Program

We develop a pseudo-spectral method to solve initial-value problems associated to PDEs involving the fractional Laplacian operator acting on the whole real line (see [1]). After a suitable representation of the operator, we perform the change of variable \(x = L\cot(s)\), \(L > 0\), to transform the real line \(\mathbb{R}\) into the interval \([0, \pi]\), where a Fourier expansion of the solution \(u(x(s))\) can be applied. We approximate the fractional Laplacian by means of the midpoint quadrature rule, improving the results with Richardson’s extrapolation, similarly as in [2]. This method deals accurately and efficiently with problems posed on \(\mathbb{R}\), and avoids truncating the domain (which requires introducing artificial
boundary conditions). In order to illustrate its applicability, we have simulated the evolution of the following non-local Fisher-KPP [3] and ZFK-Nagumo [4] models.

Keywords: Fractional Laplacian, Pseudo-spectral methods, Fourier transform, Chebyshev Polynomials.

References

[1] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017.

[2] F. de la Hoz and C. M. Cuesta. A pseudo-spectral method for a non-local KdV-Burgers equation posed on \(\mathbb{R}\). Journal of Computational Physics, 311:45–61, 2016.

[3] R. A. Fisher. The wave of advance of advantageous genes. Annals. of Eugenics, 7:355–369, 1937.

[4] Y. B. Zel’dovich and D. A. Frank-Kamenetsky. Towards the theory of uniformly propagating flames. Doklady AN SSSR, 19:693–697, 1938.

[5] J. P. Boyd. Chebyshev and Fourier Spectral Method. Springer–Verlag, XVI, 1989.

We discuss some recent results on nonlocal phase transitions
modelled by the fractional Allen-Cahn equation, also in connection with the surfaces minimising a nonlocal perimeter functional. In particular, we consider the “genuinely nonlocal regime” in which the difusion operator is of order less than 1 and present some rigidity and symmetry results.