Re: can you solve these difference equations faster?
From: Travis Willse (twillse_at_gladstone.uoregon.edu)
Date: 02/07/05
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Date: Mon, 07 Feb 2005 00:46:49 -0800
lucy,
You likely don't need to invoke matrix algebra for most difference
equation problems.
Note that your array {U_0,...,U_7} is invariant under the transformation
i -> 7-i:
Under the transformation, we have
U_7, U_0,
U_(7-i) = 0.5*U_(6-i)+0.5*U_(8-i)+1
Evidently, the array is symmetric under the transformation
(specifically, the array is the same forward and backward). Thus:
U_1 = U_6
U_2 = U_5
U_3 = U_4
Also, solving for U_(i+1) in the recursive definition, we have for i =
1,...,6 that
U_{i+1} = 2*U_i-U_{i-1}-2
In the case i = 1,
U_2 = 2*U_1-U_0-2 = 2*U_1-2
In the case i = 2,
U_3 = 2*U_2-U_1-2 = 2*(2*U_1-2)-U_1-2 = 3*U_1-6
In the case i = 3,
U_4 = 2*U_3-U_2-2
But U_4 = U_3, so
U_3 = U_2+2
Substituting with the expressions in U_1 derived above,
3*U_1-6 = (2*U_1 - 2) + 2 = 2*U_1
Solving,
U_1 = 6
You now have the first two terms of the array, so you can apply the
recursive definition directly. The solution is:
{0,6,10,12,12,10,6,0}
Cheers,
Travis
lucy wrote:
> Hi all,
>
> I have the following difference equations:
>
> U_0=0, U_7=0,
> U_i=0.5*U_(i-1)+0.5*U_(i+1)+1 for i=1, 2, 3, 4, 5, 6.
>
> Now want to find U_i for i=1.
>
> I tried to solve the matrix equations, it was really painful(this is not a
> problem that is supposed to use Matlab.)
>
> I have also tried to use Z-transform to do it. But that was still painful:
>
> I got U_i=A+B*i+C*i^2
>
> And finding the third initial condition to determine the three coefficients
> is very tedious...
>
> Anybody has some good tricks?
>
> Thanks a lot!
>
> -L
>
>
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