## Transcribed Text

Note: To solve a problem by variation of parameter, first write it in the standard form:
Problem 1 (20 points)
Solve the following differential equation by variation of parameter:
x2y"+xy'-y=lnx
Problem 2 (20 points)
Solve the following initial value problem:
= x,y(1) = 1;y'(1) = -1/2
Problem 3 (60 points)
Consider the Plug Flow Reactor problem discussed in class on 09-11-2017:
NO2 is flowing through a plug flow reactor, in which it reacts and is consumed following a 2nd order
kinetics. Assuming steady state and absence of axial diffusion, determine how the concentration of
NO2 changes along the reactor, if it enters the reactor at a concentration of Co. The consumption rate of
NO2 is given as: k CNO2' where k= 5030 mm
mole sec
Solve the developed ODE using the following numerical methods:
(a) Euler (b) Trapezoidal and (c) Backward Euler.
For each of the cases, integrate within the limits of 0 T 2. Plot = versus T by varying h (step
length) as: 0.1 0.2, 0.5, 1.0.
Present the results in the following comparative plots:
(i)
Plot the solution of explicit Euler method for different h values and compare with analytical
solution. Comment on stability and accuracy of the method.
(ii)
Plot the solution of Trapezoidal method for different h values and compare with analytical
solution. Comment on stability and accuracy of the method.
(iii)
Plot the solution of backward Euler method for different h values and compare with
analytical solution. Comment on stability and accuracy of the method.
(iv)
Plot the percentage relative error versus T for each of the methods (h=0.5). Discuss the
accuracy of each of the methods. Calculate % relative error by the following:
Numerical value Analytical value)
Analytical value
timestep
Present the plots and copy-paste the script in your answer sheet.

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clear t %Clear old time steps

clear y %Clear y values from previous runs

a=0; % Initial tau

b=2; % Final tau

f_an = @(tau) 1./(1+tau);

h=[0.1 0.2 0.5 1.0]; %Incremental time step

y0=1; % Initial value y(a)

figure('Name','Backward Euler:Numerical-Analytical solution comparison plot','NumberTitle','off');

y_an = f_an(a:h(1):b);

plot(a:h(1):b,y_an,'Color',rand(1,3),'LineWidth',2);

legendInfo{1}='Analytical solution';

hold on;

for i=1:length(h)

N=(b-a)/h(i); % Number of time steps

tau(1) = a; % assign initial time

y_n(1) = y0; % assign initial condition

for n=1:N % For loop, sets next t,y values

tau(n+1) = tau(n)+h(i);

% here newton-Raphson will be used to iteratively solve y(n+1)

% two new variables are introduced for N-R.

% ynr1 will represent the kth iteration and ynr2 the (k+1) iteration

ynr1 = y_n(n);%initialize NR

loopcount = 0; diff = 1;

while abs(diff)>0.0005

loopcount = loopcount+1;

ynr2=ynr1 - (h(i)*(ynr1^2)+ynr1-y_n(n))/(1+2*h(i)*ynr1); %N-R iteration

diff = (ynr2-ynr1)/ynr2; %evaluates convergence

ynr1=ynr2...