MATH528

MATH528 Lesson08: ODE series solution. Special functions

We now embark upon a general approach for ODEs that do not have an established analytical procedure to find a solution such as separating variables or variation of parameters. Such equations arise repeatedly in mathematical physics and leads to the topic of special functions, an extension of the elementary functions encountered in calculus such as sine or cosine.

Structure of the real numbers $(ℝ, +, \cdot)$
Series expansions for functions
Series solutions of ODEs

Real series

Remark. The real numbers are a complete, ordered field $(ℝ, +, \times)$

Remark. Power series are simply an infinite sequence of the operations defined in $ℝ$

S_{n} (x) = \sum_{j = 0}^{n} a_{j} x^{j}, n \in ℕ

Example. The following are power series expansions of common functions

\begin{array}{l} e^{x} = \frac{1}{0!} 1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{n!} x^{n} + \dots \\ \sin x = \frac{1}{1!} x - \frac{1}{3!} x^{3} + \dots + \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1} + \dots \\ \cos x = \frac{1}{0!} 1 - \frac{1}{2!} x^{2} + \dots + \frac{{(- 1)}^{k}}{(2 k)!} x^{2 k} + \dots \\ \ln x = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots + \frac{{(- 1)}^{k - 1}}{k} x^{k} + \dots + \end{array}

Remark. Power series can also be interpreted as a sequence of scalar products

S_{n} (x) = 𝒂^{T} 𝒃 (x), 𝒂^{T} = (\begin{array}{llll} a_{0} & a_{1} & \dots & a_{n} \end{array}), 𝒃 (x) = {(\begin{array}{llll} 1 & x & \dots & x^{n} \end{array})}^{T}

Elementary introduction to power series method

The power series method to solve ODEs onsist of:

Introducing a representation $y (x) =$ $\sum_{j = 0}^{\infty} a_{j} x^{j}$
Replacing the representation into the ODE of interest and identifying coefficients of powers of $x$

Example. $y^{'} = y$ , $y (0) = y_{0}$ , Try $y = a_{0} + a_{1} x + \dots + a_{n} x^{n} + \dots +$

\begin{array}{rcl} y^{'} = a_{1} + \dots + (n + 1) a_{n + 1} x^{n} & = & a_{0} + \dots + a_{n} x^{n} + \dots + = y \Rightarrow \\ a_{1} = a_{0}, & a_{2} = \frac{1}{2} a_{1}, \dots & a_{n + 1} = \frac{1}{n + 1} a_{n} \Rightarrow \\ y (x) = & (1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots) a_{0} & = e^{x} y (0), a_{0} = y (0) . \end{array}

Remark. The coefficients of the powers of $x$ can be identified term by term because they are linearly independent

W_{n} = | \begin{array}{l} \begin{array}{lllll} 1 & x & x^{2} & \dots & x^{n} \\ 0 & 1 & 2 x & \dots & n x^{n - 1} \\ 0 & 0 & 2 & \dots & n (n - 1) x^{n - 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & n! \end{array} \end{array} | = n! \cdot (n - 1)! \cdot \dots \cdot 1 \neq 0

Trigonometric functions from series solution

Example. $y^{''} = - y$ , Try $y = a_{0} + a_{1} x + \dots + a_{n} x^{n} + \dots +$

\begin{array}{rcl} y^{''} = 2 \cdot 1 \cdot a_{2} + 3 \cdot 2 \cdot a_{3} x + \dots + (n + 2) \cdot (n + 1) \cdot a_{n + 2} x^{n} & = & - (a_{0} + a_{1} x + \dots + a_{n} x^{n} + \dots) = y \Rightarrow \\ a_{2} = - \frac{1}{2 \cdot 1} a_{0}, a_{3} = - \frac{1}{3 \cdot 2} a_{1}, \\ y (x) = a_{0} (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots -) & + & a_{1} (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots -) \\ y (x) & = & a_{0} \cos (x) + a_{1} \sin (x) \end{array}

Remark. The equation $Y^{''} + Y = 0$ results from separation of variables applied to $\nabla^{2} u = 0$ , $u (x, y, z) = X (x) Y (y) Z (z)$ expressed in Cartesian coordinates $(x, y, z)$ .

Recall:

\begin{array}{rcl} \nabla & = & \partial_{x} 𝒆_{x} + \partial_{y} 𝒆_{y} + \partial_{z} 𝒆_{z} \\ \nabla u = grad u & = & \partial_{x} u 𝒆_{x} + \partial_{y} u 𝒆_{y} + \partial_{z} u 𝒆_{z} = 𝒗 = v_{x} 𝒆_{x} + v_{y} 𝒆_{y} + v_{z} 𝒆_{z} \\ \nabla \cdot 𝒗 = div 𝒗 & = & \partial_{x} v_{x} + \partial_{y} v_{y} + \partial_{z} v_{z} \\ \nabla^{2} u & = & \partial_{x}^{2} u + \partial_{y}^{2} u + \partial_{z}^{2} u \end{array}

div 𝒗 = {lim}_{| Ω | \to 0} \frac{\oint_{\partial Ω} 𝒗 \cdot d 𝝈}{| Ω |}, \partial Ω = boundary of Ω .

Power series convergence

The radius of convergence can be determined from series coefficients:

\begin{array}{ll} R^{- 1} = {lim}_{n \to \infty} {| a_{n} |}^{1 / n} & R^{- 1} = {lim}_{n \to \infty} {| \frac{a_{n + 1}}{a_{n}} |}_{} \end{array}

if limits exist and $R \neq 0$ .

Example. $e^{x} = 1 + x + \dots + \frac{x^{n}}{n!}$ , $R^{- 1} = {lim}_{n \to \infty} \frac{1 / (n + 1)!}{1 / n!} = {lim}_{n \to \infty} \frac{1}{n + 1} = 0 \Rightarrow R = \infty$ , converges $\forall x$

Example. $\frac{1}{1 - x} = 1 + x + x^{2} + \dots +, R^{- 1} = 1 \Rightarrow R = 1$ , converges for $| x | < 1$

In general a series solution with $R > 0$ exists for ODEs of form $y^{''} + p (x) y^{'} + q (x) y = 0$ if $p, q$ have power series representations.

Laplacian in curvilinear coordinates

Remark. $\sin (x), \cos (x)$ obtained from separation of variables applied to $\nabla^{2} u = 0$ in Cartesian coordinates

Separation of variables procedure:

Assume that $u (x, y) = X (x) Y (y)$ and replace in $\nabla^{2} u = 0$ to obtain
$X^{''} Y + X Y^{''} = 0 \Rightarrow \frac{X^{''}}{X} = - \frac{Y^{''}}{Y}$
On the left is a function only of $x$ , $f (x) = X^{''} / X$ . On the right there is a function only of $y$ , $g (y) = - Y^{''} / Y$ . The only way equality could hold for all $x, y$ is for both $f$ and $g$ to be constants
$\frac{X^{''}}{X} = - \frac{Y^{''}}{Y} = - n^{2}$
This leads to ODEs: $X^{''} + n^{2} X = 0$ , $Y^{''} - n^{2} Y = 0$ with solutions that are the trigonometric and hyperbolic functions
$\begin{array}{l} X_{n} (x) = a_{n} \cos (n x) + b_{n} \sin (n x) \\ Y_{n} (y) = c_{n} \cosh (n x) + d_{n} \sinh (n x) \end{array}$
The original PDE is linear hence a general solution is $u (x, y) = \sum_{n = 0}^{\infty} X_{n} (x) Y_{n} (y)$ .