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My Numerical Solution doesn't align with the exact solution

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This is a 2nd order differential equation and I'm using ODE45 for my numerical Solution. My Numerical Solution is way too high. Notice that the values for numerical is *10^8. Additionally, I first solve my constants in order to make my equation code at matlab simpler.
My initial conditions are vc(0) = 11; vc' = 77459.66692
Exact Solution:
clear all
close all
clc
C = 3*10^(-6);
R = 12;
L = 2*10^(-3);
a = 0;
B = (6/(sqrt( L*C )))/(sqrt((1/(L*C))-(R/(2*L))^2));
Vc_0 = [6/(sqrt( L*C )) , 11];
odefun = @(t,Vc) [Vc(2);...
((-600)*(Vc(2)))-((50000000/3)*(Vc(1)))];
tspan = linspace(0, 0.04, 4000000);
[t,Vc] = ode45(odefun, tspan, Vc_0);
subplot(2,1,1);
plot(t,Vc)
xlabel('time(s)', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
ylabel('Vc', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
title('Numerical Methods', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
texact=[0:0.00000001:0.04];
Vexact=exp((-300)*texact).*(a*cos(((sqrt(596760000)*texact)/6))+B*sin(((sqrt(596760000)*texact)/6)));
subplot(2,1,2);
plot(texact,Vexact)
xlabel('time(s)', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
ylabel('VCexact', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
title('Exact Solution', 'FontSize',16, 'FontName','Arial','FontWeight','bold')
  3 Comments
Prince Nino
Prince Nino on 20 Sep 2024 at 2:18
That's what I was wondering. Why would our instructor have initial values for our exact solution be different from the numerical solution.
I have tried inputting the C,R,L Variables but it seems it gives me different answers(idk why). I've tried solving it manually. So I opted to solve it when it is already numerically valued rather than being it variable.
And yes, all the units are correct.
Sam Chak
Sam Chak on 20 Sep 2024 at 12:09
@Prince Nino, Because you didn't use the values derived from the parameters R, L, C in the code. Instead, you entered the coefficients '600' and '50000000/3' manually.
%% Parameters
C = 3*10^(-6);
R = 12;
L = 2*10^(-3);
R/L
ans = 6000
1/(L*C)
ans = 1.6667e+08
50000000/3
ans = 1.6667e+07

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Answers (2)

James Tursa
James Tursa on 19 Sep 2024 at 19:35
Edited: James Tursa on 19 Sep 2024 at 19:38
Assuming your state is [vc,vc'] in that order, then it appears your initial conditions are backwards and should be this instead:
Vc_0 = [11, 6/(sqrt( L*C ))];
I haven't looked any deeper into your problem yet ...

Sam Chak
Sam Chak on 20 Sep 2024 at 5:19
Perhaps you should verify the results with your Professor. You can also change the initial condition.
%% Numerical Solution
C = 3*10^(-6);
R = 12;
L = 2*10^(-3);
f = @(t, x) [ x(2);
-(R/L)*x(2) - 1/(L*C)*x(1)];
tt = linspace(0, 0.004, 4001);
ic = [0; 6/sqrt(L*C)]; % initial condition
[t, x] = ode45(f, tt, ic);
%% Analytical Solution
a = 1;
b = R/L;
c = 1/(L*C);
rpart = b/(2*a);
ipart = sqrt(c/a - rpart^2);
alpha = ic(1);
beta = (ic(1) + ic(2))/ipart;
xt = exp(- rpart*tt).*(alpha*cos(ipart*tt) + beta*sin(ipart*tt));
%% Plot results
tl = tiledlayout(2, 1);
nexttile
plot(t, x(:,1)), grid on, title('Numerical Solution')
nexttile
plot(tt, xt), grid on, title('Analytical Solution')
xlabel(tl, 'Time')
ylabel(tl, 'x(t)')

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