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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

- w ' ' + q w = λ w ,

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 ρ 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

t = 0 x ρ ( α ) d α .

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

w ( t ) = y ( x ) ρ 1 / 4 ( x ) ,

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,

d w d t d t d x = 1 4 y ( x ) [ ρ ( x ) ] - 3 / 4 d ρ d x + d y d x [ ρ ( x ) ] 1 / 4

(suppressing the argument of x for now)

d w d t ρ 1 / 2 = 1 4 y ρ - 3 / 4 ρ ' + y ' ρ 1 / 4

Taking the derivative again,

d 2 w d t 2 d t d x ρ 1 / 2 + 1 2 d w d t ρ - 1 / 2 ρ ' = 1 2 y ' ρ - 3 / 4 ρ ' - 3 16 y ρ - 7 / 4 ( ρ ' ) 2 + 1 4 y ρ - 3 / 4 ρ ' ' + y ' ' ρ 1 / 4

Substitution and simplifying terms leads to

d 2 w d t 2 ρ + 1 2 d w d t ρ - 1 / 2 ρ ' = 1 2 y ' ρ - 3 / 4 ρ ' - 3 16 y ρ - 7 / 4 ( ρ ' ) 2 + 1 4 y ρ - 3 / 4 ρ ' ' + y ' ' ρ 1 / 4
d 2 w d t 2 ρ + ρ ' 2 ρ 1 / 2 y ρ ' 4 ρ 5 / 4 + y ' ρ 1 / 4 = 1 2 y ' ρ - 3 / 4 ρ ' - 3 16 y ρ - 7 / 4 ( ρ ' ) 2 + 1 4 y ρ - 3 / 4 ρ ' ' + y ' ' ρ 1 / 4
d 2 w d t 2 ρ + ρ ' 2 ρ 1 / 2 y ρ ' 4 ρ 5 / 4 + y ' ρ 1 / 4 = 1 2 y ' ρ - 3 / 4 ρ ' - 3 16 y ρ - 7 / 4 ( ρ ' ) 2 + 1 4 y ρ - 3 / 4 ρ ' ' - λ y ρ 5 / 4
d 2 w d t 2 ρ = - 5 16 y ρ - 7 / 4 ( ρ ' ) 2 + 1 4 y ρ - 3 / 4 ρ ' ' - λ y ρ 5 / 4
d 2 w d t 2 = - 5 16 y ρ - 11 / 4 ( ρ ' ) 2 + 1 4 y ρ - 7 / 4 ρ ' ' - λ y ρ 1 / 4
d 2 w d t 2 = - 5 16 y ρ 1 / 4 ρ - 3 ( ρ ' ) 2 + 1 4 y ρ 1 / 4 ρ - 2 ρ ' ' - λ y ρ 1 / 4
d 2 w d t 2 = - 5 16 w ρ - 3 ( ρ ' ) 2 + 1 4 w ρ - 2 ρ ' ' - λ w
- d 2 w d t 2 + 1 4 ρ - 2 ρ ' ' - 5 16 ρ - 3 ( ρ ' ) 2 w = λ w

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

- d 2 w d t 2 + q ( t ) w = λ w ,

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

q ( t ) = ρ ' ' ( x ) 4 ρ 2 ( x ) - 5 ρ ' ( x ) 2 16 ρ 3 ( x ) .

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, ρ , 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, ρ .

Changing from potential to density

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

q t ( x ) = ρ ' ' ( x ) 4 ρ 2 ( x ) - 5 ρ ' ( x ) 2 16 ρ 3 ( x ) ,

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

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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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