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## Adomian Decomposition Method for Fractional Diffusion-Reaction Problems

### Zulf Ahmed<fine.fine.forget.shoah@gmail.com>

8:31 AM (0 minutes ago)

 to donoho, bcc: aimee
Dear Professor Donoho,

The Adomian Decomposition Method allows one to produce series expansion solution to initial value PDE with easily invertible highest order terms.  If
Lu + Ru + Nu = g
is a (fractional) PDE with invertible highest order operator L and R is a linear differential lower order part and Nu are nonlinear terms and g is the source term, one considers series expansion solutions u =\sum_k u_k from
u_0 = L^{-1} L u + L^{-1} g
and the main issue is to consider the Adomian Decomposition of the nonlinear terms Nu = \sum_k A_k where
A_k = 1/k! (d/d\lamda)^k N (\sum_k lambda_k u_k)
which are called Adomian polynomials.  Then there is a simple iterative update formula for u_k where one uses these for the nonlinear term, the only term that needs the help.
For nonlinearities of type Nu = u^p  p>0 real,  these are explicitly calculated in the attached paper on Adomian Decomposition Method applied to nonlinear Sturm-Liouville problem (see snapshot) and gives us a general tool for handling fractional diffusion-reaction equations with nonlinear reaction terms.
This provides a general tool on R^d for fractional (Caputo or Riemann-Liouville) time derivatives and Riesz-Feller fractional space derivatives and a reaction term that is nonlinear on R^d.  With this complete toolset we have open questions for phase transitions to turbulence that could be numerically simulated and then used as the test for carrying over to complex networks where they become a system of fractional ordinary differential equations.
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