Showing papers on "Method of matched asymptotic expansions published in 1974"
TL;DR: In this article, matched asymptotic expansions were used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder, where the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders.
Abstract: The method of matched asymptotic expansions is used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder. Since the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders. The predicted meniscus height is compared with numerical data to determine the accuracy of the two-term result; the third term is found but does not improve the estimate.
148 citations
TL;DR: In this paper, the stability of systems of ordinary differential equations of the form dx dt = f(t, x, y, ϵ), ϵ dy dt is studied, where ϵ is a real parameter near zero, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied.
112 citations
TL;DR: In this article, the development of the flow in a curved tube near the inlet was analyzed by the matched asymptotic expansions method and it was shown that the cross-over between shear maxima on the inside and the outside of the tube occurs at a downstream distance which is 1·9 times the radius of tube for entry condition (i) while in the case of entry condition(ii) it is 0·95 times the distance required in case (i).
Abstract: This paper deals with the development of the flow in a curved tube near the inlet. The solution is obtained by the method of matched asymptotic expansions. Two inlet conditions are considered: (i) the condition of constant dynamic pressure at the entrance, which may be of practical interest in applications to blood flow in the aorta; and (ii) a uniform entry condition. It is shown that the geometry and the nature of the entry condition appreciably influence the initial development of the flow. The effect of the secondary flow due to the curvature on the wall shear is discussed and it is shown that the cross-over between shear maxima on the inside and the outside of the tube occurs at a downstream distance which is 1·9 times the radius of the tube for entry condition (i) while in the case of entry condition (ii) it is 0·95 times the radius, which is half the distance required in case (i). It is found that the pressure distribution is not significantly influenced by the secondary flow during the initial development of the motion. The analysis, which is developed for steady motion, can be extended to pulsatile flows, which are of greater physiological interest.
99 citations
Journal Article•
85 citations
78 citations
TL;DR: In this article, an explicit method of constructing an asymptotic power series solution of the TPBVP as the small parameter tends to zero is presented, which allows a separation of slow and fast dynamics in the problem while reducing the differential order of the equations.
38 citations
01 Jun 1974
TL;DR: In this paper, matched asymptotic expansions were used to analyze the interaction of a normal shock wave with an unseparated turbulent boundary layer on a flat surface at transonic speeds.
Abstract: The method of matched asymptotic expansions is used to analyze the interaction of a normal shock wave with an unseparated turbulent boundary layer on a flat surface at transonic speeds. The theory leads to a three-layer description of the interaction in the double limit of Reynolds number approaching infinity and Mach number approaching unity. The interaction involves an outer, inviscid rotational layer, a constant shear-stress wall layer, and a blending region between them. The pressure distribution is obtained from a numerical solution of the outer-layer equations by a mixed-flow relaxation procedure. An analytic solution for the skin friction is determined from the inner-layer equations. The significance of the mathematical model is discussed with reference to existing experimental data.
33 citations
TL;DR: In this article, a singular perturbation method is employed to attack the boundary layer of the solution in which some variables have very short response times and the numerical method affords a means of essentially determining the thickness of this boundary layer.
Abstract: There is an equivalence between stiff and singularly perturbed systems of ordinary differential equations. This feature is exploited in this paper by numerically employing recent singular perturbation methods to attack troublesome boundary layer stage of the solution in which some variables have very short response times. The numerical method affords a means of essentially determining the thickness of this boundary layer. The algorithm is capable of high stability and accuracy for the commonly occurring stiff system, whether or not it is in singularly perturbed form. Application to a singularly perturbed reaction system and a highly stiff reactor system not in singularly perturbed form demonstrate the effectiveness and utility of this approach.
28 citations
01 Mar 1974
27 citations
01 Jun 1974
TL;DR: In this article, matched asymptotic expansions were used to analyze the interaction of a normal shock wave with an unseparated turbulent boundary layer on a flat surface at transonic speeds.
Abstract: The method of matched asymptotic expansions is used to analyze the interaction of a normal shock wave with an unseparated turbulent boundary layer on a flat surface at transonic speeds. The theory leads to a three-layer description of the interaction in the double limit of Reynolds number approaching infinity and Mach number approaching unity. The interaction involves an outer, inviscid rotational layer, a constant shear-stress wall layer, and a blending region between them. The pressure distribution is obtained from a numerical solution of the outer-layer equations by a mixed-flow relaxation procedure. An analytic solution for the skin friction is determined from the inner-layer equations. The significance of the mathematical model is discussed with reference to existing experimental data.
23 citations
TL;DR: In this article, an analogue of the analytic Hamiltonian formalism in a complex phase space is presented for constructing formal asymptotic solutions of the characteristics and the transport equation.
