On closed boundary value problems for equations of mixed elliptic‐hyperbolic type

For partial differential equations of mixed elliptic‐hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet‐conormal types. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a ? b ? c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. © 2006 Wiley Periodicals, Inc.     

Finite element analysis of incompressible laminar boundary layer flows

A numerical procedure was developed to solve the two-dimensional and axisymmetric incompressible laminar boundary layer equations using the semi-discrete Galerkin finite element method. Linear Lagrangian, quadratic Lagrangian, and cubic Hermite interpolating polynomials were used for the finite element discretization; the first-order, the second-order backward difference approximation, and the Crank-Nicolson method were used for the system of non-linear ordinary differential equations; the Picard iteration and the Newton-Raphson technique were used to solve the resulting non-linear algebraic system of equations. Conservation of mass is treated as a constraint condition in the procedure; hence, it is integrated numerically along the solution line while marching along the time-like co-ordinate. Among the numerical schemes tested, the Picard iteration technique used with the quadratic Lagrangian polynomials and the second-order backward difference approximation case turned out to be the most efficient to achieve the same accuracy. The advantages of the method developed lie in its coarse grid accuracy, global computational efficiency, and wide applicability to most situations that may arise in incompressible laminar boundary layer flows.     

Asymptotic analysis of differential-delay equations and nonuniqueness of periodic solutions

We study the differential-delay equation x′(t) = ?αf(x(t–1)), where α is a positive parameter and f is an odd function which decays like x^?r at infinity. In particular, we consider the case r ? 2, and prove the existence of periodic solutions with special symmetries which are different from previously known periodic solutions of minimal period 4. For r = 2 we prove sharp asymptotic estimates for the minimal periods of these solutions. Our results disprove a conjecture of R. D. Nussbaum.     

Qualitative properties of the boundary derivative of the capacity potential for special classes of annular domains

Let U(p) denote the capacity potential in an annular domain Ω (bounded by Jordan curves). We describe the qualitative behavior of ∥Δ∥ on the line segments of ?Ω which are arbitrary, radial, or ν-directional (for some vector ν ≠ 0) in the respective cases where Ω is a convex, starlike, or ≠-convex annular domain. We apply these results to some free-boundary extremal problems in which the capacity plus weighted area functional is minimized in restricted classes of annular domains Ω.     

Radial sums of ridge functions: A characterization

We extend and sharpen the characterization of radial sums of ridge functions of two variables given in an earlier paper. More precisely, a bound on the degree of certain polynomials is given which depends on the angles between the ridges; not only is this bound best possible but also a converse result holds. Furthermore, variants of this characterization which hold for higher dimensional analogues are also given. These results are then applied to certain parallel am models in computed tomography to obtain more precise and otherwise improved characterizations of algorithms which commute with rigid motions.