Bei Hu

Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0<ρ<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=ρ. The system depends on a positive parameter μ, which describes tumor aggressiveness, and for a sequence of values μ2<μ3<…, there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factorxa0μ. By continuously changing μ using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate @m and the cell-to-cell adhesiveness @c are two parameters for characterizing aggressiveness of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of @m/@c symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.

A three-dimensional steady-state tumor system

Abstract The growth of tumors can be modeled as a free boundary problem involving partial differential equations. We consider one such model and compute steady-state solutions for this model. These solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions. Linear and nonlinear stability for these solutions are determined numerically.

On the rate of convergence of the binomial tree scheme for American options

An American put option can be modelled as a variational inequality. With a penalization approximation to this variational inequality, the convergence rate

Numerical Algebraic Geometry and Differential Equations

Obig((Delta x)^{2/3}big)

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

of the Binomial Tree Scheme is obtained in this paper.

Optimal convergence rate of the explicit finite difference scheme for American option valuation

We consider a free boundary problem for a system of partial differential equations, which arise in a model of cell cycle with a free boundary. For the quasi steady state system, it depends on a positive parameter