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 article, the authors give sufficient conditions on the shape of the equations and on the initial conditions under which there exist actual holomorphic solutions which are Gevrey asymptotic to the given formal series solutions with respect to the time variable on sectors in the complex plane.
Abstract: We construct formal power series solutions of nonlinear integro-differential equations for given initial conditions which are holomorphic functions on a strip in the complex plane C. We give sufficient conditions on the shape of the equations and on the initial conditions under which there exist actual holomorphic solutions which are Gevrey asymptotic to the given formal series solutions with respect to the time variable on sectors in C. We get 1?Gevrey asymptotic expansions on sectors of opening less than ? but not 1?summability as it is the case for linear partial differential equations. Moreover, our approach yields global analytic solutions of these equations in both time and space variables.
31 citations
TL;DR: In this paper, an analytical solution in terms of a multipole expansion is derived by means of the method of matched asymptotic expansions, which is especially well suited to compare the results obtained by the various authors mentioned above.
31 citations
TL;DR: This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction-diffusion problem, which may have multiple solutions and proves existence of a solution to the discretization and shows that it is accurate of order N -2 In 2 N, in the discrete maximum norm.
Abstract: This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction-diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains O(N) points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order N -2 In 2 N, in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter e and N. Numerical results are presented that verify this rate of convergence.
31 citations
TL;DR: In this article, the authors derived a general expression for the speed of a prolate spheroidal electrocatalytic nanomotor in terms of interfacial potential and physical properties of the motor environment.
Abstract: Using the method of matched asymptotic expansions, we derive a general expression for the speed of a prolate spheroidal electrocatalytic nanomotor in terms of interfacial potential and physical properties of the motor environment in the limit of small Debye length and Peclet number. This greatly increases the range of geometries that can be handled without resorting to numerical simulations, since a wide range of shapes from spherical to needle-like, and in particular the common cylindrical shape, can be well-approximated by prolate spheroids. For piecewise-uniform distribution of surface cation flux with fixed average absolute value, the mobility of a prolate spheroidal motor with a symmetric cation source/sink configuration is a monotonically decreasing function of eccentricity. A prolate spheroidal motor with an asymmetric sink/source configuration moves faster than its symmetric counterpart and can exhibit a non-monotonic dependence of motor speed on eccentricity for a highly asymmetric design.
31 citations
TL;DR: In this paper, matched asymptotic expansions are used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number.
Abstract: The method of matched asymptotic expansions is used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number. This technique is restricted to problems for which the dimensionless frequency e, defined as ωb/a0, is small, ω being the circular frequency, b the typical body dimension, and a0 the speed of sound. By combining Lorentz and Galilean transformations, the problem is transformed to a space in which the approximation appropriate to the inner region is found to be incompressible flow and that appropriate to the outer, classical acoustics. This approximation for the inner region is the unsteady counterpart of the Prandtl-Glauert transformation, but is not identical to use of that transformation in a straightforward quasi-steady manner. For wings in oscillatory motion, it is the same approximation as was given by Miles (1950).To illustrate the technique, two examples are treated, one involving a pulsating cylinder in a stream, the other the impinging of plane sound waves upon an elliptical wing in a stream.
31 citations