User:Egm5526.s11.team4.zou/NM HW7

=Problem 7.1.3: FE and BE Algorithm Applied to the Integrations of the Logistic Growth Equation=

From the lecture slide Mtg 40

 This problem was solved with referring to S10 homework: S10 Team4's Problem 7.5 

Given
The Verhulst Model as stated in the Problem 7.1.2,

Two cases of initial condition,


 * Case 1: initial condition $$ \displaystyle x_0 = 3 < \frac{1}{2}x_{max} $$.
 * Case 2: initial condition $$ \displaystyle x_0 = 9 < \frac{1}{2}x_{max} $$.

Find
Integrate the logistic equation with increasing step size $$\displaystyle h=\hat{h}\cdot {{2}^{k}}$$ by

(1) Foward Euler algorithm

(2) Backward Euler algorithm

Integrate using Backward Euler Algorithm
Do a scaling such that $$\displaystyle \bar{x}(t)=\frac{x(t)}$$, then the Verhulst Model turns into,

Then by Forward Euler Algorithm,

Case 1: $$\displaystyle {{x}_{0}}=3$$
Result using Forward Euler Algorithm when $$\displaystyle k=1$$



Result using Forward Euler Algorithm when $$\displaystyle k=4$$



Result using Forward Euler Algorithm when $$\displaystyle k=7$$



Case 2: $$\displaystyle {{x}_{0}}=9$$
Result using Forward Euler Algorithm when $$\displaystyle k=1$$



Result using Forward Euler Algorithm when $$\displaystyle k=4$$



Result using Forward Euler Algorithm when $$\displaystyle k=7$$



Integrate using Backward Euler Algorithm
From Eq 1.3.2 and Backward Euler Algorithm we have,

Case 1: $$\displaystyle {{x}_{0}}=3$$
Result using Forward Euler Algorithm when $$\displaystyle k=1$$



Result using Forward Euler Algorithm when $$\displaystyle k=7$$



Case 2: $$\displaystyle {{x}_{0}}=9$$
Result using Forward Euler Algorithm when $$\displaystyle k=4$$



Result using Forward Euler Algorithm when $$\displaystyle k=7$$



Comment
As we can see in the Foward Euler Algorithm the result become indented when $$\displaystyle k=7$$ reaches 7. This means the Foward Euler Algorithm's solution may become unstable with step size increasing. While the result by Backward Euler Algorithm still remain stable with $$\displaystyle k=7$$.