Remarks on the numerical treatment of Prandtl's boundary-layer equations

Summary The Crocco Transformation for the boundary-layer equations very successfully used by Oleinik and Nickel for theoretical purposes (existence of 2-D, uniqueness of 2-D and 3-D solutions) will be used here for the construction of numerical procedures. An approximate discretization of the transformed equations leads to a nicely structured system of nonlinear equations that can be devided into small parts to be solved one after the other (in the 3-D case even partially by parallel processing). The Jacobians of these parts are oftenM-matrices such that SOR iterations work. The inverse transformation can numerically be realized very simply.

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Zusammenfassung Die für theoretische Zwecke (Existenz von Lösungen im 2-D-Fall, Eindeutigkeit im 2-D- und 3-D-Fall) von Oleinik und Nickel so erfolgreich eingesetzte Crocco-Transformation der Grenzschichtgleichungen wird in der vorliegenden Arbeit zur Konstruktion numerischer Verfahren benutzt. Bei geeigneter Diskretisierung erhält man angenehm strukturierte Systeme nichtlinearer Gleichungen, die in sukzessiv aufzurufende und unabhängig zu lösende Teil-Systeme zerfallen (im 3-D-Fall können die Systeme teilweise parallel bearbeitet werden). Die Jacobi-Matrizen der Teil-Systeme sind oftmalsM-Matrizen, so daß SOR-Iterationsverfahren konvergieren. Die Rücktransformation erfordert numerisch nur geringen Aufwand.

     

Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations

The similarity equations for mixed-convection boundary-layer flow past a wedge having one of its surfaces parallel to the horizontal are derived for the latter surface. Both cases of prescribed wall temperature and heat flux are considered. It is shown that non-unique solutions exist for aiding ( > 0) as well as opposing flows ( < 0); being the buoyancy parameter. In some situations there are four simultaneous solutions. Dual solutions for two previously studied mixed-convection boundary-layer flows are shown to exist for > 0 in addition to those already reported in the literature for < 0. Namely, these correspond to vertical flat plate and vertical cylinder problems.     

On certain alternating boundary-layer solutions to singularly perturbed parabolic equations

The solution to a special singularly perturbed parabolic problem with a right-hand independent of the spatial variable is studied numerically and analytically.     

Singular and non-unique solutions of the boundary-layer equations for the flow due to free convection near a continuously moving vertical plate

In this paper it is shown, using a numerical technique, that the boundary-layer equations for the flow near a continuously moving vertical plate with free convection are not unique. Further it is observed that when the buoyancy forces dominate the viscous forces there are no solutions. However, when the buoyancy forces oppose the motion of the plate, it is found that the numerical solution terminates in a singular manner which is of a different nature to any previously published results which have been obtained by solving non-linear ordinary differential equations where singular and dual solutions exist.

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Zusammenfassung In diesem Artikel wird mittels eines numerischen Verfahrens gezeigt, daß die Grenzschichtgleichungen für die Strömung in der Nähe einer kontinuierlich bewegten vertikalen Platte mit freier Konvektion nicht eindeutig sind.Weiter wird beobachtet, daß es keine Lösungen gibt, wenn die Auftriebskräfte über die Zähigkeitskräfte dominieren.Wenn jedoch die Auftriebskräfte der Plattenbewegung entgegentreten, wird herausgefunden, daß die numerische Lösung zu einer singulären Lösung wird, welche sich vor allen bisher veröffentlichten Resultaten unterscheidet, die durch Lösung der gewöhnlichen nichtlinearen Differentialgleichungen gefunden worden sind, bei denen singuläre und zweifache Lösungen existieren.

     

Comparison of solutions of stochastic equations and applications

Pathwise comparison of solutions to a class of stochastic systems of differential equations is proved which extends the existing result of Geiβ and Manthey. When the diffusion coefficients are defferent, the Gal’?huk-Davis method is extended to establish the comparision results. We illustrate our results with several examples some of which arise in stochastic finance theory     

Comparison theory for Riccati equations

We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B ²+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.     

Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations

Under a nondegeneracy condition on the boundary, we prove a comparison principle for discontinuous viscosity sub- and supersolutions of the generalized Dirichlet boundary-value problem for a first-order Hamilton-Jacobi equation

Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations

In recent years, an area of research in computational mathematics has emerged that is associated with the numerical solution of fluid flow problems based on regularized fluid dynamics equations involving additional terms with velocity, pressure, and body force. The inclusion of these functions in the additional terms has been physically substantiated only for pressure and body force. In this paper, the continuity equation obtained geometrically by Euler is shown to involve second-order terms in time that contain Jacobians of the velocity field and are consistent with some of the additional terms in the regularized fluid dynamics equations. The same Jacobians are contained in the inhomogeneous right-hand side of the wave equation and generate waves of pressure, density, and sound. Physical interpretations of the additional terms used in the regularized fluid dynamics equations are given.     

Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations

Eigenfunctions of the $p$ -Laplace operator for $p>1$ are defined to be critical points of an associated variational problem or, equivalently, to be solutions of the corresponding Euler–Lagrange equation. In the highly degenerated limit case of the 1-Laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope. However, the associated Euler–Lagrange equation has many solutions that are not critical points and, thus, it cannot be used for an equivalent definition. The present paper provides a new necessary condition for eigenfunctions of the 1-Laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the Euler–Lagrange equation that are not eigenfunctions.     

Comparison theorems for stochastic evolution equations

This paper establishes an anticipating stochastic differential equation of parabolic type for the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components. Conversely, it is shown that any appropriately regular solution of this stochastic p.d.e. must be given by the conditional expectation. These results generalize the connection, known as the Feynman-Kac formula, between parabolic equations and expectations of functions of a diffusion. As an application, we derive an equation for the unnormalized smoothing law of a filtering problem with observation feedback.     

Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions

In this work, we study the application of the Method of Fundamental Solutions (MFS) for the calculation of eigenfrequencies and eigenmodes in two and three‐dimensional domains. We address some mathematical results about properties of the single layer operator related to the eigenfrequencies. Moreover, we propose algorithms for the distribution of the collocation and source points of the MFS in three‐dimensional domains which is an extension of the choices considered by Alves and Antunes (CMC 2(2005), 251–266) for the two‐dimensional case. Also the application of the Plane Waves Method is investigated. Several examples with Dirichlet and Neumann boundary conditions are considered to illustrate the performance of the proposed methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1525–1550, 2011     

Comparison and nonoscillation results for perturbed nonlinear differential equations

Summary Nonoscillation theorems for perturbed second order nonlinear differential equations are obtained. A nonlinear Picone type identity is introduced to obtain some Sturm-Picone type comparison theorems for nonlinear equations. Entrata in Redazione il 19 gennaio 1977. Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.     

Comparison theorems and existence theorems for ordinary differential equations

We are concerned with uniqueness and existence theorems for two point boundary value problems for the nonlinear differential equation Ly = f(x, y), where L is the classical nth order linear differential operator. In proving our results interesting comparison theorems are proven for linear differential equations.