Topic
Method of matched asymptotic expansions
About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.
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TL;DR: In this paper, first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications.
Abstract: In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized The new method is applied to an example of a constant speed minimum-time interception problem Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods
16 citations
TL;DR: In this paper, the spreading of viscous fluid injected under an elastic sheet is studied, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces.
Abstract: We study the spreading of viscous fluid injected under an elastic sheet, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces. The injected fluid forms a large blister and spreads outwards analogously to a viscous gravity current or a capillary droplet. The relative strengths of the three driving forces are determined by how the horizontal length scales of the system compare with three key transition length scales. Bending is dominant on small length scales, tension is dominant on intermediate length scales and gravity is dominant on large length scales. We show how to use the method of matched asymptotic expansions to predict the spreading rate and thickness profile of the blister of fluid in the seven possible asymptotic regimes, for both two-dimensional and axisymmetric geometries. Consideration of different physical effects at the fluid front increases the number of regimes yet further.
16 citations
TL;DR: An analytical solution for the shape of a fluid-fluid interface near a nanoscale solid sphere is presented, which is a configuration motivated by common measurements with an atomic force microscope, and it is shown that one configuration with a large effective Bond number, and thus one requiring the new solution, is a nanothick liquid film spread over a solid substrate.
Abstract: We present an analytical solution for the shape of a fluid–fluid interface near a nanoscale solid sphere, which is a configuration motivated by common measurements with an atomic force microscope. The forces considered are surface tension, gravity, and the van der Waals attraction. The nonlinear governing equation has been solved previously using the method of matched asymptotic expansions, and this requires that the surface tension forces far exceed those of gravity, i.e., the Bond number is much less than one. We first present this method using a physically relevant scaling of the equations, then offer a new analytical solution valid for all Bond numbers. We show that one configuration with a large effective Bond number, and thus one requiring our new solution, is a nanothick liquid film spread over a solid substrate. The scaling implications of both analytical methods are considered, and both are compared with numerical solutions of the full equation.
16 citations
TL;DR: In this paper, an optimal control problem with an 1D singularly perturbed differential state equation and its adjoint form is considered, where the effects of two layers arising at different boundaries of the domain are treated.
Abstract: We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection- diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.
16 citations
Journal Article•
TL;DR: After developing the tools required for application of the fixed point theory in the investigation, some general results about the long-time behavior of solutions of n-th order nonlinear differential equations are presented.
Abstract: We discuss a number of issues important for the asymptotic integration of ordinary differential equations. After developing the tools required for application of the fixed point theory in the investigation, we present some general results about the long-time behavior of solutions of n-th order nonlinear differential equations with an emphasis on the existence of polynomial-like solutions, the asymptotic representation for the derivatives and the effect of perturbations upon the asymptotic behavior of solutions.
16 citations