Mittag-Leffler Functions

If the Gamma function is the generalization of the factorial, the Mittag-Leffler function is the generalization of the exponential function. It is the "crown jewel" of fractional calculus.

Definition and Origin

The Mittag-Leffler function $E_{\alpha, \beta}(z)$ is defined by the following power series for $\alpha > 0$:

• When $\alpha = 1, \beta = 1$, it becomes the standard exponential: $E_{1,1}(z) = e^z$.

• When $\alpha = 2, \beta = 1$, it describes hyperbolic cosines: $E_{2,1}(z^2) = \cosh(z)$.

Physical Significance: Fractional Calculus

While the standard exponential function describes "normal" relaxation (like a cooling cup of coffee), the Mittag-Leffler function describes "anomalous" relaxation.

• Viscoelasticity: Used to model materials that are halfway between a liquid and a solid (like polymers or human tissue).
• Fractional Diffusion: Describes how particles move in crowded environments (like proteins moving inside a biological cell).

3. Implementation in SepalSolver

Unlike other scientific computing tools like matlab, in sepalsolver, the Mittag-Leffler function is not part of the sepcial function library. And it is exposed in the SepalSolver.Math class.

//Plotting E_{ a, 1}  (z) for varying alpha
ColVec z = Linspace(0, 4);
double[] alp = [0.5, 0.8, 1.0, 1.2];
Matrix Y = alp.Select(a => Arrayfun(x => MettagLeffler(a, 1, x), z)).ToList();
Plot(z, Y);
Legend(alp.Select(a=>"E_{" + a +",1}(x)"));
Title("Mittag-Leffler Function E_{\alpha, 1}(z)");

Key Properties

• Interpolation: It interpolates between a pure exponential and a power-law function.
• Laplace Transform: The Laplace transform of :math:E_{\alpha}( -at^\alpha ) is $\frac{s^{\alpha-1}}{s^\alpha + a}$, which is vital for solving fractional differential equations.