Leaf Peeling method for the wave equation on metric tree graphs

Sturm's oscillation theorem states that the n th eigenfunction of a Sturm–Liouville operator on the interval has n −1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. 1 , 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki 60 , 468–470 ( doi:10.1007/BF02320380 ); Schapotschnikow 2006 Waves Random Complex Media 16 , 167–178 ( doi:10.1080/1745530600702535 )) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278 , 803–819 ( doi:10.1007/S00220-007-0391-3 ); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54 , 1346–1349 ( doi:10.1103/PhysRevLett.54.1346 ); Fiedler 1975 Czechoslovak Math. J. 25 , 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n , the n th eigenfunction of the graph has n −1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE 6 , 1213–1233 ( doi:10.2140/apde.2013.6.1213 ); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372 , 20120522 ( doi:10.1098/rsta.2012.0522 ); Colin de Verdière 2013 Anal. PDE 6 , 1235–1242 ( doi:10.2140/apde.2013.6.1235 )). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.

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An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions

Vibration ◽

10.3390/vibration3040028 ◽

2020 ◽

Vol 3 (4) ◽

pp. 448-463

Author(s):

Sergei Avdonin ◽

Julian Edward

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

Boundary Vertex ◽

Tree Graph ◽

Matching Conditions ◽

Metric Tree

In this paper, we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that the so-called delta-prime matching conditions are satisfied at the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Neumann-to-Dirichlet map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph together with the coefficients of the equations.

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The Three-Dimensional Wave Equation

Fundamentals of Geophysics ◽

10.1017/9781108685917.014 ◽

2020 ◽

pp. 394-396

Keyword(s):

Wave Equation ◽

Three Dimensional ◽

Dimensional Wave

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The wave equation in spherical coordinates

Theory of Reflectance and Emittance Spectroscopy ◽

10.1017/cbo9780511524998.017 ◽

1993 ◽

pp. 420-427

Keyword(s):

Wave Equation ◽

Spherical Coordinates

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Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

Al-Jabar Jurnal Pendidikan Matematika ◽

10.24042/ajpm.v11i1.5153 ◽

2020 ◽

Vol 11 (1) ◽

pp. 93-100

Author(s):

Vina Apriliani ◽

Ikhsan Maulidi ◽

Budi Azhari

Keyword(s):

Wave Equation ◽

Exact Solutions ◽

Internal Waves ◽

Water Waves ◽

Wave Equations ◽

Elliptic Functions ◽

Expansion Method ◽

Mkdv Equation ◽

Innovative Methods ◽

De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations.

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