Linear Partial Differential Equations

Debnath, Lokenath

doi:10.1007/978-0-8176-8265-1_1

Cited by 12 publications

(13 citation statements)

References 10 publications

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“…We observe that the problem described in (5.2), (5.3) and (5.4) is invariant under the one-parameter family of scaling transformations thus we know that there is a similarity solution of the form (e.g. Bluman & Kumei 2013; Debnath 2005) Applying the transformation produces the ODE with the conditions and Following an argument (see Appendix B for details) similar to that presented by Romero & Yost (1996) for capillary flow into a sharp horizontal groove, the inequality in (5.8 b ) can be used to show that the liquid column extends to a finite altitude (or in dimensionless terms) at which the cross-section reduces to a point, as well as to derive a boundary condition at this location, Analogous conditions were also adopted by Weislogel & Lichter (1998). Higuera et al.…”

Section: Scaling and A Self-similar Solutionmentioning

confidence: 99%

Capillary rise in sharp corners: not quite universal

Wu,

Duprat,

Stone

2024

J. Fluid Mech.

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We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws $h_i(x) = c_i x^n$ , $i = 1,2$ , where $c_2 > c_1$ for $n \geq 1$ . Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as $t^{1/3}$ , a result that is universal, i.e. applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for $n>1$ , there exist a scaling and a similarity transformation that are independent of $c_i$ and $n$ , which gives rise to the universal $t^{1/3}$ power law for capillary rise. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of $n$ , and it is shown to be bounded and monotonically decreasing as $n\to \infty$ . Accordingly, the profile of the interface radius as a function of altitude is also independent of $c_i$ and exhibits slight variations with $n$ . Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude.

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Section: Scaling and A Self-similar Solutionmentioning

confidence: 99%

Capillary rise in sharp corners: not quite universal

Wu,

Duprat,

Stone

2024

J. Fluid Mech.

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“…Soliton pada umumnya dideskripsikan melalui persamaan diferensial parsial nonlinear. Beberapa persamaan yang dapat mendeskripsikan fenomena soliton adalah persamaan Korteweg-de Vries (Theodorakopoulus, 2006), persamaan Camassa-Holm (Debnath, 2012), persamaan Sine-Gordon (Zhou, 2017), persamaan Degasperis-Procesi (Debnath, 2012), dan persamaan Schrödinger nonlinear (Agrawal, 2001). Penelitian ini akan difokuskan pada persamaan Schrödinger Nonlinear / Nonlinear Schrödinger (NLS).…”

Section: Pendahuluanunclassified

Analisis Solusi Spatial dan Temporal Persamaan NLS (Nonlinear SchrÃ¶dinger) Sebagai Solusi Soliton Menggunakan Metode DVR (Discrete Variable Representation

Maulana

Abdullah

2020

JFU

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Analisis solusi persamaan Schrödinger Nonlinear (NLS) dengan menggunakan metode Representasi Variabel Diskrit (DVR) telah dilakukan. Penggunaan metode DVR ini dilakukan dengan mengkontruksikan fungsi basis melalui interpolasi Lagrange dan koefisien spektral yang berdasarkan polinomial Chebyshey pada titik Chebyshev-Gauss-Lobatto (CGL). Hasil simulasi dari metode DVR ini memiliki bentuk yang sama dengan 2 jenis solusi eksak persamaan NLS yaitu Cubic Nonlinear Schrödinger (CNLS) dan Generalized Nonlinear Schrödinger (GNLS), namun memiliki rentang penjalaran yang berbeda. Perbedaan rentang tersebut dikarenakan adanya kondisi awal yang ditetapkan pada metode DVR ini. Penggunaan metode DVR pada pengkondisian CNLS menghasilkan bentuk soliton onsite, sedangkan pada pengkondisian GNLS dihasilkan bentuk soliton onsite namun memiliki amplitudo yang berbeda pada setiap perubahan posisi dan waktu. Analysis of Nonlinear Schrödinger (NLS) equation by using Discrete Variable Representation (DVR) method has been carried out. The DVR method done with built basic function from Lagrange interpolation and spectral coefficient from Chebyshev polynomial in Chebyshev-Gauss-Lobatto (CGL) point. The result of DVR method simulation has same form with 2 type of exact solution of NLS equation are Cubic Nonlinear Schrödinger (CNLS) and Generalized Nonlinear Schrödinger (GNLS), however the result has different distance of propagation. That difference distance caused by initial condition which set on this method. The using of DVR method in CLNS condition produce soliton onsite form, but for GNLS condition produce soliton onsite form too but it has different amplitude for every change of position and time.

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“…It should be noted that the assumed form of (1a) can always be obtained from the general case, in which Λ(z) and A(z) are arbitrary matrices with elements in C 1 [0, 1]. This is possible by making use of the transformations given in (Debnath, 2012;Lamare & Bekiaris-Liberis, 2015;Vazquez & Krstic, 2014, Ch. 6.9).…”

Section: Problem Formulationmentioning

confidence: 99%

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

Deutscher

Gehring

Kern

2018

International Journal of Control

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This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients. Thereby, the coupling in the PDE is in-domain and at the uncontrolled boundary, whereby the ODE is coupled with the latter boundary. For the state feedback design a two-step backstepping approach is developed, that yields the conventional kernel equations and additional decoupling equations of simple form. The latter can be traced back to simple Volterra integral equations of the second kind, which are directly solvable with a successive approximation. In order to implement the state feedback controller, the design of observers for the ODE-PDE systems in question is considered, whereby anticollocated measurements are assumed. Simple conditions for the existence of the resulting observer-based compensator are formulated, that can be evaluated in terms of the plant transfer behaviour. The resulting systematic compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third order ODE modelling a dynamic boundary condition.

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Linear Partial Differential Equations

Cited by 12 publications

References 10 publications

Capillary rise in sharp corners: not quite universal

Capillary rise in sharp corners: not quite universal

Analisis Solusi Spatial dan Temporal Persamaan NLS (Nonlinear SchrÃ¶dinger) Sebagai Solusi Soliton Menggunakan Metode DVR (Discrete Variable Representation

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

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