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subsection 'Difference equation' in Strang's linear algebra section 6.3

Mathematics Asked by oscarmetal break on December 15, 2021

In this section, Strang converts the constant-coefficient differential equation into linear algebra in order to solve them. I was in trouble reading the difference equation in this section which demands to provide an alternative solution to Example 3 which is a second differential equation of the motion around a circle.

Here is Example 3:

Motion around a circle with $y” + y = 0$ and $y = cos{t}$.

And I have no problem to solve this by applying the usual method that converts the equation into linear algebra. However, I can’t understand the following that applying the difference equation and converts it to linear algebra for solving the same example. Here is the description:

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I can’t understand this entire section perhaps it is because I have never studied the difference equation before but I really want to understand this part because it looks important. Therefore I wonder if anyone can provide me a clear explanation. Thanks a lot.

One Answer

The author is using finite differences to explain its usefulness and limitations by leveraging on the concepts described previously. The example is still $y''+y=0$ which was re-written as $y''=-y$ in the first paragraph of the section titled "Difference Equations".

The value of $y''$ is approximated by numerically by the Eq. 11 (FCB). Then there is a renaming of variables $Z$ stands for the discretized first derivative (not the second) of the solution, while $Y$ is the discretized solution (this is no different from the vector $textbf{u}$ in Eq. 10 -and I mention the vector, not the equation as a whole).

The author abandon the 2nd derivative in favor of the phase portrait diagram that contains $u=(y,y')$, see Fig. 6.3. as a qualitative representation of the solution space of the ODE. Then, he uses the forward scheme to determine $Z$ and $Y$ at the time step $n+1$ knowing the solution at the step $n$ in an incremental fashion in Eq. 11 F (which can be obtain from Taylor expansion by getting rid of higher order terms). Eq. 12 contains the same information as Eq. 11 in matrix form where $U_n=(Y_n : Z_n)^T$, the vector on the LHS stands for the homologous vector $U_{n+1}$ and the matrix contains the coefficients multiplying $Y_n$ and $Z_n$ on the RHS of Eq. 11F.

The construction of the diagrams is essentially analogous to an integration where the pair $(Y_n,Z_n)$ denotes the point $n$ in the phase portrait (See figures 6.3 and 6.4) The author also claims that there are numerical errors that produce inaccurate results (ideally all points should lie in a circle) for a fixed steplength equivalent to 32 pts along the perimeter of the circle. The errors are reduced if the number of points are increased.

Answered by Basco on December 15, 2021

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