Help to approximate this integral...

Hello,

I am wondering if it is possible to have an approximation of this integral

I have only an approximation

of f which is a solution of an ODE.

Many thanks,

Maher.

[darova]

f is a constant or function?

[Peter Clifford]

No f is asolution of an ODE equation, I have an approximation of it by ode45 method.

[darova]

but what f depends on? If we calculate integral x = [0, 3], what will be f (for example y = 1.56)

have you relation ?

Answer 1

[David Goodmanson]

Hi M^2,

I am assuming that y(t) is strictly monotonic increasing or decreasing so that the inverse exists. It's good if the y and t arrays are not too sparse. Depending on how time comsuming the ode solver is, in the case of ode45, for example, you can output a lot of points by specifying a grid of points in the tspan input variable rather than just an upper and lower limit.

Assuming that t and y are reasonably dense an that the y array starts at 0, then

V(x) = cumtrapz(exp(-t),y) 

There are better ways to do an indefinite integral, but this gives a reasonable result if exp(-t) is fairly smooth,

[Peter Clifford]

Thanks for the help.

You took ?

what are the relations between and t in

V(x) = cumtrapz(exp(-t),y)

I have this

[t, f] = ode45(@odefunc, tspan, [0; 1], options);

How to deduce an approximation of V defined by :

where is the inverse function of f which is strictly increasing and

Maher.

[David Goodmanson]

HI Maher,

I should have been more explicit. Yes, I took y = f(t). Then

t = f^-1(y)

This is consistent with just using the line

[t, y] = ode45(@odefunc, tspan, [0; 1], options);

There are two side-by-side output arrays, t and y = f(t). These match up point-by-point. But if both of these arrays are monotonic there is no reason you can't consider y to be the independent variable, t the dependent variable and redefine the arrays as y and t(y), the inverse function of y. Since t and y match up point-by point, t is the same as t(y) and

Int {0,x} e-t(y) dy ---> Int {0,x} e-t dy.

With y as the independent variable you can do the definite integral from 0 to x by trapz (assuming the y array starts at 0) and the indefinite integral by cumtrapz to give the function V(x).

Matlab has plenty of integral evaluation capability when given a function f(x) that can be evaluated at arbitrary points in x.

I hope I'm wrong, but if what you have is an independent and a dependent array at fixed points, I don't know that Matlab has anything easily available that's better than the lamentably primitive trapz and cumtrapz.