Solving system of four second order ODEs

Hi,

I am trying to solve a system of four second order ordinary differential equations with the initial conditions . However after running the following code

syms U1(t) V1(t) U2(t) V2(t)

%Constants
alpha = 0.5;

dU1 = diff(U1, t);
dU2 = diff(U2, t);
dV1 = diff(V1, t);
dV2 = diff(V2, t);

t0 = 0; %initial time
tf = 5; %final time

%initial conditions 
c1 = 1; %U(0)
c2 = 0; %U'(0)
c3 = 1; %V(0)
c4 = 0; %V'(0)

y0 = [c1 c2 c3 c4];

eq1 = diff(U2, 2) == alpha*dV2;
eq2 = diff(V2, 2) == -alpha*dU2;
eq3 = diff(U1, 2) == alpha*dV1;
eq4 = diff(V1, 2) == -alpha*dU1;

vars = [U1(t); V1(t); U2(t); V2(t)];
V = odeToVectorField([eq1,eq2,eq3,eq4]);
M = matlabFunction(V, 'vars', {'t','Y'});
interval = [t0 tf]; %time interval
ySol = ode45(M,interval,y0);
tValues = linspace(interval(1),interval(2),1000);
yValues1 = deval(ySol,tValues,1); %U(t)
yValues2 = deval(ySol,tValues,3); %V(t)
plot(yValues1, yValues2)

I get this error

Index exceeds the number of array elements. Index must not exceed 4.

Error in symengine>@(t,Y)[Y(2);Y(4).*(-1.0./2.0);Y(4);Y(2)./2.0;Y(6);Y(8)./2.0;Y(8);Y(6).*(-1.0./2.0)]

Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

Error in ode45 (line 106)
  odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Could you please help? If there is another way to solve this sytem then it will be much appreciated if you can share it. Thank you.

Answer 1

[Torsten]

4 second-order ODEs need 8 initial conditions, not 4.

[Jake Barlow]

Hi Torsten, thank you for your answer. I have added four more initial conditions, but does this affect the order of the ouput ? and is the order of the initial conditions applied correctly to ?

[Torsten]

Use

[V,S] = odeToVectorField([eq1,eq2,eq3,eq4])

instead of

V = odeToVectorField([eq1,eq2,eq3,eq4])

to see in which order the solution variables appear.

Specify your initial conditions in this order, too.

[Jake Barlow]

Hi Torsten, thank you very much. I specified the initial conditions in the correct order.