Over the last months, I had been seeking ideas for understanding the instability properties of nonlinear fractional diffusions, of which the KPP equation is a well-studied example with known phase transition properties the nonlinear equation is

Lu + F(u) = 0

where L is the operator (d/dt)^alpha + (-Laplacian)^(beta/2) say in R^d; the time fractional derivative is defined by a modified Riemann-Liouville or Caputo derivative and the Laplacian power is Riesz-Feller derivative. The Green’s function for L is known exactly in terms of the higher transcendental Mittag-Leffler function studied by Mainardi, Gorenflo et. al.

A general rigorous method for solving nonlinear equations is the Homotopy Analysis method and its variants. It is a general method for solving nonlinear equations described in Section 3 of the attached paper. I am trying to understand what happens in the iteration step (12) when F(z) is a nonlinear function with a large complex Julia set on the complex plane.

When Nu = Lu + F(u) one would expect that the iteration step

L(u_m – u_{m-1}) = h H(x,t) R_m(v_{m-1}) (*)

where v_{m-1} = (u_0,…,u_{m-1}) a vector which the numerical analysts solve by computation in our case is just integration with respect to a Green’s function that is known, and

R_m(v_{m-1}) = (d/dp)^{m-1} N(phi(x,t,p)) (**)

The step (*) could be transformed by integration by the Green’s function for L leaving us with the problem of understanding whether in (**) we can observe iterations of F(.). So this is still cloudy for me but if we can observe iterations of F(.) in (**) then we have some direct way to show something like: whenever F has a large Julia set on the complex plane, then these nonlinear fractional diffusions with F as nonlinearity will have unstable solutions. This would be a very useful result if true since these nonlinear fractional diffusions are better models for global financial volatility than linear diffusion sample paths.

Screenshot 2015-08-07 18.34.56.png

Preview attachment mahmood-homotopy-analysis-nonlinear-diffusion.pdf

mahmood-homotopy-analysis-nonlinear-diffusion.pdf

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