10.5 There and back again - an exploration of the liouville  (Page 3/10)

The inverse problem can be attacked by many means, including several numerical schemes. In order to apply the techniques described in [link] , it is necessary to transform [link] into the Sturm-Liouville equation

where $q$ is called the potential . It is the transformation between these equations that captures the focus of our research and of this paper.

The transformations

Changing from mass density to sturm-liouville potential

To transform [link] into the Sturm-Liouville equation [link] , it is necessary to employ a change of variables that is due to Joseph Liouville [link] . Liouville's transformation requires $\rho \in {\mathcal{C}}^2$ [0, L]. If that condition is satisfied, the transformation involves a change of dependent and independent variables. The independent variable $x$ is transformed into a new independent variable $t$ according to

(NB: This $t$ is completely unrelated to the $t$ variable used to denote the dependence on time in [link] .) The new dependent variable $w\left(t\right)$ is given by

where the variable $x$ on the right-hand side of this equation is meant to be interpreted as a function of $t$ whose definition is given implicitly by the integral [link] .

Taking the derivative once,

(suppressing the argument of x for now)

Taking the derivative again,

Substitution and simplifying terms leads to

This equation is now in the form of the Sturm-Liouville equation

where the Sturm-Liouville Dirichlet potential function , $q\left(t\right)$ , is given by

As in [link] above, the variable $x$ on the right side of this expression is to be interpreted as a function of $t$ defined by [link] .

This section describes what is needed to change from the wave equation with a mass density, $\rho$ , to a Sturm-Liouville equation with a potential, $q$ . The next section describes the reverse, changing from the Sturm-Liouville equation with a potential, $q$ , to a wave equation with a mass density, $\rho$ .

Changing from potential to density

Converting a given Dirichlet potential function $q\left(t\right)$ into corresponding mass density $\rho \left(x\right)$ amounts to solving the nonlinear ordinary differential equation, or ODE, in [link] for $\rho$ ; however, one must be careful. Using the integral [link] and abusing notation a bit, it is tempting to write $t=t\left(x\right)$ and arrive at the ODE

which one would then attempt to solve numerically by selecting some initial conditions for $\rho \left(x\right)$ , picking a grid in the $x$ -domain and applying one's ODE solver of choice, using some quadrature scheme to estimate $t\left(x\right)$ by approximating [link] . This approach, however, suffers from a somewhat subtle flaw: it presumes that one already knows the domain ofdefinition of $\rho \left(x\right)$ , i.e., that the length $L$ of the string can be known a priori.