Abstract: In this paper we give a method for constructing formal asymptotic solutions. This method uses in some sense "approximate solutions" of the equation of the characteristics and the transport equation. The construction of approximate solutions is brought abount by means of an analogue of the analytic Hamiltonian formalism in a complex phase space. Bibliography: 19 items.
TL;DR: In this article, the steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory.
Abstract: The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.
TL;DR: In this article, the amplitude of the Van der Pol equation was determined up to a higher order accuracy in v than has been done so far, by applying the method of intermediate matching for connecting the four local asymptotic solutions.
Abstract: The purpose of this paper is to apply the method of intermediate matching for connecting the four local asymptotic solutions of the Van der Pol equation, given by Dorodnicyn [1]. It turns out that for the approximation of the periodic solution a fifth local solution is needed. The present approach results in a reduction of the computational work. The amplitude of the periodic solution is determined up to a higher order accuracy in v than has been done so far.
01 Nov 1974
TL;DR: In this paper, an asymptotic power series solution of the optimal control of a class of high-order time delay systems is developed with respect to the scalar parameter, and the method is applied to an eighth order nonlinear time delay model of a coupled-core nuclear reactor system by reducing it to a second order ordinary model.
Abstract: Optimal control of a class of high-order time delay systems is considered. The mathematical description of these systems is written in such a way that whenever a scalar parameter is perturbed, the dimensionality of the system is reduced (singular perturbation). Further, the time delayed variables also will disappear from the system equations. An asymptotic power series solution of the optimal control is then developed with respect to the scalar parameter. This asymptotic control is valid for the high-order time delay system, while it is calculated on the basis of only a reduced order ordinary system. At first a scalar example illustrates the method of constructing the asymptotic expansions. Then the method is applied to an eighth order nonlinear time delay model of a coupled-core nuclear reactor system by reducing it to a second order ordinary model. Computationally, the asymptotic approximation method is compared with the second variation method and is shown to be superior.
TL;DR: In this paper, the authors considered the non-Newtonian flow between parallel porous stationary disks due to uniform suction at the disks and found that the combined effect of cross viscosity and visco-elastic co-efficients is to increase the maximum velocity at the centre of the channel and to decrease the magnitude of the velocity gradient at the disk.
TL;DR: In this article, matched asymptotic expansions have been used to obtain a low-frequency solution for the diffraction of a plane compressional wave by a rigid spheroid embedded in an infinite homogeneous isotropic elastic medium.
Abstract: The method of matched asymptotic expansions has been used to obtain a low-frequency solution for the diffraction of a plane compressional wave by a rigid spheroid embedded in an infinite homogeneous isotropic elastic medium. In contrast to standard techniques such as separation of variables which are not effective in this case, the present method gives, in a systematic manner, both the “inner” and “outer” fields. Expansions have been carried out which are correct to $O(\varepsilon ^2 ),\varepsilon $ being a suitably defined nondimensional wave number. These results are then used to determine the motion of the inclusion as well as the stress distribution on the spheroid and the far-field displacement components. Numerical studies for these properties are presented in graphical form and the results compared with known results in various limiting cases.
TL;DR: A new computational method for calculating the asymptotic solution to the matrix Riccati equation is described, which is fast, efficient and gives all possible solutions to the Matrix quadratic form.
Abstract: This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation This method is fast, efficient and gives all possible solutions to the matrix quadratic form Matrix sign functions are used to find the asymptotic solutions
01 Oct 1974
TL;DR: In this article, the authors derive asymptotic expansions associated with two particular latent root distributions to derive moments and significance points of distributions, and powers of tests for multivariate analysis.
Abstract: : Some of the difficulties in multivariate analysis involve the calculation of moments and significance points of distributions, and of powers of tests. In many cases, due to the complexity of the distributions involved, exact determination of these quantities appears all but impossible, and it makes sense to look for approximate solutions. This thesis is concerned essentially with deriving asymptotic expansions associated with two particular latent root distributions.
TL;DR: Asymptotic expansions for Bessel functions starting from the differential-difference equations they satisfy are derived in this paper, which are the well known Green-Liouville expansions obtainable from either the pure differential or pure difference equations satisfied by Bessel function.
Abstract: Asymptotic expansions are derived for Bessel functions starting from the differential-difference equations they satisfy. These are just the well known Green-Liouville expansions obtainable from either the pure differential or pure difference equations satisfied by Bessel functions. The Bessel functions are considered because of their well known properties but the method described should be applicable to many more general situations.
TL;DR: In this article, the authors used the matched asymptotic expansions to establish a two-term formula for the speed of propagation of the front of an exothermic reaction in a condensed medium whose thermophysical characteristics depend on the concentration of the reacting matter and the temperature.
