Non-Linear PDE's: Reaction-Diffusion Equation: Fisher's Equation w/ harvesting?
Ok so I am dealing with a spatially dependent logistic equation:
$u_t = u_{xx} + u(1-u) - h,$ on 0 ≤ x ≤ L w/ homogenous Dirichlet at x = 0 & homogenous Neumann at x = L boundary conditions $u(0,t) = 0, u_x(L,t) = 0$
Now I converted this pde (partial differential equation) to system of 2 ode (ordinary differential equation)
so $u_x = v$ & $v_x = -u(1 - u) - h$
The question I am having trouble w/ asks.
Find steady state solution in the phase plane for small h > 0. What is the range for h > 0 that supports a steady state solution
Now I am solving for the steady state solutions
So setting both $u_x$ & $v_x$ to zero
$u_x = v = 0. $
$v_x = u = -u(1 - u) - h = 0,$ solve for u
i realized that this is a quadratic equation, so i expanded it to $u^{2} - u - h = 0$
$u = \frac{-b ± \sqrt{b^{2}-4*a*c}}{2}$
$u = \frac{1 ± \sqrt{1+4*h}}{2}$
I get the steady state solutions as $u = \frac{1 - \sqrt{1+4*h}}{2}$ , $v = 0$ or $u = \frac{1 + \sqrt{1+4*h}}{2}$ , $v = 0$
exact possible range of h so that steady state solution becomes real values (for both of the steady states) is
range of h is (0, ∞]
However, when I crossed-check with the instructor she said my u is incorrect.
Is there somewhere I went wrong in my calculations because i reworked the problem but keep getting the same answer again.
Any help appreciated