Taylor and Volterra Series ⤶

Sep 20, 2017

In the theory of computational mechanics, there are a few operations used that are not taught in Calculus 101, which can be confusing without taking a lecture in calculus of variations. One of them is taking variations (a.k.a. Gateaux derivatives), akin to taking directional derivatives, but with functions of functions called functionals.

You need to take variations when you are linearizing a nonlinear problem for the purpose of solving with a numerical scheme. Linearization is the process of expanding a function or functional into a series, and discarding terms that are of order higher than linear—i.e. quadratic, cubic, quartic, etc. These expansions are called Taylor for functions, and Volterra for functionals.

Taylor Series

A function $f:\IR\to\IR$ can be expanded about a point $\bar{x}$ as a power series:

\begin{aligned} f(x) &= f(\bar{x}) + \frac{\dif f}{\dif x}\evat_{\bar{x}} \frac{(x-\bar{x})}{1!} + \frac{\dif^2 f}{\dif x^2}\evat_{\bar{x}}\frac{(x-\bar{x})^2}{2!} + \frac{\dif^3 f}{\dif x^3}\evat_{\bar{x}}\frac{(x-\bar{x})^3}{3!} + \cdots \\ &= \suml{n=0}{\infty} \frac{\dif^n f}{\dif x^n} \evat_{\bar{x}}\frac{(x-\bar{x})^n}{n!} \end{aligned}

Letting $x$ be a perturbation $\var x$ from the expansion point $\bar{x}$, that is $x\to\bar{x}+\var x$, the series can also be phrased as follows

\begin{aligned} f(\bar{x}+\var x) &= f(\bar{x}) + \frac{\dif f}{\dif x}\evat_{\bar{x}} \frac{\var x}{1!} + \frac{\dif^2 f}{\dif x^2}\evat_{\bar{x}}\frac{\var x^2}{2!} + \frac{\dif^3 f}{\dif x^3}\evat_{\bar{x}}\frac{\var x^3}{3!} + \cdots \\ &= \suml{n=0}{\infty} \frac{\dif^n f}{\dif x^n} \evat_{\bar{x}}\frac{\var x^n}{n!} \end{aligned} \label{eq:2}

This is what is taught in Calculus 101 and everyone knows. Now for the part that you may have missed:

Variation

Let $X$ be the space of functions $\IR\to\IR$. The variation of a functional $F\in X$ is defined as

$$$\boxed{ \varn{F(u)}{u}{v} := \lim_{\eps\to 0} \frac{F(u+\eps v) - F(u)}{\eps} \equiv \deriv{}{\eps} F(u + \eps v) \evat_{\eps = 0} }$$$

where $v \in X$ is called the perturbation of the variation. This operation is analogous to taking the directional derivative of a function.

Shorthand notation

When working with variational formulations, writing out variations can be a bit of a hassle if there are many symbols involved. Therefore we use the following shorthand for variations:

$$$\Var F := \varn{F(u)}{u}{v}$$$

Here, we assume that there is no chance of confusing the varied function or perturbation. We use this shorthand in contexts where the perturbation does not play an important role.

The shorthand for evaluation is

$$$\bar{F} := F(\bar{u}) \eqand \bar{\Var} F := \varn{F(u)}{u}{v}\evat_{\bar{u}}$$$

where there is no risk of confusion for $\bar{u}\in X$.

Volterra Series

Let $X$ be the space of functions $\IR\to\IR$. Analogous to the Taylor series, a functional $F\in X$ can be expanded about a point $\bar{u}$ as a power series:

\boxed{ \begin{aligned} F(\bar{u}+v) &= F(\bar{u}) + \frac{1}{1!} \varn{F(u)}{u}{v}\evat_{\bar{u}} + \frac{1}{2!} D^2_u F(u) \dtp v^2 \evat_{\bar{u}} + \frac{1}{3!} D^3_u F(u) \dtp v^3 \evat_{\bar{u}} + \cdots \\ &= \suml{n=0}{\infty} \frac{1}{n!} D^n_u F(u) \dtp v^n \evat_{\bar{u}} \end{aligned} } \label{eq:6}

where $v\in X$ is the perturbation of the expansion. This is called the Volterra series expansion of $F$. Verbally, the Volterra series expansion of a functional about a function is the infinite sum of the variations of the functional with increasing degree, evaluated at that function, each divided by the factorial of the degree.

In shorthand notation, the expansion is rendered

$$$\boxed{ F = \bar{F} + \frac{\bar{\Var} F}{1!} + \frac{\bar{\Var}^2 F}{2!} + \frac{\bar{\Var}^3 F}{3!} + \cdots } \label{eq:7}$$$

To me, there is an elegance in \eqref{eq:7} that is not reflected in \eqref{eq:2} or \eqref{eq:6}.