How to integrate linear system of vectorial equations?

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!

[Wan Ji]

Use ode45 or other ode tools

Answer 1

[Star Strider]

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

.

[Anshuman Pal]

This is great! But I should have been clearer. κ is not a constant, it is a function of s: whose values are stored in some array. How would the above code need to be changed to accommodate that?

[Star Strider]

Thank you!

If κ is a vector, use interp1 to interpolate it to the current value of s using the technique described in detail in ODE with Time-Dependent Terms (using s instead of t as the variable to interpolate), however if it is a matrix, this becomes more complicated because the array sizes have to match, and that could be a problem.

.

[Anshuman Pal]

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]

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.

.