How to solve two differential equations using ode45.
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My system is this
x"+ x'+ x + y'=0;
y"+ y'+ y + x'=0;
i need to solve these differential equations using ode's.
thanks in advance.
2 个评论
Torsten
2018-2-22
You will need four initial conditions (for x,x',y,y') if you want to use ODE45 as numerical solver.
采纳的回答
Torsten
2018-2-22
fun=@(t,y)[y(2);-y(2)-y(1)-y(4);y(4);-y(4)-y(3)-y(2)];
x0=...; %supply x(t0)
x0prime=...; %supply x'(t0)
y0=...; %supply y(t0)
y0prime=...; %supply y'(t0)
Y0=[x0; x0prime; y0; y0prime];
tspan=[0 5];
[T,Y]=ode45(fun,tspan,Y0);
plot(T,Y(:,1)) %plot x
plot(T,Y(:,3)) %plot y
Best wishes
Torsten.
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Andreas Zimmermann
2020-4-16
编辑:Andreas Zimmermann
2020-4-16
Hi Joe,
Using Torsten's relations
x = y(1)
x' = y(2)
y = y(3)
y' = y(4)
the first line
fun=@(t,y)[y(2); -y(2)-y(1)-y(4); y(4); -y(4)-y(3)-y(2)];
can be translated to the following, using Torsten's relations:
[x'; -x'-x-y'; y'; -y'-y-x']
which is essentially
[x'; x"; y'; y"].
He got these relations by solving for x" and y":
x"+ x'+ x + y'=0; => x" = -x' -x -y' = -y(2) -y(1) -y(4)
y"+ y'+ y + x'=0; => y" = -y' -y +x' = -y(4) -y(3) -y(2)
I have tried to use ode45 to solve the SIR model from this fantastic Geogebra Tutorial: (https://youtu.be/k6nLfCbAzgo)
% Just some start parameters and coefficients
Istart = 0.01;
Sstart = 0.99;
Rstart = 0;
transm = 3.2;
recov = 0.23;
maxT = 20;
% define S',I',R'
% S' = - transm * S * I
% I' = transm * S I - recov * I
% R' = recov * I
% S is y(1)
% I is y(2)
% R is y(3)
%so here fun = @(t,y)[S'; I'; R'];
fun = @(t,y)[-transm*y(1)*y(2); (transm*y(1)*y(2))-(recov*y(2)); recov*y(2)];
% Provide the starting parameters
Y0 = [Sstart; Istart; Rstart;];
% Define the range of t
tspan = [0 maxT];
% Magic happens and matrix Y contains S,I,R
[T,Y] = ode45(fun,tspan,Y0);
% Plot plot
figure
plot(T,abs(Y(:,1)),'b-') % S
hold on
plot(T,abs(Y(:,2)),'r-') % I
plot(T,abs(Y(:,3)),'g-') % R
xlim([0 23])
%Hope this helps :)
% Many thanks to Torsten and Ebraheem for your very very useful discussion!
更多回答(3 个)
Abraham Boayue
2018-2-24
Hey Ebraheem There are many excellent methods that you can use to solve your problem, for instance, the finite difference method is a very powerful method to use. I can try with that.The ode45 function is a matlab built in function and was designed to solve certain ode problems, it may not be suitable for a number of problems. What are the initial values of your equations? Do you have any plot of the solution that one can use as a guide?
Abraham Boayue
2018-2-24
The finite difference method is used to solve differential and partial equations. It is easier to implement in matlab. You can do the coding in any version of matlab, I have taken a course in numerical mathematics before and have a fairly good knowledge of how to solve such problems.
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