Homotopy Methods for Nonlinear Partial Differential Equation Systems

Doctoral Dissertation


Homotopy methods are efficient tools to compute multiple solutions, bifurcations and singularities of nonlinear partial differential equations (PDEs) arising from biology and physics. New and efficient methods based on the homotopy approach are presented in this thesis for computing multiple solutions, bifurcation points, and for solving steady states of hyperbolic conservation law. These new approaches make use of polynomial systems (with thousands of variables) arising by discretization. Examples from hyperbolic systems and tumor growth models will be used to demonstrate the ideas. The algorithms presented in this thesis can be applied to other problems arising in nonlinear PDEs and dynamic systems


Attribute NameValues
  • etd-06262013-101537

Author Wenrui Hao
Advisor Andrew Sommese
Contributor Bei Hu, Committee Co-Chair
Contributor Zhiliang Xu, Committee Member
Contributor Yongtao Zhang, Committee Member
Contributor Andrew Sommese, Committee Chair
Degree Level Doctoral Dissertation
Degree Discipline Applied and Computational Mathematics and Statistics
Degree Name PhD
Defense Date
  • 2013-06-25

Submission Date 2013-06-26
  • United States of America

  • free boundary

  • Homotopy methods

  • hyperbolic system

  • multiple solution

  • bifurcation

  • differential equation system

  • University of Notre Dame

  • English

Record Visibility Public
Content License
  • All rights reserved

Departments and Units


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