Hi i've been asked to solve SIR model using fsolve command in MATLAB, and Euler 3 point backward. I'm really confused on how to proceed, please help. This is what i have so far. I created a function for 3BDF scheme but i'm not sure how to proceed with fsolve and solve the system of nonlinear ODEs. The SIR model is shown as and 3BDF scheme is formulated as
clc
clear all
gamma=1/7;
beta=1/3;
ode1= #(R,S,I) -(beta*I*S)/(S+I+R);
ode2= #(R,S,I) (beta*I*S)/(S+I+R)-I*gamma;
ode3= #(I) gamma*I;
f(t,[S,I,R]) = [-(beta*I*S)/(S+I+R); (beta*I*S)/(S+I+R)-I*gamma; gamma*I];
R0=0;
I0=10;
S0=8e6;
odes={ode1;ode2;ode3}
fun = #root2d;
x0 = [0,0];
x = fsolve(fun,x0)
function [xs,yb] = ThreePointBDF(f,x0, xmax, h, y0)
% This function should return the numerical solution of y at x = xmax.
% (It should not return the entire time history of y.)
% TO BE COMPLETED
xs=x0:h:xmax;
y=zeros(1,length(xs));
y(1)=y0;
yb(1)=y0+f(x0,y0)*h;
for i=1:length(xs)-1
R =R0;
y1(i+1,:) = fsolve(#(u) u-2*h/3*f(t(i+1),u) - R, y1(i-1,:)+2*h*F(i,:))
S = S0;
y2(i+1,:) = fsolve(#(u) u-2*h/3*f(t(i+1),u) - S, y2(i-1,:)+2*h*F(i,:))
I= I0;
y3(i+1,:) = fsolve(#(u) u-2*h/3*f(t(i+1),u) - I, y3(i-1,:)+2*h*F(i,:))
end
end
You have an implicit equation
y(i+1) - 2*h/3*f(t(i+1),y(i+1)) = G = (4*y(i) - y(i-1))/3
where the right-side term G is constant in the call to fsolve, that is, during the solution of the implicit step equation.
Note that this is for the vector valued system y'(t)=f(t,y(t)) where
f(t,[S,I,R]) = [-(beta*I*S)/(S+I+R); (beta*I*S)/(S+I+R)-I*gamma; gamma*I];
To solve this write
G = (4*y(i,:) - y(i-1,:))/3
y(i+1,:) = fsolve(#(u) u-2*h/3*f(t(i+1),u) - G, y(i-1,:)+2*h*F(i,:))
where a midpoint step is used to get an order 2 approximation as initial guess, F(i,:)=f(t(i),y(i,:)). Add solver options for error tolerances as necessary, you want the error in the implicit equation smaller than the truncation error O(h^3) of the step. One can also keep only a short array of function values, then one has to be careful for the correspondence of the position in the short array to the time index.
Using all that and a reference solution by a higher order standard solver produces the following error graphs for the components
where one can see that the first order error of the constant first step results in a first order global error, while with a second order error in the first step using the Euler method results in a clear second order global error.