Abstract: In this paper we use the method of matched asymptotic expansions to establish a two-term formula for the speed of propagation of the front of an exothermic reaction in a condensed medium whose thermophysical characteristics depend on the concentration of the reacting matter and the temperature. As the parameter of the expansion we use the ratio of the activation temperature to the adiabatic combustion temperature. The results are applied to the case of the combustion of nonvolatile condensed systems. We compare the approximate formula obtained with the results of a numerical integration.
TL;DR: In this article, the authors focus on the first order matching process for singular functions and consider the techniques of matching that were proposed to yield a relationship between inner and outer expansion of a singular function.
Abstract: Publisher Summary This chapter focuses on the first order matching process for singular functions and considers the techniques of matching that were proposed to yield a relationship between inner and outer expansion of a singular function. Many problems, especially in fluid mechanics, have been treated using the method of matched asymptotic approximation. However, the method has no actual justification, and there is no rigorous mathematical proof for it. Using the ideas of Eckhaus, the chapter shows how Van Dyke's matching rule is the best. Some asymptotic definitions are recalled and the extension theorem is considered. Matching principle and uniform approximation of singular functions are studied by using some simple examples. The chapter discusses the foundations of the method of matched asymptotic expansions; and it was possible to state a rule that is easy to apply.
TL;DR: In this paper, a flow of a grey radiating gas induced by buoyant force in a semi-infinite space bounded by a flat plate, which is parallel to the direction of the gravity, is considered.
Abstract: This paper deals theoretically with a flow of a grey radiating gas induced by buoyant force in a semi-infinite space bounded by a flat plate, which is parallel to the direction of the gravity, An external beam radiation is imposed normally through the transparent plate. The gas is assumed to be viscous, non-heat conducting and to have constant physical properties. The radiation field assumed to be in local thermodynamic equilibrium. The case is considered that the temperature and the flow velocity are uniform in a region away from the plate. The solution is obtained by method of matched asymptotic expansions. A steady flow is obtained in which the buoyant force balances against the viscous force. The flow is induced not only in the neighbourhood of the plate, but also in a region away from the plate, though the temperature change is localized in the former.
TL;DR: In this article, the structure of a shock wave in a vibrationally relaxing gas undergoing reflexion from a plane wall is examined, and different asymptotic expansions are determined for the various regions through the method of matched ASP expansion.
Abstract: The structure of a shock wave in a vibrationally relaxing gas undergoing reflexion from a plane wall is examined. The shock wave is assumed to be weak, and departures from thermodynamic equilibrium are assumed small; both an adiabatic and an isothermal wall are considered. The flow field is divided into three regions: a far-field region, an interaction region, and, for the isothermal-wall case, a thermal boundary layer. Different asymptotic expansions are determined for the various regions through the method of matched asymptotic expansions. In the region far from the wall, a non-equilibrium Burgers equation governs the motion and the incident and the reflected shock wave structures. During reflexion, a non-equilibrium wave equation applies; its first-order terms are equivalent to an acoustic approximation. Heat conduction to the wall is modelled by an isothermal wall boundary condition which requires the introduction of a thermal boundary layer adjacent to the wall. This thermal boundary layer is thin and the adiabatic-wall result provides the outer solution for treating this layer. This thermal layer affects the structure of the reflected wave.
TL;DR: The wave packet formalism is applied to the Vlasov-Poisson equations to derive a K-dV equation and also to the problem of the wave modulation by taking an example of the Bussinesque equation as discussed by the authors.
Abstract: Some asymptotic methods except for the reductive perturbation method are presented. The wave packet formalism is applied to the Vlasov-Poisson equations to derive a K-dV equation and also to the problem of the wave modulation by taking an example of the Bussinesque equation. The derivative expansion method and the extended Krylov Bogoliubov-Mitropolsky method are also discussed.
TL;DR: In this article, two forms of a unified type of Krylov-Bogoliubov method are considered for the purpose of deriving approximate solutions of non-linear ordinary differential equations that describe the underdamped and overdamped motion of systems subjected to step function excitation.
01 Jan 1974
TL;DR: In this article, the authors present a construction of asymptotic expansions in descending powers of u for solutions of the differential equation, where the leading term in each expansion is one of the LG functions.
Abstract: This chapter discusses expansions in elementary functions. Zeros of f ( z ) are called turning points or transition points of a differential equation, where z is an independent variable ranging over a real interval or a complex domain, neither of which need be bounded . The name transition point is extended to include singularities of f ( z ) or g ( z ). The forms of the approximate solutions depend on the number and nature of the transition points. The chapter presents a construction of asymptotic expansions in descending powers of u for solutions of the differential equation. The leading term in each expansion is one of the LG functions. For large values of the order v and fixed values of the argument z , the asymptotic behavior of any Bessel function is immediately available from its series expansion in ascending powers of z .
TL;DR: In this article, the Schrodinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied and eigenstate expansions for solving each equation are explicitly introduced and their properties discussed.