
2:37 PM (44 minutes ago)



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.
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
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}
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)

3:21 PM (0 minutes ago)



A_0 = u_0^alpha
For A_1,
from which we get, setting z=0 that
A_1 = alpha*u_0^{alpha1}*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 + …)^{alpha1} (u_1 + 2 u_2 z + …) + alpha * (u_0+u_1 z + …)^{alpha1} d/dz(u_1 + 2 u_2 z + …)
so
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