulations of IVGTT glucose-insulin Interaction Models with two Time delays","authors":"S. Saber, Ahmad Alalyani","doi":"10.3846/mma.2022.14007","DOIUrl":"https://doi.org/10.3846/mma.2022.14007","url":null,"abstract":"This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77936948","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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