Airy Turning-Point Solutions

Airy Equation and Turning Points

The Airy differential equation, given by y'' = x * y, is the simplest second-order linear differential equation with a turning point. A turning point occurs where the coefficient of the unknown function changes sign. For this equation, the turning point is at x = 0.

The character of the solutions changes fundamentally at the turning point. For x > 0, the solutions behave exponentially (either growing or decaying), representing the "exponential coefficient" regime. For x < 0, the solutions exhibit oscillatory behavior, known as the "oscillatory coefficient" regime. The two linearly independent solutions to this equation are traditionally denoted as Ai(x) and Bi(x).

Interactive Exploration

You can interactively explore the behavior of both Airy solutions using the Airy Turning-Point Solutions module. Follow these ordered instructions:

1. Select a basis function from the Basis control.
2. Adjust the probe point `x` across the interval [-3, 3] from the Probe x control.
3. Observe how the function's value, derivative, and second derivative evolve as you pass through the turning point at x=0. The visual traces also highlight the transition between the oscillatory regime on the left and the exponential regime on the right.

Worked Example

Let us calculate the value of the Ai(x) function and its derivatives at the probe point x = 1 using its Taylor series expansion around x=0, which is characterized by the following initial values:

Ai(0) ≈ 0.3550280538878172 Ai'(0) ≈ -0.2588194037928068

Using the recurrence relation a_{k+3} = a_k / ((k+3)(k+2)), we can determine the coefficients of the series Ai(x) = a_0 + a_1 x + a_2 x^2 + .... We can verify this computationally using the module interface:

1. Select the Ai option from the Basis control.
2. Select the 1 option from the Probe x control.
3. Observe the Value result is approximately 0.13529241631288141.
4. Observe the Derivative result is approximately -0.1591474412967932.
5. Observe the Second derivative result is approximately 0.13529241631288141.
6. Observe the Wronskian result is approximately 0.3183098861837907.
7. Observe the Regime result is Exponential coefficient.

Since we are at x = 1 > 0, we are in the "Exponential coefficient" regime. The Wronskian of Ai and Bi is constant and equals 1/π, which is approximately 0.3183098861837907.

Limitations

The power series used in this interactive module is evaluated up to around 130 terms. While this provides excellent accuracy up to 15 decimal places for inputs in the range |x| ≤ 3, the Taylor series convergence slows down significantly for larger values of |x|. Due to finite precision floating-point arithmetic (round-off error) and the high number of terms needed, evaluating the solution far from the turning point with a naive power series becomes impractical. In those regimes, asymptotic expansions are typically employed instead.

References

For comprehensive formal definitions, asymptotic behaviors, and tabulated values of the Airy solutions Ai(x) and Bi(x), consult standard reference texts and digital libraries for mathematical functions.

• NIST Digital Library of Mathematical Functions: 9.2 Airy Functions
• NIST Digital Library of Mathematical Functions: 9.4 Wronskians and Cross Products