Write a computer program to solve a set of $3$ decay equations $(N_A{N}_B{N}_C,N_C$ is stable$)$ analytically

and numerically. Use a forward difference approximation $($explicit scheme$)$ for the numerical solution.

Your computer program should do the following:

Read input file with the input data:

decay constants:

initial conditions: $,N_{AO},N_{BO},N_{Co}$

numerical parameters: $t,t_{\text{f}\text{i}\text{n}\text{a}\text{l}}$

Perform the analytical solution of the decay equations for $+10\%+10\%+20\%N_B(+20\%)t_{\frac{1}{2}A}=1\mathrm{.}97h,t_{\frac{1}{2}B}=9\mathrm{.}91h$

$N_{Ao}=100,N_{Bo}=o,N_{Co}=o$

$t_{\text{f}\text{i}\text{n}\text{a}\text{l}}=50ht=1ht{N}_B(t)t+10\%N_A(t),N_B(t),N_C(t)N_A(t)+N_B(t)+N_C(t)t+10\%N_B\frac{1}{}tt{N}_B+20\%+-t-10\%o.$

Write results $toan$ output file that can later $be$ used $to$ plot the results. The output file should

also contain the parameters read from the input file.

Submit a brief report with your results. See separate instructions $on$ the report format. Minimum

content $of$ the report:

Part $1,$ theory

Show and describe differential equations you are solving $(+10\%).$

Show analytic solution $of$ the differential equations $(+10\%).$

Show complete derivation $of$ numerical solution $(+20\%).$

Show complete derivation for the time $of$ maximum $N_B(+20\%).$

Part $2,$ solution $of$ radioactive decay chain

Half$-$life:

Initial conditions:

Solution time:

$t_{\frac{1}{2}A}=1\mathrm{.}97h,t_{\frac{1}{2}B}=9\mathrm{.}91h$

$N_{Ao}=100,N_{Bo}=o,N_{Co}=o$

$t_{\text{f}\text{i}\text{n}\text{a}\text{l}}=50h$

For the numerical solution, start with $t=1h$ and keep reducing $itby$ the factor $of$ two until you get $a$

reliable solution $(solution$ does not change significantly with deceasing $t).$ Show the following results:

Plot numerical $N_B(t)vs.$ time for $3$ different values $oft(coarse,$ medium, fine$),$ all $of$ them $on$

the same graph. Add analytical solution $on$ the same graph $(+10\%).$

Plot numerical $N_A(t),N_B(t),N_C(t)$ and $N_A(t)+N_B(t)+N_C(t)as$ a function $of$ time, all $on$ the same

graph, use $t$ that gives reliable solution $(+10\%).$

Using numerical solution, plot time $of$ maximum $N_{B}vs.\frac{1}{}t$ for several different $t.$ Use

analytical solution $to$ determine time $of$ maximum $N_B,$ add that value $to$ the graph $(+20\%).$

Note: numerical solution with large $t$ might $be$ unstable $(e.g.,$ solution oscillates, goes toward $+or-,$

other unphysical behavior$).Do$ not report such results. Continue reducing $t$ until you get physically

realistic solution $(-10\%).o.$

Perform the numerical solution $of$ the decay equations for $o.$

Write results $toan$ output file that can later $be$ used $to$ plot the results. The output file should

also contain the parameters read from the input file.

Submit a brief report with your results. See separate instructions $on$ the report format. Minimum

content $of$ the report:

Part $1,$ theory

Show and describe differential equations you are solving $(+10\%).$

Show analytic solution $of$ the differential equations $(+10\%).$

Show complete derivation $of$ numerical solution $(+20\%).$

Show complete derivation for the time $of$ maximum $N_B(+20\%).$

Part $2,$ solution $of$ radioactive decay chain

Half$-$life:

Initial conditions:

Solution time:

$t_{\frac{1}{2}A}=1\mathrm{.}97h,t_{\frac{1}{2}B}=9\mathrm{.}91h$

$N_{Ao}=100,N_{Bo}=o,N_{Co}=o$

$t_{\text{f}\text{i}\text{n}\text{a}\text{l}}=50h$

For the numerical solution, start with $t=1h$ and keep reducing $itby$ the factor $of$ two until you get $a$

reliable solution $(solution$ does not change significantly with deceasing $t).$ Show the following results:

Plot numerical $N_B(t)vs.$ time for $3$ different values $oft(coarse,$ medium, fine$),$ all $of$ them $on$

the same graph. Add analytical solution $on$ the same graph $(+10\%).$

Plot numerical $N_A(t),N_B(t),N_C(t)$ and $N_A(t)+N_B(t)+N_C(t)as$ a function $of$ time, all $on$ the same

graph, use $t$ that gives reliable solution $(+10\%).$

Using numerical solution, plot time $of$ maximum $N_{B}vs.\frac{1}{}t$ for several different $t.$ Use

analytical solution $to$ determine time $of$ maximum $N_B,$ add that value $to$ the graph $(+20\%).$

Note: numerical solution with large $t$ might $be$ unstable $(e.g.,$ solution oscillates, goes toward $+or-,$

other unphysical behavior$).Do$ not report such results. Continue reducing $t$ until you get physically

realistic solution $(-10\%).$