How to integrate linear system of vectorial equations?

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Hello,
I have the following system of vectorial equations that describe a curve in space r, its tangent vector t and its normal $n$, parametrised by the arc length s:
Suppose I have the function given as a vector of values, as well as initial values of r, t and n. Then is there some simple way (or package) for numerically integrating this system of vectorial equations?
Thank you very much!

Accepted Answer

Star Strider
Star Strider on 20 Aug 2021
syms kappa n(s) r(s) s t(s) r0 t0 n0 Y
Eqn = [diff(r) == t; diff(t) == -r + kappa*n; diff(n) == -kappa*t];
rtn = dsolve(Eqn, r(0)==r0, t(0)==t0, n(0)==n0)
rtn = struct with fields:
t: [1×1 sym] r: [1×1 sym] n: [1×1 sym]
t = simplify(rtn.t, 500)
t = 
r = simplify(rtn.r, 500)
r = 
n = simplify(rtn.n, 500)
n = 
Alternatively:
[VF,Subs] = odeToVectorField(Eqn)
VF = 
Subs = 
rtnfcn = matlabFunction(VF, 'Vars',{s,Y,kappa})
rtnfcn = function_handle with value:
@(s,Y,kappa)[kappa.*Y(3)-Y(2);Y(1);-kappa.*Y(1)]
Use ‘rtnfcn’ with the approopriate numeric ODE solver (for example 0de45, ode15s) depending on the magnitude of κ.
For example:
sspan = linspace(0,10); % Vector Of 's' Values
initconds = rand(3,1); % Initial Conditions
[s,rtn] = ode15s(@(s,rtn,kappa), sspan, initconds); % Integrate
figure
plot(s, rtn)
grid
.
  4 Comments
Anshuman Pal
Anshuman Pal on 22 Aug 2021
Thank you! A couple more comments:
1) Is there a typo in `ode15s(@(s,rtn,kappa), sspan, initconds)`? Shouldn't it be `ode15s(@rtnfcn, sspan, initconds)`?
2) Can I use `rtnfcn` with a boundary-value solver like `bvp4c`?
Star Strider
Star Strider on 22 Aug 2021
As always, my pleasure!
Is there a typo in `ode15s(@(s,rtn,kappa), sspan, initconds)`?
There is. It should be:
[s,rtn] = ode15s(@(s,rtn) rtnfcn(s,rtn,kappa), sspan, initconds); % Integrate
Can I use `rtnfcn` with a boundary-value solver like `bvp4c`?
Probably. One addition would be to create ‘kappa’ as an anonymous function, for example with ‘s’ as the independent variable and ‘kapa’ as the dependent variable:
kapamtx = [s(:) kapa(:)];
kappa = @(s) interp1(kapamtx(:,1), kapamtx(:,2), s);
It would be necessary either to keep ‘s’ within the limits of ‘kapamtx(:,1)’ in order to avoid either extrapolating or returning NaN values for ‘s’ outside that range.
.

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