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HOMOTOPY ANALYSIS METHOD FOR SOLVING NONLINEAR EQUATIONS

The Homotopy Analysis Method was introduced in S. Liao’s Ph.D. thesis in 1992 and has had great success in solving nonlinear differential equations since.  For the nonlinear equation $N[u(z)]=0$ one introduces auxiliary linear operator $L$ with $L(0)=0$ and a new parameter $q$ and considers the family of equations $(1-q)L(U(x;q) - u_0(x)) + q NU(x;q) = 0$.  This new family has formally the property that its solution deforms from $u_0$ of the linear operator $L$ to that of $Nu=0$ at $q=1$.  This is a very general infrastructure where then the solution of the original equation $N[u(x)] = 0$ is obtained by a Taylor expansion in $q$

$u(x) = u_0(x) + \sum_{k=1}^\infty u_k(x) q^k$

and then the solution to the original problem is obtained by determining $u_k(x)$ by differentiations in $q$ and setting $q=0$.

small step toward understanding instability of solutions of nonlinear fractional diffusions

The problem is still that of generalizing phase transitions to turbulence for nonlinear fractional diffusions from what is now known F(z) = z^2-z.  Formally, nonlinear differential equations have been handled by various relatively new methods such as J. Shiao’s Homotopy Analysis Method.  I’m attempting to understand how this method works for a related problem (2009 work by Mahmood et. al.)

The solutions is just a matter of hard computations it seems.  We want to get the computations near a point where iterations of F on the complex plane gets us to a Julia set.  This is still messy in my mind, but looks promising …
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Preview attachment Screenshot 2015-08-07 15.41.17.png

Screenshot 2015-08-07 15.41.17.png

Preview attachment mahmood-homotopy-analysis-nonlinear-diffusion.pdf

mahmood-homotopy-analysis-nonlinear-diffusion.pdf