EDGE-BASED LINEAR WAVE EQUATION ON DISCRETE STRUCTURE

• Main Author: Januar Ismail Burhan, Mohammad
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/83673
• Institution: Institut Teknologi Bandung

This dissertation is a study of the solution to the edge-based linear wave equations in a quantum graph. The wave equation on a graph is constructed by an edge-based Laplacian, which depends on the vertex conditions and the shape or structure of the graph. The main motivation for this research comes from the problem of sending wave packets in complex networks. Mathematically, this problem is translated into the study of wave propagation in discrete structures, namely quantum graphs. Specifically, the aim of this research is to study the wave equation in graphs through the spectrum of edge-based Laplacian in quantum graphs accompanied by various vertex conditions. We will also review aspects of the above application in solving problems of initial conditions and vertex conditions of edge-based linear wave equations in quantum graphs. The underlying theory for this research is the wave model in graphs introduced by Friedman and Tillich (2004), and research on wave packet markers in complex networks by Azis, Wilson, and Hancock (2019). There are two main topics discusses. The first is constructing alternative methods for determining the spectrum of the Laplacian on an equilateral quantum tree with end-vertex in Dirichlet conditions; follow by a discussion on the scattering matrix in quantum graphs for both Dirichlet and Robin conditions. The second topic discusses several linear wave equations in equilateral quantum graphs given Dirichlet and Neumann vertex conditions. The alternative method discussed in the first topic is a modification of the adjacency calculus method for spectral analysis of Laplacian on quantum graphs. Adjacency Calculus methods which was introduced by Von Below (1984), can only be done on quantum graphs with Neumann and Neumann Kirchhoff conditions on interior vertices. In this section, the process of modifying the adjacency calculus is carried out for the Laplacian problem on quantum trees where the leaf vertices have Dirichlet conditions. Next, the results obtained were compared with the secular determinant spectral analysis method using a scattering matrix introduced by Jean- Pierre Roth (1984). The discussion of this topic refers to research on the spectral comparison of the Laplace operator on finite metric graphs by Klawonn (2019) and quantum graph theory by Berkolaiko (2016). In this section, the form of the scattering matrix for quantum graphs with Dirichlet and Robin conditions is obtained. In the second topic, the results obtained from the discussion of the first topic used to solve edge-based wave equation models using the Laplacian. In this section, four linear wave equation models are discussed. Namely, ordinary linear wave equation models on quantum trees with Dirichlet vertex conditions, damped linear wave equation models on cycle graphs, linear wave equation models with elastic forces, and nonhomogeneous linear wave equation models on star graphs.