Simulation of Fractional Laplacian

To start this tutorial, write the following command in the MATLAB console

open T0000_FractionalLaplacianDynamics

We use DyCon Toolbox to solve numerically the following fractional heat equation:

Here, for all $s\in(0,1)$, $(-d_x^2)^s$ denotes the one-dimensional fractional Laplace operator, defined as the following singular integral

Discretization of the problem

As a first thing, we need to discretize \eqref{frac_heat}. Hence, let us consider a uniform N-points mesh on the interval $(-1,1)$.

N = 50;
xi = -1; xf = 1;
xline = linspace(xi,xf,N+2);
xline = xline(2:end-1);

Out of that, we can construct the FE approxiamtion of the fractional Lapalcian, using the program FEFractionalLaplacian developped by our team, which implements the methodology described in [1].

s = 0.8;
A = -FEFractionalLaplacian(s,1,N);

Moreover, we build the mass matrix $M$ associated to our mesh.

M = MassMatrix(xline);

We can then define a time horizon for our simulations and an initial datum

FinalTime = 0.5;
Y0 =cos(pi*xline');
Y0(xline > 0.2) = 0;
Y0(xline < -0.2) = 0;

and construct the system

dynamics = pde('A',A,'InitialCondition',Y0,'FinalTime',FinalTime,'Nt',100);
dynamics.mesh = xline;
dynamics.MassMatrix = M;
dynamics.InitialCondition = Y0;
solve(dynamics);

We can launch several simulations for different values of “s”

ssline = linspace(0.01,0.99,14);


iter = 0;
for s = ssline
    iter = iter + 1;
    A = -FEFractionalLaplacian(s,1,N);
    dynamics.A = A;
    [tspan,Ysolution] = solve(dynamics);
    Data(iter).Y = Ysolution;
end
animation_FLD(ssline,Data,xline)

This simulation clearly show how the diffusivity of the equation is strongly affected by the power of the fractional Laplacian being, in particular, very low for small values of $s$.

This behavior then affects the controllability properties of \eqref{frac_heat}, as it is discussed in [1].

References

[1] U. Biccari and V. Hernández-Santamaría - Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects, IMA J. Math. Control. Inf., to appear