2.1 Algorithm flow chart

This chart gives the algorithm for second order ode where expansion point is regular singular point. This uses Frobenius series. There are 4 different cases to consider.

Figure 3: Series algorithm for second order ode for regular singular point

This gives the algorithm for second order ode where expansion point is ordinary.

if \(f(x,y,y')\) is analytic at expansion point \(x_0\) then this means \(x_0\) is an ordinary point. We Apply Taylor series defintion directly to find the series expansion. Let \(y_0=y(x_0),y'(x_0)=y'_{0}\) and

\begin{align*} y &= y_0 + y'_0 + \sum _{n=0}^{\infty } \frac { x^{n+2}}{(n+2)!} \, F_n(x,y) \bigg \vert _{\substack {x_0\\y_0\\y'_0}} \end{align*}

Where

\begin{align*} F_{0} & =f\left ( x,y,y^{\prime }\right ) \\ F_{n} & =\frac {d}{dx}\left ( F_{n-1}\right )\\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) F_{0} \end{align*}

Ordinary point and regular singular point are supported. irregular singular point support will be added in the future. Expansion around point other than zero is also supported, including initial conditions. All 4 cases of regular point are supported, these are when the roots on indicial equation are repeated, or differ by an integer, or differ by non integer. case of Complex roots of indicial equation is also supported. Only second order and first order series solution is supported. Higher order ode support will be added in the future.