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, critical trajectories for an autonomous nonlinear differential equation, perturbed by slowly varying coefficients, are analyzed in the case which Kevorkian has shown describes sustained roll resonance.
Abstract: Critical trajectories for an autonomous nonlinear differential equation, perturbed by slowly varying coefficients, are analyzed in the case which Kevorkian has shown describes sustained roll resonance. The method of matched asymptotic expansions is used to calculate initial conditions for which a previously inaccessible center is captured. The results are significantly simplified using the energy dissipated along the critical trajectories.
27 citations
TL;DR: In this paper, an alternative integral representation for the radiated pressure field is applied which is different from the generally used integral representation introduced by Lighthill and Curle, where there is a linear dependence of the integrand on the time derivative of the vorticity fluctuations in the hydrodynamic near field; instead of the ordinary Green function a "vector Green function" is used.
27 citations
TL;DR: In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of lin- ear and nonlinear second-order boundary value problems.
Abstract: In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of lin- ear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefficients of the sought-for spectral solutions. Linear differential equations are treated by applying the Petrov-Galerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specific illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efficiency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.
27 citations
TL;DR: In this article, the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are k + o(1) as t tends to infinity.
27 citations
TL;DR: In this article, an initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint, which is very easy to use and to implement.
Abstract: An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.
27 citations