Mathematica 9 is now available
Student Support Forum
-----
Student Support Forum: 'Integrating parametric function' topicStudent Support Forum > General > Archives > "Integrating parametric function"

Help | Reply To Topic
Author Comment/Response
Nicola
01/15/13 09:24am

I have to solve a double integral with parametrized functions, that is (written as a latex equation)
$$
\int_{-\infty}^\infty \int_0^\infty u(t)u(t-\Delta)\frac{1+e^{-2\int_{t-\tau-\Delta}^{t-\tau} y(s)^2 ds }}{2} d\Delta \dt
$$
that depends upon:
- the function u(t), that in my case is a squared pulse between 0 and 1, defined in Mathematica as
u[t_] = Piecewise[{{1, 0 <= t <= 1}}, 0];
- the function y(s), that in my case is a piecewise constant function between 0 and 1 with a discontinuity in t1, defined in Mathematica as
y[t_, y1_, y2_, t1_] =
Piecewise[{{y1, 0 <= t <= t1}, {y2, t1 < t <= 1}}, 0];
- the parameter \tau.

Consider the assumptions
$Assumptions =
1 > \[Tau] > 0 && y1 \[Element] Reals && y2 \[Element] Reals && 1 > t1 > 0
and also t1 + \[Tau] < 1. I'd like to solve the integral to have a function of \tau, y1, y2, t1.

Now, if I just substitute the definition of u(t) and y(s) and use Mathematica, I get I can't get the integral solved, as you can check with
Assuming[t1 + \[Tau] < 1,
Integrate[
Integrate[
u[t] u[t - \[CapitalDelta]] (1 +
Exp[-2 Integrate[
y[s, y1, y2, t1]^2, {s, t - \[CapitalDelta] - \[Tau],
t - \[Tau]}]])/2, {\[CapitalDelta],
0, +Infinity}], {t, -Infinity, +Infinity}]] // Simplify.

So, I have done "by hand" part of the integral, in particular the integral at the exponential argument, defining the funtion
IntY2[a_, b_, y1_, y2_, t1_] = Piecewise[{
{y1^2 b, a < 0 && 0 <= b < t1}, {y1^2 t1 + y2^2 (b - t1),
a < 0 && t1 <= b <= 1}, {y1^2 t1 + y2^2 (1 - t1), a < 0 && 1 < b},
{y1^2 (b - a),
0 <= a < t1 && 0 <= b < t1}, {y1^2 (t1 - a) + y2^2 (b - t1),
0 <= a < t1 && t1 <= b <= 1}, {y1^2 (t1 - a) + y2^2 (1 - t1),
0 <= a < t1 && 1 < b},
{y2^2 (b - a), t1 <= a <= 1 && t1 <= b <= 1}, {y2^2 (1 - a),
t1 <= a <= 1 && 1 < b}}, 0]
and calculating
Assuming[t1 + \[Tau] < 1,
Integrate[
Integrate[
u[t] u[t - \[CapitalDelta]] (1 +
Exp[-2 IntY2[t - \[CapitalDelta] - \[Tau], t - \[Tau], y1, y2,
t1]])/2, {\[CapitalDelta],
0, +Infinity}], {t, -Infinity, +Infinity}]] // Simplify

This elaboration works, and I get a solution. But doing by hand the substitution of u(t) and y(t) with some algebra I get
Integrate[(1 + Exp[-2 y1^2 \[CapitalDelta]])/2, {\[CapitalDelta], 0,
t - \[Tau]}, {t, \[Tau], t1 + \[Tau]}] +
Integrate[(1 + Exp[-2 y1^2 (t - \[Tau])])/2, {\[CapitalDelta],
t - \[Tau], t}, {t, \[Tau], t1 + \[Tau]}] +
Integrate[(1 + Exp[-2 y2^2 (t - \[Tau] - t1) - 2 y1^2 t1])/
2, {\[CapitalDelta], t - \[Tau], t}, {t, t1 + \[Tau], 1}] +
Integrate[(1 +
Exp[-2 y2^2 (t - \[Tau] - t1) -
2 y1^2 (t1 - (t - \[Tau] - \[CapitalDelta]))])/
2, {\[CapitalDelta], t - t1 - \[Tau], t - \[Tau]}, {t,
t1 + \[Tau], 1}] +
Integrate[(1 + Exp[-2 y2^2 \[CapitalDelta]])/2, {\[CapitalDelta], 0,
t - t1 - \[Tau]}, {t, t1 + \[Tau], 1}] // Simplify
that is different from the one calculated by Mathematica. Of course, probably I made a mistake somewhere that I cannot find. Anyway, I tried to substitute some parts of the original integral to get my formula, for example sustituting the constraint given by u(t), then u(t-\Delta), ecc checking each step of Mathematica. For example, with
Assuming[t1 + \[Tau] < 1,
Integrate[
Integrate[u[t]
u[t - \[CapitalDelta]] (1 +
Exp[-2 IntY2[t - \[CapitalDelta] - \[Tau], t - \[Tau], y1, y2,
t1]])/2, {\[CapitalDelta], 0, +Infinity}], {t, 0,
1}]] // Simplify
I get the same (Mathematica's, not mine) solution.
When I get to
Assuming[t1 + \[Tau] < 1,
Integrate[
Integrate[
u[t - \[CapitalDelta]] (1 +
Exp[-2 IntY2[t - \[CapitalDelta] - \[Tau], t - \[Tau], y1, y2,
t1]])/2, {\[CapitalDelta], 0, +Infinity}], {t, 0,
1}]] // Simplify
and I tried to make that solved, Mathematica cannot solve it (or at least it takes a lot of time, the previous integral it takes 20 s but in this one after 5 minutes it's not solved). Note that I just delete u[t] in the integral argument, but from the mathematical point of view it should be indifferent since t is integrated between 0 and 1.

I'd like to understand why Mathematica cannot solve this last formulation of the integral (and how can I fix it), because I want to check my solutions and also I have some more complex integrals that Mathematica cannot solve and I think the problem is the same.

Sorry for the long post,
thanks,
Nicola

URL: ,
Help | Reply To Topic