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MathGroup Archive 2001

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Re: Integration problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27452] Re: [mg27417] Integration problem
  • From: BobHanlon at aol.com
  • Date: Sun, 25 Feb 2001 20:55:59 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Tx1[i_] := ((1/a)*(1 - Abs[x - i*a]/a))*(UnitStep[x - (i - 1)*a] - 
          UnitStep[x - (i + 1)*a]);

Ty1[j_] := (UnitStep[y - (j - 1)*b] - UnitStep[y - j*b]);

You need to define L and W.

L = 5; W = 9;

JsxT = Table[Ty1[p]*Tx1[m], {m, L-1}, {p, W}];

JsxSum = Simplify[Sum[JsxT[[m, p]], {m, 1, 4}, {p, 1, 9}]];

The Simplify is used to speed up calculation of the following integral

Clear[soln];

soln[a_ /; a != 0, b_] :=  
    Evaluate[FullSimplify[Integrate[JsxSum, {x, 0, a}, {y, 0, b}], 
        Element[a, Reals]]] ;

Table[soln[a, b], {a, -3, 3}, {b, -3, 3}]

{{63/2, 21, 21/2, 0, 21/2, 21, 63/2}, {63/2, 21, 21/2, 0, 21/2, 21, 63/2}, 
{63/2, 21, 21/2, 0, 21/2, 21, 63/2}, 
  {soln[0, -3], soln[0, -2], soln[0, -1], soln[0, 0], soln[0, 1], soln[0, 2], 
soln[0, 3]}, {3/2, 1, 1/2, 0, 1/2, 1, 3/2}, 
  {3/2, 1, 1/2, 0, 1/2, 1, 3/2}, {3/2, 1, 1/2, 0, 1/2, 1, 3/2}}

a /: a != 0 = True;

For a > 0,

Simplify[soln[a, b], a > 0] /. {UnitStep[a, b_] -> UnitStep[b], 
      UnitStep[-a, b_] -> 0} /. UnitStep[-b] -> UnitStep[b] - Sign[b]

(1/2)*b*Sign[b]

Since

FullSimplify[b*Sign[b] == Abs[b], Element[b, Reals]]

True

Then,  soln[a_?Positive, b_] := Abs[b]/2;

For a <0,

FullSimplify[soln[a, b], a < 0] /. {UnitStep[a, b_] -> 0, 
        UnitStep[-a, b_] -> UnitStep[b]} /. 
    UnitStep[-b] -> UnitStep[b] - Sign[b]//Simplify

(21/2)*b*Sign[b]

Then, 

Clear[soln];

soln[a_?Positive, b_] := Abs[b]/2;

soln[a_?Negative, b_] := 21*Abs[b]/2;


Bob Hanlon

In a message dated 2001/2/25 1:29:29 AM, drek1976 at yahoo.com writes:

>I am presently experiencing some problems with the use of integration in
>Mathematica, and I do not understand what could be wrong.
>I defined 2 functions as follow:
>
>Tx1[i_] := ((1/a)*(1 - Abs[x - i*a]/a))*(UnitStep[x - (i - 1)*a] -
>UnitStep[x - (i + 1)*a])
>Ty1[j_] := (UnitStep[y - (j - 1)*b] - UnitStep[y - j*b])
>
>where a, b are any assigned real values.
>
>I then formed a matrix as follows:
>
>JsxT := Table[Ty1[p]*Tx1[m], {m, L - 1}, {p, W}]
>
>which is then followed by:
>
>JsxSum = Sum[JsxT[[m,p]], {m, 1, 4}, {p, 1, 9}]
>
>I then try to integrate JsxSum with respect to x and y in the following
>way:
>
>Integrate[JsxSum, {x, 0, a}, {y, 0, b}]
>
>When Mathematica (I'm using version 4) tries to work out this last
>expression, it outputs the following message:
>
>"Unique::usym : -0.43661971830985913 is not a symbol or a valid sumbol
>name"
>
>before giving an output.
>
>This problem occurs whatever values I may use for a and b.
>Does someone know why?
>


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