Question

In the last portfolio task, we modelled water draining from a tank with a small hole in the bottom. To slow the volume decrease of the tank, a hose is placed in the top of tank with flow rate Q(t). This results in the following modified Torcelli's law: dV Given A 4 m2, a -0.1 m2, g 9.81 m/s2, Q(t) 0.7e-0.11 m3/s and V(0) 10 m3

Answer 1

1.method:

Mat lab Code:

Given data:

A=4m^2, a=0.1m^2, g=9.81m/s^2, Q(t)=0.7e^-0.1t m^3/s and V(0)=10m^3

3. %%Matlab function for Question 3. function [tyl modifierEuler(ODEFUN,TSPAN,YO,N) tu TSPAN(1):1/N:TSPAN(2): % Euler steps y(1) YO; t(1) TSPANO1):h /N; for i-1:length(tn)-1 y(i+1)-y(i)+h(ODE FUN(t(i),y(i))); end end

Copyable code:

3.

%%Matlab function for Question 3.

function [t,y]=modifierEuler(ODEFUN,TSPAN,Y0,N)

tn=TSPAN(1):1/N:TSPAN(2);

% Euler steps

y(1)=Y0;

t(1)=TSPAN(1);h=1/N;

for i=1:length(tn)-1

t(i+1)= t(i)+h;

y(i+1)=y(i)+h*(ODEFUN(t(i),y(i)));

end

end

2.method:

%Matlab code for plotting ode45 and Euler method
clear all
clear all
%Constant terms
A=4;a=0.1;g=9.81;
v0 = 0;
tspan = [0 50];
%Quetion 2.
%code for solving using ode45
[t,v] = ode45(@vdt1, tspan, v0);
%Plotting the data using ode45
plot(t,v,'-o')
xlabel('t')
ylabel('v(t)')
title('t vs v(t) plot')
%Code for Euler solution
%all step size
N=[10 20 40 80];
h=1./N;
%Initial values
v0=0;
t0=0;
%t end values
tend=50;
fprintf('\n Due to nonlinear equation we cant solve it analytically.\n')
for nn=1:length(N)
    [t1,y1]=modifierEuler(@vdt1,tspan,v0,N(nn));
    hold on
    plot(t1,y1)
    yy= spline(t,v,t1);
    err_norm=norm(y1-yy);
    fprintf('The error in Euler for N=%d is %f\n',N(nn),err_norm)
end
legend('ode45','euler N=10','euler N=20','euler N=40','euler N=80')
fprintf('\nBy increasing step length, error used to decrease.\n')

Result
Due to nonlinear equation we cant solve it analytically.
The error in Euler for N=10 is 0.257780
The error in Euler for N=20 is 0.181609
The error in Euler for N=40 is 0.129477
The error in Euler for N=80 is 0.094032

By increasing step length, error used to decrease.

%Function for dvdt of modified Torcelli's law
function dvdt = vdt1(t,v)
    A=4;a=0.1;g=9.81;
    dvdt = 0.7*exp(-0.1*t)-a*sqrt((2*g*v)/A);
end

%%Matlab function for Question 2.
function [t,v]=tankVolume()
    A=4;a=0.1;g=9.81;
    v0 = 0;
    tspan = [0 50];
    %Quetion 2.
    %solving using ode45
    [t,v] = ode45(@(t,v) 0.7*exp(-0.1*t)-a*sqrt((2*g*v)/A), tspan, v0);
end

%%Matlab function for Question 3.

function [t,y]=modifierEuler(ODEFUN,TSPAN,Y0,N)
tn=TSPAN(1):1/N:TSPAN(2);
    % Euler steps
    y(1)=Y0;
    t(1)=TSPAN(1);h=1/N;
    for i=1:length(tn)-1
        t(i+1)= t(i)+h;
        y(i+1)=y(i)+h*(ODEFUN(t(i),y(i)));
    end
end

3.method with graph: