Fokas’s Unified Transform Method for linear systems

Deconinck, Bernard; Guo, Qi; Shlizerman, Eli; Vasan, Vishal

doi:10.1090/qam/1484

Cited by 24 publications

(27 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As is typical for problems posed on a semi-infinite domain, we assume the field q(x, t) vanishes as x → ∞ uniformly for all t. Though this problem may be solved using classical techniques, we employ the Unified Transform Method for reasons that will become evident in subsequent sections. We begin by rewriting the diffusion equation in its divergence form using the 'local relation' (Deconinck et al 2014;Fokas 2008;Deconinck et al 2017)…”

Section: Diffusion Equation On Half-linementioning

confidence: 99%

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

2019

Self Cite

View full text Add to dashboard Cite

The Maxey-Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey-Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the Unified Transform Method.We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as stretchedexponential at short time while slowing it down to a power-law relaxation, ∼ t −3/2 , at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (∼ t −1/2 ), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of rain drops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a rain drop can be gross overestimated when history effects are not accounted. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone.For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects.Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover our results indicate that the Basset history can affect dynamics significantly.

show abstract

Section: Diffusion Equation On Half-linementioning

confidence: 99%

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…The same dispersion relation arises in the Boussinesq approximations to the free boundary problem for water waves, [23]. A regularized version 4) in which two of the x derivatives are replace by t derivatives, has the same order of approximation to the full physical model, and has also been proposed as the linearization of a model for DNA dynamics, [19]. Equation (4.4) has dispersion relation…”

Section: Bidirectional Dispersive Lamb Modelsmentioning

confidence: 98%

“…However, the space-dependent coefficient places the system outside the class of equations currently solvable by the UTM. A second possible way to approach such problems is to view Lamb's original formulation as an interface problem, as in [5,20], and combine this with recent work applying the UTM to systems of equations, [4]. However, the Lamb interface condition is more complicated than those considered to date, and extending current work on interface problems remains an interesting challenge.Remark: The paper includes still shots of a variety of solutions at selected times.…”

mentioning

confidence: 99%

Dispersive Lamb systems

Olver¹,

Sheils

2019

Journal of Geometric Mechanics

View full text Add to dashboard Cite

Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profiles emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is exactly the opposite of the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium induced by discontinuous initial conditions. solution to the periodic initial-boundary value problem produced by rough initial data, e.g., a step function, is "quantized" as a discontinuous but piecewise constant profile or, more generally, piecewise smooth or even piecewise fractal, at times which are rational multiples of L n /π n−1 , where L is the length of the interval, but exhibits a continuous but non-differentiable fractal profile at all other times. (An interesting question is whether such fractal profiles enjoy selfsimilarity properties similar to the Riemann and Weierstrass non-differentiable functions, as explored in [6]. However, the Fourier series used to construct the latter are quite different in character, and so the problem remains open and challenging.)Dispersive quantization relies on the slow, conditional convergence of the Fourier series solutions, and requires the dispersion relation to be asymptotically polynomial at large wave number.The key references include the 1990's discovery of Michael Berry and collaborators, [1], in the context of optics and quantum mechanics, and the subsequent analytical work of Oskolkov, [18]. In particular, this effect underlies the experimentally observed phenomenon of quantum revival, in which an electron that is initially concentrated at a single location of its orbital shell is, at rational times, re-concentrated at a finite number of orbital locations. The subsequent rediscovery by the first author, [2,16], showed that the phenomenon appears in a range of linear dispersive partial differential equations, while other models arising in fluid mechanics, plasma dynamics, elasticity, DNA dynamics, etc. exhibit a fascinating range of as yet poorly understood behaviors, whose qualitative features are tied to the large wave number asymptotics of the dispersion relation. These studies were then extended, through careful numerical simulations, to nonlinear dispersive equations, including both integrable models, such as the cubic nonlinear Schrödinger, Korteweg-deVries, and modified Korteweg-deVries equations, as well as non-integrable generalizations with higher degree nonlinearities. Some of these numerical observations were subsequently rigorously confirmed in papers of Chousionis, Erdogan, and Tzirakis, [3,9,10], and, further, in the very recent paper by Erdogan and Shakan, [8], which extends the analysis to non-polynomial dispersion relations, but much more work remains to be completed, including extensions to other types of boundary conditions and dispersiv...

show abstract

“…In 1997, Fokas announced a new unified approach based on the Riemann-Hilbert factorization problem to analysis the IBV problems for linear and nonlinear integrable PDEs [1,2,3], we call that Fokas unified transform method. This method provides an important generalization of the IST formalism from initial value to IBV problems, and over the last 20 years, this method has been used to analyse boundary value problems for several of the most important integrable equations possessing 2 × 2 Lax pairs, such as the KdV, the nonlinear Schrödinger(NLS), the sine-Gordon and the stationary axisymmetric Einstein equations and so on [4][5][6][7][8][9][10][11][12][13][14][15]. In 2012, Lenells first extended the Fokas unified transform method to the IBV problem for the 3 × 3 matrix Lax pair [16,17].…”

Section: Introductionmentioning

confidence: 99%

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

Xia

2019

Complex Variables and Elliptic Equations

View full text Add to dashboard Cite

In this paper, we consider the initial-boundary value problem of the Kundu-Eckhaus equation on the half-line by using of the Fokas unified transform method. Assuming that the solution u(x, t) exists, we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.Assume that the solution u(x; t) of the KE equation exists, and the initial-boundary values data are defined as follows( 1.2) We will show that u(x, t) can be expressed in terms of the unique solution of a matrix RHP formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions a(λ), b(λ) and A(λ), B(λ), which obtained from the initial data u 0 (x) = u(x, 0) and the boundary data g 0 (t) = u(0, t), g 1 (t) = u x (0, t), respectively. The problem has the jump across {Imλ 2 = 0}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation, which is an algebraic equation coupling a(λ), b(λ) and A(λ), B(λ). Where a different RHP was formulated and a different representation of the solution of the KE equation was given, which are more convenient for studying the long-time asymptotic behavior for the solutions of the KE equation with the decay initial and boundary values lie in the Schwartz class.Organization of this paper is as follows. In section 2, some summary results and the basic RHP of the KE equation are given. In section 3, the spectral functions a(λ), b(λ) and A(λ), B(λ) are investigated and the RHP is presented. The last section is devoted to conclusions and discussions.

show abstract

Fokas’s Unified Transform Method for linear systems

Cited by 24 publications

References 15 publications

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

Dispersive Lamb systems

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

Contact Info

Product

Resources

About