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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: A numerical method to solve Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered.
Abstract: Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.

32 citations

Journal ArticleDOI
TL;DR: In this article, general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large-scale systems possibly containing non-differentiable nonlinearities, which often appear in control systems.
Abstract: New general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large-scale systems possibly containing non-differentiable non-linearities, which often appear in control systems. For systems with multiple time scales caused by several small parameters, or for large-scale systems, an essential order reduction of aggregation matrices is achieved. The results are conceptually and numerically applied to the absolute stability analysis of singularly perturbed Lurie-Postnikov systems.

32 citations

Journal ArticleDOI
TL;DR: In this paper, the boundary value problem for the singularly perturbed second-order differential equation was studied and it was shown that the multiplicity of the root of the degenerate equation significantly affects the asymptotics of the solution, especially in the boundary layer.
Abstract: Using the boundary-value problem for the singularly perturbed second-order differential equation as an example, we show that the multiplicity of the root of the degenerate equation significantly affects the asymptotics of the solution, especially in the boundary layer.

32 citations

Journal ArticleDOI
TL;DR: In this article, the steady and pulsating modes of flame propagation through a premixed combustible mixture are studied for the case in which the flame is characterized by the sequential production and depletion of a significant intermediate species.
Abstract: Steady and pulsating modes of flame propagation through a premixed combustible mixture are studied for the case in which the flame is characterized by the sequential production and depletion of a significant intermediate species. We employ the method of matched asymptotic expansions to derive a model valid for large activation energies, and show that the pulsating solution is the result of a supercritical Hopf bifurcation from the steadily propagating solution (which becomes unstable). Through a nonlinear bifurcation analysis, we calculate the pulsation amplitude and other characteristics of the flame along the bifurcated branch. It is shown that the average thickness of the pulsating flame, by which we mean the average effective separation distance between production and depletion of the intermediate species, is greater than that predicted by a steady-state theory. In addition, we find that the mean propagation speed is less than that of the steadily propagating solution, but that the instantane...

32 citations

Journal ArticleDOI
TL;DR: In this paper, the mathematical modeling of the formation of a pointed drop in a four-roller mill was addressed, using matched asymptotic expansions (MAAE) method.
Abstract: The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

32 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202321
202244
202110
202023
201913
201835