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Return to Part IV of the course APMA0340
Introduction to Linear Algebra
The most differential equations can’t be solved explicitly in terms of finite combinations of simple familiar functions. In this section, we develop an algorithm for solving a certan class of differential equations based on the
power series method. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
Power series solution method has been traditionally used to solve linear differential equations.
Although the power series method is not, generally speaking, suitable for
direct integration of many differential equations, it has many modifications
and it is a part of theoretical analysis and practical numerical procedures.
The main obstacle in its direct application lies in difficulty of
determination of the radius of convergence. It is well known that a power
series \( S(x) = \sum_{n\ge 0} a_n \left( x- x_0 \right)^n
\) converges within a circle
\( \left\vert x - x_0 \right\vert < R \) of
radius R. However, its value is determined by behavior of
coefficients \( a_n \) at infinity. Since
\( R^{-1} = \lim_{n\to \infty} \sqrt[n]{a_n} , \)
the radius of convergence depends on how fast the coefficients grow. However,
if sum-function S(x) is unknown because in our applications it
is usually the unknown solution of the differential equation, it is impossible
to find the radius of convergence.
Nevertheless, the power series method can be used practically at least is some
small neighborhood of the center. We are going to demonstrate this approach in
a sequence of examples.
Example:
We analyze the rotation of a free rigid body that is suspended in space, for example, a satellite or a planet. Recall that a rigid body is a collection of finite points that the distance between the points is fixed. So we consider a body that is rotated about some point, which could take as the center of mass. Then the position of a particle in the body can be written as
where Δr_{i} is
the position measured from the center of mass. For any point r in the
body
\begin{align*}
\frac{{\text d} {\bf r}}{{\text d}t} &=r_a \,\frac{{\text d}R_{ab}}{{\text d}t}\, {\bf e}_b \qquad \mbox{in the space frame} \\
&= r_a \,\frac{{\text d}{\bf e}_1 (t)}{{\text d}t} \qquad \mbox{in the body frame} .
\end{align*}
The components of the orthogonal matrix
\( \displaystyle \left[ R_{ab} \right] \) satisfy
\( \displaystyle {\bf e}_a (t) = R_{ab} (t)\,{\bf e}_b . \) The angular velocity vector ω(t) has components
ω_{a}e_{a}.
Then examining the
kinetic energy, we get
because \( \displaystyle \sum_i m_i \Delta {\bf r}_i = 0. \)
So we find that the dynamics separates into the motion of the center of mass
R, together with rotation about the center of mass. For rotational process, the angular momentum must be conserved:
\[
\frac{{\text d} {\bf L}}{{\text d}t} = 0 ,
\]
where \( \displaystyle {\bf L} = L_1 {\bf e}_1 + L_2 {\bf e}_2 + L_3 {\bf e}_3 = I_1 \omega_1 {\bf e}_1 + I_2 \omega_2 {\bf e}_2 + I_3 \omega_3 {\bf e}_3 , \) I_{k} are the principal moments of inertia and ω_{k} are the components of the angular velocity about the principal axes.
Using \( \displaystyle L_a = I_{ab} \omega_b , \)
we apply to every point and obtain the equations of motion (Euler's equations)
The spin of the Earth causes it to bulge at the equator so it is no longer a
sphere but can be treated as a symmetric top.
It is an oblate ellipsoid, with I_{3} > I_{1}
and is spherical to roughly 1 part in 300, meaning
Of course, we know the magnitude of the spin ω_{3} is just
1 (day)^{-1}. This information is
enough to calculate the frequency of the Earth’s wobble, which takes 433 days
and is known as the Chandler wobble.
Suppose that the body is freely rotating symmetrically about the z-axis.
Then I_{xx} = I_{yy} = I_{⊥}.
Likewise, we can write I_{zz} = I_{∥}
≠ I_{⊥}, in general. In this case, Euler's equations
become
where \( \displaystyle \omega_z \left( I_{\parallel}/I_{\perp} -1 \right) . \) Since this system of equations is linear, its solution is not hard to determine:
where ω is a constant.
Thus, the projection of the angular velocity vector onto the x-y plane has the fixed length ω, and rotates steadily about the z-axis with angular velocity Ω.
■
Example:
The famous Lorenz system of equations is given as
where x, y, and z are respectively proportional to the
convective velocity, the temperature difference between descending and
ascending flows, and the mean convective heat flow; r is the bifurcation parameter, and the real coefficients σ and b are taken the numerical values r = 28, σ = 10, and b = 8/3 where
the system exhibits chaotic behavior.
The explicit solution to the Lorenz system we assume to be represented by
power series:
These recurrence relations are used for algorithmic computations over time mesh.
■
Abbasbandy, S. and ervillier, C., Analytic continuations of Taylor series and the two-point boundary value problem of some nonlinear ordinary differential equations, [arXiv:1104.5073v1] (2011).
Asaithambi, A., A series solution of the Falkner–Skan equation using the crocco–wang transformation, International Journal of Modern Physics, 2017, Vol. 28, No. 11, 1750139; https://doi.org/10.1142/S012918311750139X
Noor, N.F.M., Hashim, I., Noorani, M.S.M., A Note On The Accuracy Of The Adomian
Decomposition Method Applied To The Chaotic Lorenz System, Proceedings of the 2nd IMT-GT Regional Conference on mathematics and Applications, University Saint Malaysia, Penang, June 13--15, 2006.
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