We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.
{"title":"Nonlinear elliptic problems with the method of finite volumes","authors":"S. Khattri","doi":"10.1155/DENM/2006/31797","DOIUrl":"https://doi.org/10.1155/DENM/2006/31797","url":null,"abstract":"We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-16"},"PeriodicalIF":0.0,"publicationDate":"2006-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/31797","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The steady flow of an incompressible electrically conducting fluid over a semi-infinite moving vertical cylinder in the presence of a uniform transverse magnetic field is analyzed. The partial differential equations governing the flow are reduced to an ordinary differential equation, using the self-similarity transformation. The analysis deals with the existence of an exact solution to the boundary value problem by a shooting method.
{"title":"On a similarity solution of MHD boundary layer flow over a moving vertical cylinder","authors":"M. Amkadni, A. Azzouzi","doi":"10.1155/DENM/2006/52765","DOIUrl":"https://doi.org/10.1155/DENM/2006/52765","url":null,"abstract":"The steady flow of an incompressible electrically conducting fluid over a semi-infinite moving vertical cylinder in the presence of a uniform transverse magnetic field is analyzed. The partial differential equations governing the flow are reduced to an ordinary differential equation, using the self-similarity transformation. The analysis deals with the existence of an exact solution to the boundary value problem by a shooting method.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-9"},"PeriodicalIF":0.0,"publicationDate":"2006-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/52765","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Complex vibration of flexible elastic shells subjected to transversal and sign-changeable local load in the frame of nonlinear classical theory is studied. A transition from partial to ordinary differential equations is carried out using the higher-order Bubnov-Galerkin approach. Numerical analysis is performed applying theoretical background of nonlinear dynamics and qualitative theory of differential equations. Mainly the so-called Sharkovskiy periodicity is studied.
{"title":"Dynamics of flexible shells and Sharkovskiy's periodicity","authors":"V. Krysko, J. Awrejcewicz, N. Saveleva, A. Krysko","doi":"10.1155/DENM/2006/59709","DOIUrl":"https://doi.org/10.1155/DENM/2006/59709","url":null,"abstract":"Complex vibration of flexible elastic shells subjected to transversal and sign-changeable local load in the frame of nonlinear classical theory is studied. A transition from partial to ordinary differential equations is carried out using the higher-order Bubnov-Galerkin approach. Numerical analysis is performed applying theoretical background of nonlinear dynamics and qualitative theory of differential equations. Mainly the so-called Sharkovskiy periodicity is studied.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-8"},"PeriodicalIF":0.0,"publicationDate":"2006-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/59709","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The object of considerations is a thin linear-elastic cylindrical shell having a periodic structure along one direction tangent to the shell midsurface. The aim of this paper is to propose a new averaged nonasymptotic model of such shells, which makes it possible to investigate free and forced vibrations, parametric vibrations, and dynamical stability of the shells under consideration. As a tool of modeling we will apply the tolerance averaging technique. The resulting equations have constant coefficients in the periodicity direction. Moreover, in contrast with models obtained by the known asymptotic homogenization technique, the proposed one makes it possible to describe the effect of the period length on the overall shell behavior, called a length-scale effect.
{"title":"On dynamics and stability of thin periodic cylindrical shells","authors":"B. Tomczyk","doi":"10.1155/DENM/2006/79853","DOIUrl":"https://doi.org/10.1155/DENM/2006/79853","url":null,"abstract":"The object of considerations is a thin linear-elastic cylindrical \u0000shell having a periodic structure along one direction tangent to \u0000the shell midsurface. The aim of this paper is to propose a new \u0000averaged nonasymptotic model of such shells, which makes it \u0000possible to investigate free and forced vibrations, parametric \u0000vibrations, and dynamical stability of the shells under \u0000consideration. As a tool of modeling we will apply the \u0000tolerance averaging technique. The resulting equations have \u0000constant coefficients in the periodicity direction. Moreover, in \u0000contrast with models obtained by the known asymptotic \u0000homogenization technique, the proposed one makes it possible to \u0000describe the effect of the period length on the overall shell \u0000behavior, called a length-scale effect.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-23"},"PeriodicalIF":0.0,"publicationDate":"2006-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/79853","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dynamic investigations of multimass discrete-continuous systems having variable moment of inertia are performed. The systems are torsionally deformed and consist of an arbitrary number of elastic elements connected by rigid bodies. The problem is nonlinear and it is linearized after appropriate transformations. It is shown that such problems can be investigated using the wave approach. Some analytical considerations and numerical calculations are done for a two-mass system with a special case of boundary conditions.
{"title":"Modeling of multimass systems torsionally deformed with variable inertia","authors":"A. Pielorz, M. Skóra","doi":"10.1155/DENM/2006/20758","DOIUrl":"https://doi.org/10.1155/DENM/2006/20758","url":null,"abstract":"Dynamic investigations of multimass discrete-continuous systems \u0000having variable moment of inertia are performed. The systems are \u0000torsionally deformed and consist of an arbitrary number of elastic \u0000elements connected by rigid bodies. The problem is nonlinear and \u0000it is linearized after appropriate transformations. It is shown \u0000that such problems can be investigated using the wave approach. \u0000Some analytical considerations and numerical calculations are done \u0000for a two-mass system with a special case of boundary conditions.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2006-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/20758","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the effect of variable viscosity on the transient Couette flow of dusty fluid with heat transfer between parallel plates. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates. The parallel plates are assumed to be porous and subjected to a uniform suction from above and injection from below. The upper plate is moving with a uniform velocity while the lower is kept stationary. The governing nonlinear partial differential equations are solved numerically and some important effects for the variable viscosity and the uniform magnetic field on the transient flow and heat transfer of both the fluid and dust particles are indicated.
