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## plan coding of nonlinear diffusion using Taylor-Riemann formula

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

6:28 PM (11 minutes ago)

 to donoho, bcc: aimee
We want to take the fractional Taylor expansion of Riemann for our forcing term F (snapshot formula 1.3) and then use the Adomian polynomials we computed for fractional powers for each of the terms in the expansion and use the sum of Adomian polynomials in the iterative solution of a Fractional Nonlinear Diffusion to solve the equation.  Suppose p(t,x) is the d-dimensional symmetric stable distribution and assume that it can be generalized without problems to

((d/dt)^alpha + (-Laplacian)^beta )u = F = sum_k a^k u^{q_k} (*)
where the sum on the right side is from the Taylor-Riemann expansion in terms of fractional derivatives encoded in a^k, and let the Adomian polynomials that correspond to these be A^k_n, so that the iteration solution consists of updating u_n to u_n with \sum_k a^k A^k_n in place of the nonlinear term F in (*).
We can use the symbolic python package Sage perhaps to do this programmatically in python.
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Preview attachment Screenshot 2015-07-14 18.16.20.png

Screenshot 2015-07-14 18.16.20.png

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

6:39 PM (0 minutes ago)

 to donoho, bcc: aimee
We can also solve the equation with a different approach using the same Adomian polynomials by plugging the Adomian terms for F in the equation (**) below if we have available the fundamental solution of the ‘diffusion’ part of the equation.

If p(t,x) is the fundamental solution of (*) with F==0 then we can solve the Cauchy initial value problem with u(0,x) = a(x) in the integral form

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 (**)