How to solve two differential equations using ode45.

 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.

[Torsten]

You will need four initial conditions (for x,x',y,y') if you want to use ODE45 as numerical solver.

[Ebraheem Menda]

Thank you Torsten. i have the initial conditions. but my question is how to convey these equations to ode45 or any other solver. Because they are coupled equations. thanks for your help.

Answer 1

[Torsten]

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.

[Ebraheem Menda]

Thank you Mr.Torsten. i will try it and give reply.

[Ebraheem Menda]

Hello Torsten it seems you have applied change of variable. like x'=y(1),x''=y(2) and y'=y(4) then what is 'y(3)'. if what i understood from your answer is right then the equations become fun=@(t,y)[y(2);-y(2)-y(1)-y(4);y(4);-y(4)-y(1)-y(2)] not fun=@(t,y)[y(2);-y(2)-y(1)-y(4);y(4);-y(4)-y(3)-y(2)] am i correct? thank you in advance.

[Ebraheem Menda]

Thank you Torsten.

[Torsten]

y1=x, y2=x', y3=y, y4=y'.

Thus my ODE-function looks correct, doesn't it ?

Best wishes

Torsten.

[Ebraheem Menda]

Yes you are correct Sir. Thank you.

[Joe Byrne]

Hi Torsten, I was just wodering what was going on in the first line of code here? I am attempting a similar problem but with different ODEs, was just wondering what the relevance of this line was to the rest of the code, i.e. what variables were assigned to what, etc.

Cheers

[Joe Byrne]

I mean to say what is the significance of y(2) and y(4) given that they have their own row in the array?

[Andreas Zimmermann]

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!

[Ebraheem Menda]

👍

Answer 2

[Abraham Boayue]

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?

[Ebraheem Menda]

thank you Abraham for your response. i am yet to solve those equations and obtain initial conditions. once i got them i will post the equations. Meanwhile what is this finite difference method ? is it available in matlab 2009b ?

Answer 3

[Abraham Boayue]

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.

[Ebraheem Menda]

👍

[Ebraheem Menda]

👏

[Ebraheem Menda]

please post your email Abraham

[Ebraheem Menda]

will contact you personally.

[madhan ravi]

You can email him by clicking his profile.

Answer 4

[Abraham Boayue]

Hi Ebraheem Is there anything specific that you want me to do for you?

[Ebraheem Menda]

Thank you. If I need will let you know for sure.

[Ebraheem Menda]

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

hi Abraham i want to put this code in for loop and run it for 2 or more tspan. Could you please help out on this ?