MATLAB: solving a 2nd order ODE which has a parameter which evolves in time
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Hi, I'm trying to solve the equation: y'' + k(t)y = 0 where k(t) is an array of values and y(0) = 1/2 and y'(0) = 1 in the range t = 0 to 20.
I understand it is possible to solve this as 2 coupled first order differential equations using ode45 as follows when k is constant:
k = 2
syms y(t)
[System] = odeToVectorField(diff(y, 2) == -(k).*y);
M = matlabFunction(System,'vars', {'t','Y'});
sol = ode45(M,[0 20],[1/2 1]);
fplot(@(x)deval(sol,x,1), [0, 20])
Please will you help me introduce the dependance on t of k(t), thanks!
1 个评论
darova
2020-2-24
- Please will you help me introduce the dependance on t of k(t), thanks!
Can you be more specific? What is it? Where is dependance?
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Star Strider
2020-2-24
Try this:
syms k(t) y(t) % Declare ‘k(t)’ Here
[System,Subs] = odeToVectorField(diff(y, 2) == -(k).*y); % Note ‘k(t)’ Included In Code
M = matlabFunction(System,'vars', {'t','Y'})
kv = rand(1,20); % ‘k’ Vector
tv = linspace(0, 20, numel(kv)); % Time Vector Matching ‘k’
k = @(t) interp1(tv, kv, t, 'linear', 'extrap'); % Function Interpolating ‘k’ To Specific Time
M = @(t,Y)[Y(2);-k(t).*Y(1)]; % Output Of 'matlabFunction’ (Needs To Be Copied, And Pasted In The Code After The 'k' Function Is Defined)
[T,Y] = ode45(M,[0 20],[1/2 1]);
figure
plot(T, Y)
grid
xlabel('T')
ylabel('Amplitude')
legend(string(Subs))
Use your own ‘kv’ vector. The rest of the code should adapt to it, whatever it is.
6 个评论
Mark Anslow
2020-2-24
Star Strider
2020-2-24
As always, my pleasure!
J. Alex Lee
2020-2-24
As long as you are allowed to make k a function handle, can you simply encode the equation that generates kv and tv?
k = @(t)(exp(t))
For example, would be better than
tv = linspace(0,10,100);
kv = exp(tv);
k = @(t) interp1(tv, kv, t, 'linear', 'extrap'); % Function Interpolating ‘k’ To Specific Time
Or is there something about symbolic toolbox that it won't work this way?
Star Strider
2020-2-24
@J. Alex Lee —
‘... where k(t) is an array of values ...’
So your idea is not appropriate here.
J. Alex Lee
2020-2-24
oops guess i need help with reading skills. i saw a similar question before, where "k(t)" in the function sense was being confused with "k(t)" in the index sense. Hope this isn't one of those cases and one really needs the discretized time-dependence.
Star Strider
2020-2-24
Noted.
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