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## the fundamental solution of the fractional diffusion-wave equation

### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

9:55 AM (0 minutes ago)

 to donoho, bcc: aimee
Sir,

Francesco Mainardi’s elegant paper attached provides us with a unified Green function or fundamental solution of the Cauchy initial value problem for the diffusion equation and the wave equation interpolating them via (d/dt)^{2*beta} for the time derivative with beta=1/2 reproducing the diffusion and beta=1 reproducing the wave equation.  He uses the Laplace transform representation of these fundamental solutions.  Formulae (3.2) and (3.4) in the snapshot tell us using his notation:

The diffusion-wave equation is, minus initial conditions:

(d/dt)^{2*beta} = D (d/dx)^2 u
Laplace transform of the fundamental solution for beta in (0,1/2) is called Gc, for beta in (1/2,1) is called Gs, and
Gc(x,s; beta) = (2*sqrt(D)*s^{1-beta}) * exp( s^beta * abs(x)/sqrt(D))
(1)

Gs(x,s; beta) = exp( – x*s^beta/sqrt(D) )
(2)

This is for the one-dimensional case, but there are later results for higher dimensions and Riesz-Feller fractionalizing of the Laplacian as well.

We can now return to our setting in R^d where we are interested in modifications of the century-long tradition in quantitative finance of modeling volatility using minor variations of sample paths of diffusions by adding a nonlinear forcing term F.  We can use these formulae directly in the solution of the NONLINEAR diffusion equations whose sample paths I believe will ultimately address two important shortcomings of this state of affairs:  (a) possibly provide quantitative basis for phase transition to turbulence in global financial volatility,  (b) fit empirical data of global financial volatility in a large complex network.  Goals (a) and (b) are ambitious goals but at this stage of the Volatility Storms project, I am focusing on getting some solid mathematical foundations for nonlinear fractional diffusions in R^d.

Let p(t,x) be the Green function (1) or (2) or their generalization to R^d with Riesz-Feller fractional derivative for the space part.  Then the equation  (please forgive the switching of meaning of alpha, beta, etc. which are due to mapping to our notation but the ideas are clear)

For

((d/dt)^alpha + (-Laplacian/2)^beta )u= F(u) (3)

yesterday we looked at Adomian polynomials to address solving this for alpha=1 which can be done with p(t,x) the symmetric Levy stable density.  We can now replace p(t,x) with the Green functions above or their generalizations in the integral representation of (3)

u(t,x) = \int p(t,x-y) a(y) dy + \int_0^t ds \int p(t-s,x-y) F(u(s,y))dy
The idea is clear but we are interested in simulations for phase transition to turbulence with some clearer solid R^d case first.

2 Attachments

Preview attachment Screenshot 2015-07-15 09.27.58.png

Screenshot 2015-07-15 09.27.58.png
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Preview attachment mainardi-fundamental-solutions-diffusion-wave.pdf

mainardi-fundamental-solutions-diffusion-wave.pdf