Showing papers on "Method of matched asymptotic expansions published in 2021"

A Novel Method for Singularly Perturbed Delay Differential Equations of Reaction-Diffusion Type

Abstract: In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction–diffusion type. A fitted operator finite difference scheme based on Numerov’s method is constructed. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

Abstract: This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

Steady thermocapillary droplet migration under thermal radiation with a uniform flux

Abstract: Thermocapillary migration of a droplet under thermal radiation with a uniform flux is numerically investigated and theoretically analyzed. By using the front-tracking method, it is observed that thermocapillary droplet migration at small Reynolds numbers and moderate Marangoni numbers reaches a steady process. The steady migration velocity decreases as Marangoni number increases. The time-evolution behavior of temperature fields in the steady migration process is found to be a quadratic function, in which the linear rise of the steady state temperatures with the relative time is a main characteristics. The quadratic function behavior of the temperature fields is further used to derive the steady energy equations. From the steady momentum and energy equations, an analytical result at small Reynolds number and zero Marangoni number is determined by using the method of matched asymptotic expansions. The steady migration velocity decreases as Reynolds number increases, which is in qualitatively agreement with the numerical result.

Multi-spike Patterns in the Gierer–Meinhardt System with a Nonzero Activator Boundary Flux

Abstract: The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction–diffusion equations have been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems, we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer–Meinhardt reaction–diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions, we construct boundary layer spikes and derive a class of shifted nonlocal eigenvalue problems analogous to those studied in Maini et al. (Chaos 17(3):037106, 2007) for which we rigorously prove partial stability results. Moreover, by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular, we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.

Near-exact explicit asymptotic solution of the SIR model well above the epidemic threshold

Abstract: A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers R0. The proposed formula yields good qualitative agreement already when R0 ≥ 3 and rapidly becomes quantitatively accurate as larger values of R0 are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given.

Compressibility Effect on Markstein Number for a Flame Front in Long-Wavelength Approximation

Abstract: The effect of compressibility on the Markstein number for a planar front of a premixed flame is examined, at small Mach numbers, in the form of M2-expansions. The method of matched asymptotic expansions is used to analyze the solution in the preheat zone in a power series in two small parameters, the relative thickness of the preheat zone and the Mach number. We employ a specific form of perturbations, valid at long wavelengths, for the thermodynamic variables, which produces the correction term, to the Markstein number, of second order in the Mach number in drastically simple form. Our analysis accounts for the pressure variation as a source term in the heat-conduction equation and calls for the Navier–Stokes equation. The suppression effect of the front curvature on the Darrieus-Landau instability is enhanced by the viscous effect if Pr > 4/3, but is weakened if otherwise.

The triple-deck stage of marginal separation

Abstract: The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar–turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous–inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler–Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed.

Asymptotic Analysis on the Sharp Interface Limit of the Time-Fractional Cahn--Hilliard Equation

Abstract: In this paper, we aim to study the motions of interfaces and coarsening rates governed by the time-fractional Cahn--Hilliard equation (TFCHE). It is observed by many numerical experiments that the microstructure evolution described by the TFCHE displays quite different dynamical processes comparing with the classical Cahn--Hilliard equation, in particular, regarding motions of interfaces and coarsening rates. By using the method of matched asymptotic expansions, we first derive the sharp interface limit models. Then we can theoretically analyze the motions of interfaces with respect to different timescales. For instance, for the TFCHE with the constant diffusion mobility, the sharp interface limit model is a fractional Stefan problem at the time scale $t = O(1)$. However, on the time scale $t = O({\varepsilon}^{1/{\alpha}} )$ the sharp interface limit model is a fractional Mullins--Sekerka model. Similar asymptotic regime results are also obtained for the case with one-sided degenerated mobility. Moreover, scaling invariant property of the sharp interface models suggests that the TFCHE with constant mobility preserves an ${\alpha}/3$ coarsening rate and a crossover of the coarsening rates from ${\alpha}/3$ to ${\alpha}/4$ is obtained for the case with one-sided degenerated mobility, which are in good agreement with the numerical experiments.

Viscous Vortex Rings

Abstract: We present a viscous vortex ring model based on a solution to the time-dependent Stokes-flow equation. The solution has a quasi-isotropic Gaussian vorticity distribution, which is more realistic compared to the one assumed in steady inviscid vortex rings models discussed in the previous chapter. We derive closed formulae for the stream function, translational velocity of the vorticity centroid and integral characteristics (impulse, circulation, kinetic energy) of the viscous vortex ring. Well-known results for the translational velocity are recovered for the limiting regimes of short-time evolution (Saffman) and long-time evolution (Rott and Cantwell). The model is then generalised to include a time-evolving eddy viscosity scale, offering a unified framework to describe both laminar and turbulent vortex rings. A Reynolds-number correction of the model is derived using the method of matched asymptotic expansions. A comparison with experimental results and direct numerical simulations shows that the model describes well the integral quantities and provides a more accurate description of the vortex ring geometry than inviscid models.

Multi-Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain

Abstract: In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of symmetric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain $2\times 2$ matrix. Our rigorous results are complemented by formal calculations of $N$-spike solutions using the method of matched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds. By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.