Solve systems of algebraic equations with parameters obtained by discretizing non-linear partial differential equations. Study how solutioins depend on parameters (local and global bifurcation problems).

Here is one test case. Consider the following reaction-diffusion equation.

Here, a and b are parameters. Set b=1 without loss of generality.
If 0 < a is small, there exists only trivial solution $u=0$ (extinction). If a >> 0, there exists non-trivial solution (stable population). Finding the border of the set is the first step of the bifurcation problem. Bifircation value is about 9.85.

Time evolution of u(t,x) for a=1, a=11, a=20. Here, u(0,x)=if (x < 0.5) then 2x else 2-2x. u(t,x) converges to a solution of the equation for the stable state when t goes to infinity.
Let us consider a difference scheme to solve this differential equation for stable state numerically. Devide [0,1] into N+1 segments.

We call the system of algebraic equations parter-N . The unknowns are u_1, ... , u_(N).
u_k=0 is a trivial solution. A bifurcation question is to find a non-trivial solution of the parter-N.
As to this questioin, Parter gave the following answer.
Theorem (Parter, 1965):
Put a_0 = 4 (N+1)^2 sin^2(\pi/(2 (N+1))).
If a < a_0 b, then there exists only trivial real solution. If a > a_0 b, then there exist two non-trivial real solutions.

Here are questions.
(子曰. 工欲善其事, 必先利其器.)
  1. Find a shape basis of parter-N. How far can we compute? Find rational univariate expression.
  2. Compute a primary (prime) ideal decomposition of parter-N.
  3. Can we rediscover Parter's result by our software systems?
  4. What is the function u_k(a,b)? Describe the global behavior of the function u_k(a,b).
parter-10   (d is b)



Remark: It is known that the function u_k(a,b) can be expressed in terms hypergeometric functions of several variables. As to details, see here