TWO TIME-SCALES PERTURBATION METHOD FOR MOVING BOUNDARY PROBLEMS

• Main Author: Firman Ihsan, Aditya
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/69680
• Institution: Institut Teknologi Bandung

In this dissertation we study two types of linear partial differential equations, i.e. the parabolic and the hyperbolic type, with moving boundary conditions, by means of constructing an approximate solution. This type of problems was originally studied by J Stefan, hence the name Stefan problem. By using the multiple time-scale method, we construct an approximation which valid for a rather long timescale. For the parabolic type of equations, we consider the linear one dimensional heat equation with various small initial conditions and moving boundary condition. We study both types of boundary conditions, i.e. the Neumann and the Dirichlet types. As usual, we apply transformation to dimension-less variables and also boundary fixing transformation. In comparison with the existing study in the literature, the approach in this dissertation involves time rescaling transformation which reformulate the problem as nonlinear perturbation of heat equation instead of Laplace equation. The advantage is we can include more type of initial condition to our analysis. Comparing the two types of boundary condition, the Dirichlet type turns out to be more complicated, namely the slowly varying function to be chosen to remove the secular term cannot be determined in the closed form. In this case, we also apply truncation as part of the approximation. For the hyperbolic type of equations, we consider string problem with an obstacle in one of the string end. Nondimesionalization and boundary fixing transformations are also applied. We use characteristic coordinates to avoid the infinite-dimensional system defining the interaction between nodes that appears in the original coordinate. The characteristic system obtained is then computed using numerical approach, which gives results close to the analytical solution on individual mode.