Implement the method in general terms
from scipy.optimize import fsolve
def BDF2(f,t,y0,y1):
N, h = len(t)-1, t[1]-t[0];
y = (N+1)*[np.asarray(y0)];
y[1] = y1;
for i in range(1,N):
t1, G = t[i+1], (4*y[i]-y[i-1])/3
y[i+1] = fsolve(lambda u: u-2*h/3*f(t1,u)-G, y[i-1]+2*h*f(t[i],y[i]), xtol=1e-3*h**3)
return np.vstack(y)
Set up the model to be solved
gamma=1/7;
beta=1/3;
print beta, gamma
y0 = np.array([8e6, 10, 0])
P = sum(y0); y0 = y0/P
def f(t,y): S,I,R = y; trns = beta*S*I/(S+I+R); recv=gamma*I; return np.array([-trns, trns-recv, recv])
Compute a reference solution and method solutions for the two initialization variants
from scipy.integrate import odeint
tg = np.linspace(0,120,25*128)
yg = odeint(f,y0,tg,atol=1e-12, rtol=1e-14, tfirst=True)
M = 16; # 8,4
t = tg[::M];
h = t[1]-t[0];
y1 = BDF2(f,t,y0,y0)
e1 = y1-yg[::M]
y2 = BDF2(f,t,y0,y0+h*f(0,y0))
e2 = y2-yg[::M]
Plot the errors, computation as above, but embedded in the plot commands, could be separated in principle by first computing a list of solutions
fig,ax = plt.subplots(3,2,figsize=(12,6))
for M in [16, 8, 4]:
t = tg[::M];
h = t[1]-t[0];
y = BDF2(f,t,y0,y0)
e = (y-yg[::M])
for k in range(3): ax[k,0].plot(t,e[:,k],'-o', ms=1, lw=0.5, label = "h=%.3f"%h)
y = BDF2(f,t,y0,y0+h*f(0,y0))
e = (y-yg[::M])
for k in range(3): ax[k,1].plot(t,e[:,k],'-o', ms=1, lw=0.5, label = "h=%.3f"%h)
for k in range(3):
for j in range(2): ax[k,j].set_ylabel(["$e_S$","$e_I$","$e_R$"][k]); ax[k,j].legend(); ax[k,j].grid()
ax[0,0].set_title("Errors: first step constant");
ax[0,1].set_title("Errors: first step Euler")
I'm trying to solve a system of two second order non linear Odes and since it is a boundary valued problem I suppose I need to use the bvp4c function.
The system I'm talking about is the following:
f''(x) = F(f,f',x);
s''(x) = G(f, f',s,s',x)
with the conditions f(0) = pi, f(inf = 35) = s(inf = 35) = 0. The F and G functions are known and I assumed that 35 would be a decent replacement for infinity.
It is separable and I have already solved for f but I don't know how to solve it for s either.
The code that allegedly solves for f is the following:
options = bvpset('RelTol', 1e-5);
Xstart = 0.01;
Xend = 35;
solinit = bvpinit(linspace(Xstart, Xend, 1000), [0, 1]);
sol = bvp4c(#twoode, #twobc, solinit, options);
x = linspace(Xstart,Xend);
y = deval(sol,x);
figure(1)
plot(x,y(1,:))
figure(2)
plot(x,y(2,:))
function dydx = twoode(x,y)
dydx = [y(2); ((-1/(x^2 + 2+sin(y(1))^2))*(2*x*y(2) + sin(2*y(1))*y(2)^2 -
2*sin(2*y(1)) - (sin(y(1)^2)*sin(2*y(1)))/x^2) )];
end
function res = twobc(ya,yb)
res = [ya(1) - pi
yb(2)];
end
So my question is how can I use the results I obtained for f in order to solve the equation for s? I have tried doing the same things I did for f but if I define a function for s that uses y(1,:) and y(2,:) it gives me an error message that says y is not defined.
Since I am quite new to solving dfferential equations with Matlab and to using Matlab in general I am probably making some trivial mistake but I have been looking for answers and couldn't find any. I hope someone with enough patience can help me.
Thanks in advance for any useful advice.
I want to integrate a differential equation dc/dt. Below is the code and the values of the variables.
clear all;
c1=.185;c0=2*10^-6;k3=.1*10^-6;
v1=6;v2=.11;v3=.09*10^-6;
Ca_ER=10*10^-6;Ca_cyto=1.7*10^-6;
p_open3=0.15;c=15*10^-6;
dcdt= (c1*(v1*(p_open3)+v2)*(Ca_ER)-c)-v3*((c)^2)/(c^2+(k3)^2);
I know there is an integral function but I am not sure how to apply for this equation. How do I proceed from here? Please help. The value of initial c, if needed, can be taken as 0.15*10^-6. Also, I need to plot the obtained result versus time. So will get an array of values or just a single value?
the link to the article. the equation i have used comes under Calcium Oscillations section
You could use Euler method to solve this problem to get a rough idea regarding the solution yet not accurate.
clear all
clc
t = 0;
dt = 0.0001;
c1 = 0.185;
c0 = 2*10^-6;
k3 = 0.1*10^-6;
v1 =6;
v2 =.11;
v3 =.09*10^-6;
Ca_ER =10*10^-6;
Ca_cyto =1.7*10^-6;
p_open3 =0.15;
c = 15*10^-6;
%store initial values
C(1) = c;
T(1) = t;
for i = 1:40000
dc = ( (c1*(v1*(p_open3)+v2)*(Ca_ER)-c)- v3*( c^2 /( c^2+(k3)^2) ) );
c = c + dt*dc;
t = t + dt;
%store data
C(i+1) = c;
T(i+1) = t;
end
plot(T,C, 'LineWidth',2)
xlabel('time (sec)')
ylabel('c(t)')
grid on
The result is
You can also use Wolfram which gives same result.
