Homotopy Methods for Nonlinear Partial Differential Equation Systems

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