Traveling waves arising in the approximation and control of PDEs
Open Access
- Author:
- Zhang, Minyan
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 22, 2023
- Committee Members:
- Alberto Bressan, Chair & Dissertation Advisor
Wen Shen, Major Field Member
Yuxi Zheng, Major Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies
Francesco Costanzo, Outside Unit & Field Member - Keywords:
- Traveling wave
diffusion equation
control theory
conservation laws
Backward Euler method - Abstract:
- This dissertation is in the area of partial differential equations. It contains two parts: (1). The backward Euler approximation to conservation laws. (2). The traveling profiles for invasive species Model. In the first part, we study a hyperbolic system of conservation laws and construct an approximation by a backward Euler scheme, where time is discretized while space is still described by a continuous variable x ∈ IR. We prove the global existence and uniqueness of these approximate solutions, and the invariance of suitable subdomains. Furthermore, given a left and a right state u_l, u_r connected by an entropy-admissible shock, we construct a traveling wave profile for the backward Euler scheme connecting these two asymptotic states in two main cases. Namely: (i) a scalar conservation law, where the jump u_l–u_r can be arbitrarily large, and (ii) a strictly hyperbolic system, assuming that the jump u_l–u_r occurs in a genuinely nonlinear family and is sufficiently small. In the second part, we study a parabolic system of PDEs describing the spreading of an invasive species. The two models are worked on: (i) all insects carry the infection and contaminate the trees, and (ii) only the infected insects contaminate the tree. We try to find the traveling front for these two models with some proper speed. In the first model (i), we can get the optimal control added to evolution equations for the given cost function. We discover that the more we slow down the traveling front speed, the more cost we will pay. In second model (ii), the smallest traveling speed for the existence of the traveling front could be found, and in this case, no matter what controls are added to the system, the smallest traveling speed will not decrease unless the density of all insects decreases at the uninfected location, or the density of all insects decreases to zero at some places.