public void OdeAiry()
{
// This is the airy differential equation
Func <double, double, double> f = (double x, double y) => x * y;
// Solutions should be of the form f(x) = a Ai(x) + b Bi(x).
// Given initial value of f and f', this equation plus the Wronskian can be solved to give
// a = \pi ( f Bi' - f' Bi ) b = \pi ( f' Ai - f Ai' )
// Start with some initial conditions
double x0 = 0.0;
double y0 = 0.0;
double yp0 = 1.0;
// Find the a and b coefficients consistent with those values
SolutionPair s0 = AdvancedMath.Airy(x0);
double a = Math.PI * (y0 * s0.SecondSolutionDerivative - yp0 * s0.SecondSolutionValue);
double b = Math.PI * (yp0 * s0.FirstSolutionValue - y0 * s0.FirstSolutionDerivative);
Assert.IsTrue(TestUtilities.IsNearlyEqual(y0, a * s0.FirstSolutionValue + b * s0.SecondSolutionValue));
Assert.IsTrue(TestUtilities.IsNearlyEqual(yp0, a * s0.FirstSolutionDerivative + b * s0.SecondSolutionDerivative));
// Integrate to a new point (pick a negative one so we test left integration)
double x1 = -5.0;
OdeResult result = FunctionMath.IntegrateConservativeOde(f, x0, y0, yp0, x1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.X, x1));
Console.WriteLine(result.EvaluationCount);
// The solution should still hold
SolutionPair s1 = AdvancedMath.Airy(x1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Y, a * s1.FirstSolutionValue + b * s1.SecondSolutionValue, result.Settings));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.YPrime, a * s1.FirstSolutionDerivative + b * s1.SecondSolutionDerivative, result.Settings));
}
public void AiryAgreement()
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 1.0E2, 4))
{
SolutionPair s = AdvancedMath.Airy(-x);
Assert.IsTrue(TestUtilities.IsNearlyEqual(s.FirstSolutionValue, AdvancedMath.AiryAi(-x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(s.SecondSolutionValue, AdvancedMath.AiryBi(-x)));
}
}
public void AiryWronskian()
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 1.0E2, 8))
{
SolutionPair p = AdvancedMath.Airy(x);
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(p.FirstSolutionValue * p.SecondSolutionDerivative, -p.SecondSolutionValue * p.FirstSolutionDerivative, 1.0 / Math.PI));
SolutionPair q = AdvancedMath.Airy(-x);
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(q.FirstSolutionValue * q.SecondSolutionDerivative, -q.SecondSolutionValue * q.FirstSolutionDerivative, 1.0 / Math.PI));
}
}
public void AiryPowerIntegrals()
{
// Bernard Laurenzi, "Moment Integrals of Powers of Airy Functions", ZAMP 44 (1993) 891
double I2 = FunctionMath.Integrate(t => MoreMath.Pow(AdvancedMath.AiryAi(t), 2), 0.0, Double.PositiveInfinity);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I2, MoreMath.Sqr(AdvancedMath.Airy(0.0).FirstSolutionDerivative)));
// I3 involves difference of 2F1 functions
double I4 = FunctionMath.Integrate(t => MoreMath.Pow(AdvancedMath.AiryAi(t), 4), 0.0, Double.PositiveInfinity);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I4, Math.Log(3.0) / (24.0 * Math.PI * Math.PI)));
}
public void AiryModulusMomentIntegrals()
{
// https://math.stackexchange.com/questions/507425/an-integral-involving-airy-functions-int-0-infty-fracxp-operatornameai
double I0 = FunctionMath.Integrate(t => {
SolutionPair s = AdvancedMath.Airy(t);
return(1.0 / (MoreMath.Sqr(s.FirstSolutionValue) + MoreMath.Sqr(s.SecondSolutionValue)));
}, 0.0, Double.PositiveInfinity);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I0, Math.PI * Math.PI / 6.0));
double I3 = FunctionMath.Integrate(t => {
SolutionPair s = AdvancedMath.Airy(t);
return(MoreMath.Pow(t, 3) / (MoreMath.Sqr(s.FirstSolutionValue) + MoreMath.Sqr(s.SecondSolutionValue)));
}, 0.0, Double.PositiveInfinity);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I3, 5.0 * Math.PI * Math.PI / 32.0));
}
