3. A chemical compound decays over time when exposed to air, at a rate proportional to its

concentration to the power of 3/2. At the same time, the compound is produced by another

process. The differential equation for the instantaneous concentration is,

where C(t) is the instantaneous concentration, Cinitial = 2000 is the initial concentration at

t = 0.

(a) First clearly write down your finite difference approximation for a general time, using

the Forward Euler (Forward Difference) method.

(b) Solve the differential equation (by writing a computer program) to find the concentration

as a function of time from t = 0 until t = 0.5 s, using the Forward Euler method. Use

a step size of h = 0.002 s and plot C(t) versus t. Provide your computer program and

clearly indicate your algorithm with sufficient comments on the program.

(c) How will you verify that your predicted answer is a reasonable estimate using this particu-

lar numerical scheme? Accordingly, verify your answer and show proof of your verification

study.

(d) Now solve the problem using Backward Difference (Backward Euler) method. For that,

write down the finite-difference approximation and then write a computer program to

solve that finite-difference approximation. Note that, you will get an implicit relation

and this would have to be solved iteratively. Provide your computer program and clearly

indicate your algorithm with sufficient comments on the program. Use the same step

size as above and plot C(t) versus t.

(e) Do the Backward and Forward Euler schemes provide similar solutions for the step size

chosen? Is one more accurate than the other? Why or why not?