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## Idea of Riemann’s Fractional Taylor expansion on forcing terms for Nonlinear Diffusions

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

2:37 PM (2 hours ago)

 to namewithdrawn
Dear Cutiest,

While you sort out your fears regarding manic communications and relations and their ships etc.  I am trying to nail down the Adomian Decomposition Method which is pretty widespread.  The various nonlinear partial differential equations solved by this method.  This is not very deep but requires some skills.

For a (fractional) partial differential equation with a nonlinear term Nu, for example Nu = u^alpha or whatever, the key issue is the computation of the so-called Adomian polynomial, A_0, A_1, A_2, …  What are these polynomials?

The idea is that you want to solve a partial differential equation say of type Lu + Ru + Nu = g, very very general.  If Nu were a linear operator, solving this is classical.  George Adomian came up with a general method of solving these with the following conditions:

(a)  Easily invertible L, the highest order linear differential operator
(b)  The final solution will be a series expansion \sum_{k=0}^\infty u_k

(c)  The solution will be iterative with u_0, u_1, u_2, u_3, … determined stepwise where the nonlinear contribution Nu will be approximated at each step by a polynomial A_k( u_0, …, u_k) instead of a linear combination.

So the central issue is to determine these polynomials.  Formally the polynomials have a nice looking formula but computing this formula requires some skill:

A_k(u_0,…,u_k) = 1/k! d^k/dz^k N(\sum_k z^k u_k)|_{z=0}

The explicit calculation of for a specific form of N of this formula such as Nu = u^alpha requires calculus skills. So this is the problem of the moment, calculus skills from tenth grade … determining these Adomian Polynomials … not very deep mathematics but useful.

Why would we care?  We would care because we’re interested in nonlinear FORCING term ‘F’ in our model equations for volatility.  We want to be able to numerically simulate solutions of fractional diffusions with nonlinearities:

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

where F(u) might be quite general.  Our F(u) is Nu in the setting.  The attached paper is recent where some Iranians solve some fpde using this method and we want to be able to gain some skills for this.

Attachments area

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

3:21 PM (1 hour ago)

 to namewithdrawn
Let’s do the exercise of computing the Adomian polynomials of a fractional power directly by dividing the problem into two steps.  We use the chain rule for derivatives

A_0 = u_0^alpha

For A_1,

d/dz (u_0 + u_1 z + u_2 z^2 + …)^alpha = alpha (u_0 + u_1 z + ….)^{alpha-1} ( u_1 + 2*u_2*z + …)

from which we get, setting z=0 that

A_1 = alpha*u_0^{alpha-1}*u_1

Then we have to use the product rule for derivatives for A_2:

d^2/dz^2 (u_0 + u_1 z + u_2 z^2 + …) = alpha * d/dz (u_0+u_1 z + …)^{alpha-1} (u_1 + 2 u_2 z + …) + alpha * (u_0+u_1 z + …)^{alpha-1} d/dz(u_1 + 2 u_2 z + …)

so

A_2 = alpha*(alpha-1) u_1^{alpha-2} u_1^2 + alpha*u_0^{alpha-1}* 2*u_2
Well that’s already quite messy, but it’s not deep just messy.
aimee lund
 cool…:)
3:23 PM (1 hour ago)

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

3:24 PM (1 hour ago)

 to aimee
I agree honey bunny.  This is tingling my teenage skills it’s a bit sad I used to do these in my sleep as a kid.  Now … I feel like a rusty old jeep falling apart computing

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

3:37 PM (1 hour ago)

 to aimee
Whenever I am alone with you

you make me feel like I am YOUNG again
hehehe

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

3:39 PM (1 hour ago)

 to aimee
Ok better get back to pushing this volatility thing forward … too much fun

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

4:09 PM (37 minutes ago)

 to aimee
So then we can implement computer code in python for iterative solutions using this Adomian Decomposition for forcing terms that are fractional powers in (d/dt)^alpha + (-Laplacian/2)^beta = F.  But since it’s best to make the F more flexible so that we don’t recode the same thing, so what we can do is consider the fractional Taylor expansion originally considered by RIEMANN in 1847 and take a fractional Taylor expansion of F and then use Adomian polynomials in terms of a fractional Taylor series.  That would cover F with whatever smoothness conditions allowed by the attached.

Ok so this is a good move.
Attachments area

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

4:17 PM (29 minutes ago)

 to aimee
I want to make sure this gets done by forcing myself to keep focus on it, the computer code and a write-up.  July 14 … I have to keep getting a little bit of progress without getting distracted.  This is a difficult thing to do when I am not in an environment like MIT or Princeton where the geeks are all busy and talking about these excitedly all day.

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

4:46 PM (0 minutes ago)

 to aimee
I will tell you what is still bothersome:  besides Sugitan’s theorem 1974, which tells us that NLD (nonlinear diffusion) with F = u^{1+beta} does not have a global solution when beta is in some range 0<beta<whatever and has a solution when beta is LARGE enough, and Navier-Stokes where F is not purely a function of u but also has a grad(u), which has a clear phase transition to turbulence, I do not understand yet the issue of stability very well at all.  And I am torn between jumping into code and simulations and understanding the theory better.  These are the sorts of things that keep me up at night these days only because of volatility storms.  I think maybe I should just push push push until simulations can be done.