Differentiation of an Integral Function

Hello,

I have this problem at hand to solve but it's taking longer than I envisaged to solve.

Let A be a function with respect to x,y,z i.e A(x,y,z)

B is the integration of A(x,y,z) w.r.t to time t from t_0 to t_f.

I need to solve dz/dt = B( x(t),y,z(t)).

A(x,y,z), where A can be anything, if possible a constant but a function of x,y,z.
x is a function of time, i.e x(t)
z also is a function of time i.e z(t)

Let just say B = integral(@(t) A,t_0,t_f)

Then I need to solve dz/dt = B(x,y,z);

I have tried both numerical means to solve this but what I am getting is not making sense.

Please advise what I can do.

[Walter Roberson]

Do you mean A is A(x(t), y(t), z(t)) and B = int(A(x(t), y(t), z(t)), t, t_0, t_f) ?

Is the question how to find the function A(x(t), y(t), z(t)) such that dz(t)/dt = int(A(x(t), y(t), z(t)), t, t_0, t_f) ? If so, then for arbitrary x(t), y(t), z(t) ?

[Shozeal]

Hi Walter, Yes, that is my question. I wrote a numerical code for it but I don't think it's correct.

if true
 m=1;
t_0=0;
t_final=1;
NT=100;
delta_t = (t_final-t_0)/NT;
f(1)=20;  % the fist value of f(m)
F(1)=0;  % The initial value of f at t=0
disp([' delta_t','  f(1)','  F(1)'])
disp([delta_t,f(1),F(1)]);
while m<=NT
    f(m+1)=F(m)+f(m);
   F(m+1)=F(m)+f(m+1)*delta_t
  m=m+1;
  end
t_m=t_0:delta_t:t_final;
plot(t_m,f)
end

I also saw something on functional derivative, but I don't know if it can be applied here. Not sure how to even apply to the problem at hand.

[Walter Roberson]

I do not see any x(t), y(t) or z(t) there?

There is not going to be a closed form solution for arbitrary x(t), y(t), z(t) . Perhaps there are some solutions for particular x(t), y(t), z(t)

[Shozeal]

Yes, there are some solutions for particular x(t), y(t), z(t) at each particular point in time. The differential of z (dz/dt) is to be determined w.r.t to the integral of A( x(t),y(t),z(t) ) at this particular time.

If possible, can we just pick a particular solution for x(t),y(t),z(t) at this particular time in order to solve the equation?

What I wrote up there is a general way to see if something can be done numerically.

[Walter Roberson]

If we look at

dz(t)/dt = int(A(x(t), y(t), z(t)), t, t_0, t_f)

then the right hand side is going to be constant relative to t, because of the definite integral that substitutes t_0 and t_f for t.

But the left hand side, dz(t)/dt would generally be dependent on t, except in the case where dz(t)/dt is a constant, in which case z(t) would have to be constant1*t+constant2 in form.

So the situation is not possible unless z(t) is of that form.

[Shozeal]

"_So the situation is not possible unless z(t) is of that form._" With this, I can differentiate z(t) w.r.t time t. But since the right hand side is a constant, won't that give an error?

[Walter Roberson]

syms x(t) y(t) z(t) C1 C2
z(t) = C1 * t + C2;
lhs = diff(z(t),t);  %would be C1
syms A(X, Y, Z) t_0 t_f
rhs = int(A(x(t), y(t), z(t)), t, t_0, t_f);
eqn = lhs == rhs

No error (but also not much you can do with this.)

Note that for this purpose, C1 and C2 might be related to additional variables other than t: they just have to be independent of t, not of any other variable.

[Shozeal]

Thank you, Walter.

Let me see how I can develop this to make it compatible with what I am working on.