Liouville-Green approximation

Suppose you have a differential equation of the form

\frac{d^2y}{dx^2} = f(x)y

If the function f(x) is constant, the differential equation is easy to solve in closed form. But when it is not constant, it is very unlikely that closed form solutions exist. But there may be useful closed form approximate solutions.

There is no linear term in the equation above, but there is a standard change of variables to eliminate a linear term and put any second order linear equation with variable coefficients in the form above.

The Liouville-Green approximate solutions are linear combinations of the functions

f^{-1/4}(x)\exp\left(\pm\int f^{1/2}(x)\, dx \right )

The constant of integration doesn’t matter. It would only change the solution by a constant multiple, so any indefinite integral will do.

To try out the Liouville-Green approximation, let’s look at the equation

\frac{d^2y}{dx^2} = x^2y

Here f(x) = x² and so the LG approximation solutions are

A\, \frac{\exp\left(x^2/2 \right )}{ \sqrt{x} } + B\, \frac{\exp\left(-x^2/2 \right )}{ \sqrt{x} }

Let’s see how these solutions behave compare to a numerical solution of the equation. To make things more definite, we need initial conditions. Let’s say A B = 1, which corresponds to initial conditions y(1) = 2.25525 and y‘(1) = −0.0854354.

Here’s a plot of the L-G approximation and a numerical solution.

Plot of Liouville-Green approximation and numerical solution

Here’s the Mathematica code that was used to create the plot above.

s = NDSolve[
       y''[x] == x^2 y[x], 
       y[1] == 2.25525, 
       y'[1] == -0.0854354
    {x, 1, 3}
g[x_] = (Exp[-x^2/2] + Exp[x^2/2])/Sqrt[x]
    {Evaluate[y[x] /. s], g[x]}, 
    {x, 1, 3}, 
    PlotLabels -> {"Numerical", "LG"}

For more on Liouville-Green approximations, see Olver’s classic book Asymptotics and Special Functions.

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