{"title":"Influence of temperature-dependent viscosity on the MHD Couette flow of dusty fluid with heat transfer.","authors":"H. A. Attia","doi":"10.1155/DENM/2006/75290","DOIUrl":"https://doi.org/10.1155/DENM/2006/75290","url":null,"abstract":"This paper studies the effect of variable viscosity on the \u0000transient Couette flow of dusty fluid with heat transfer between \u0000parallel plates. The fluid is acted upon by a constant pressure \u0000gradient and an external uniform magnetic field is applied \u0000perpendicular to the plates. The parallel plates are assumed to be \u0000porous and subjected to a uniform suction from above and injection \u0000from below. The upper plate is moving with a uniform velocity \u0000while the lower is kept stationary. The governing nonlinear \u0000partial differential equations are solved numerically and some \u0000important effects for the variable viscosity and the uniform \u0000magnetic field on the transient flow and heat transfer of both the \u0000fluid and dust particles are indicated.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-14"},"PeriodicalIF":0.0,"publicationDate":"2006-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/75290","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.
{"title":"On the Navier-Stokes equations with temperature-dependent transport coefficients","authors":"E. Feireisl, J. Málek","doi":"10.1155/DENM/2006/90616","DOIUrl":"https://doi.org/10.1155/DENM/2006/90616","url":null,"abstract":"We establish long-time and large-data existence of a weak solution\u0000to the problem describing three-dimensional unsteady flows of an\u0000incompressible fluid, where the viscosity and heat-conductivity\u0000coefficients vary with the temperature. The approach reposes on\u0000considering the equation for the total energy rather than the\u0000equation for the temperature. We consider the spatially periodic\u0000problem.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"45 1","pages":"1-14"},"PeriodicalIF":0.0,"publicationDate":"2006-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/90616","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Solutions for a class of nonlinear second-order differential equations arising in steady Poiseuille flow of an Oldroyd six-constant model are obtained using the quasilinearization technique. Existence, uniqueness, and analyticity results are established using Schauder theory. Numerical results are presented graphically and salient features of the solutions are discussed.
{"title":"Existence, uniqueness, and quasilinearization results for nonlinear differential equations arising in viscoelastic fluid flow","authors":"F. Akyildiz, K. Vajravelu","doi":"10.1155/DENM/2006/71717","DOIUrl":"https://doi.org/10.1155/DENM/2006/71717","url":null,"abstract":"Solutions for a class of nonlinear second-order differential \u0000equations arising in steady Poiseuille flow of an Oldroyd \u0000six-constant model are obtained using the quasilinearization \u0000technique. Existence, uniqueness, and analyticity results are \u0000established using Schauder theory. Numerical results \u0000are presented graphically and salient features of the solutions \u0000are discussed.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-9"},"PeriodicalIF":0.0,"publicationDate":"2006-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/71717","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the modelling of complex sociopsychological games and reciprocal feelings involving interacting individuals. The modelling is based on suitable developments of the methods of mathematical kinetic theory of active particles with special attention to modelling multiple interactions. A first approach to complexity analysis is proposed referring to both computational and modelling aspects.
{"title":"On the modelling of complex sociopsychological systems with some reasoning about Kate, Jules, and Jim","authors":"N. Bellomo, B. Carbonaro","doi":"10.1155/DENM/2006/86816","DOIUrl":"https://doi.org/10.1155/DENM/2006/86816","url":null,"abstract":"This paper deals with the modelling of complex sociopsychological games and reciprocal feelings involving interacting individuals. The modelling is based on suitable developments of the methods of mathematical kinetic theory of active particles with special attention to modelling multiple interactions. A first approach to complexity analysis is proposed referring to both computational and modelling aspects.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"232 1","pages":"1-26"},"PeriodicalIF":0.0,"publicationDate":"2006-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/86816","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In classical particle mechanics, it is well understood that while working with nonholonomic and nonideal constraints, one cannot expect that the constraint be workless. It is curious that in continuum mechanics, however, the implications of the result in classical mechanics have not been clearly understood. In this paper, we show that in dealing with the response of dissipative systems, one cannot require that constraints do no work or ignore the fact that the material response functions depend on the constraint reaction. An example of this is the viscosity of a fluid depending on the pressure.
{"title":"On internal constraints in continuum mechanics","authors":"K. Rajagopal, G. Saccomandi","doi":"10.1155/DENM/2006/18572","DOIUrl":"https://doi.org/10.1155/DENM/2006/18572","url":null,"abstract":"In classical particle mechanics, it is well understood that while\u0000working with nonholonomic and nonideal constraints, one cannot\u0000expect that the constraint be workless. It is curious that in\u0000continuum mechanics, however, the implications of the result in\u0000classical mechanics have not been clearly understood. In this paper,\u0000we show that in dealing with the response of dissipative systems,\u0000one cannot require that constraints do no work or ignore the fact\u0000that the material response functions depend on the constraint\u0000reaction. An example of this is the viscosity of a fluid depending\u0000on the pressure.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-12"},"PeriodicalIF":0.0,"publicationDate":"2006-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/18572","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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