I am trying to compute the value of this integral using Matlab
Here the other parameters have been defined or computed in the earlier part of the program as follows
N = 2;
sigma = [0.01 0.1];
l = [15];
meu = 4*pi*10^(-7);
f = logspace ( 1, 6, 500);
w=2*pi.*f;
for j = 1 : length(f)
q2(j)= sqrt(sqrt(-1)*2*pi*f(j)*meu*sigma(2));
q1(j)= sqrt(sqrt(-1)*2*pi*f(j)*meu*sigma(1));
C2(j)= 1/(q2(j));
C1(j)= (q1(j)*C2(j) + tanh(q1(j)*l))/(q1(j)*(1+q1(j)*C2(j)*tanh(q1(j)*l)));
Z(j) = sqrt(-1)*2*pi*f(j)*C1(j);
Apprho(j) = meu*(1/(2*pi*f(j))*(abs(Z(j))^2));
Phi(j) = atan(imag(Z(j))/real(Z(j)));
end
%integration part
c1=w./(2*pi);
rho0=1;
fun = #(x) log(Apprho(x)/rho0)/(x.^2-w^2);
c2= integral(fun,0,Inf);
phin=pi/4-c1.*c2;
I am getting an error like this
could anyone help and tell me where i am going wrong.thanks in advance
Define Apprho in a separate *.m function file, instead of storing it in an array:
function [ result ] = Apprho(x)
%
% Calculate f and Z based on input argument x
%
% ...
%
meu = 4*pi*10^(-7);
result = meu*(1/(2*pi*f)*(abs(Z)^2));
end
How you calculate f and Z is up to you.
MATLAB's integral works by calling the function (in this case, Apprho) repeatedly at many different x values. The x values called by integral don't necessarily correspond to the 1: length(f) values used in your original code, which is why you received errors.
Hello I am relatively new to MATLAB and have received and assignment in which we could use any programming language. I would like to continue MATLAB and have decided to use it for this assignment. The questions has to do with the following formula:
x(t) = A[1+a1*E(t)]*sin{w[1+a2*E(t)]*t+y}(+/-)a3*E(t)
The first question we have is to develop an appropriate discretization of x(t) with a time step h. I think i understand how to do this using step but because there is a +/- in the end I am running into errors. Here is what I have (I have simplified the equation by assigning arbitrary values to each variable):
A = 1;
E = 1;
a1 = 1;
a2 = 2;
a3 = 3;
w = 1;
y = 0;
% ts = .1;
% t = 0:ts:10;
t = 1:1:10;
x1(t) = A*(1+a1*E)*sin(w*(1+a2*E)*t+y);
x2(t) = a3*E;
y(t) = [x1(t)+x2(t), x1(t)-x2(t)]
plot(y)
The problem is I keep getting the following error because of the +/-:
In an assignment A(I) = B, the number of elements in B and I must be the same.
Error in Try1 (line 21)
y(t) = [x1(t)+x2(t), x1(t)-x2(t)]
Any help?? Thanks!
You can remove the (t) from the left-hand side of all three assignments.
y = [x1+x2, x1-x2]
MATLAB knows what to do with vectors and matrices.
Or, if you want to write it out the long way, tell MATLAB there will be two columns:
y(t, 1:2) = [x1(t)'+x2(t)', x1(t)'-x2(t)']
or two rows:
y(1:2, t) = [x1(t)+x2(t); x1(t)-x2(t)]
But this won't work when you have fractional values of t. The value in parentheses is required to be the index, not a dependent variable. If you want the whole vector, just leave it out.