Fractional Calculus and Dynamic Approach to Complexity Page: 71
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IFPPeq(X) -0
We introduce the auxiliary function 4 (x, t) by the expression
P(x, t) = Peq(x)) (x, t) ,
which when substituted into the FPE reduces to
&G(x) 04(x, t) 2(Xt)
+ D
L (x, t).The auxiliary function 1(x, t) can be expanded in a complete set of eigenfunctions for the
FP operator :L (.X) = Ek#k(X)
00
4(x, t) L ak e-k- / ,(X),
k=owhich when inserted into Eq. (169) yields a tautology. The coefficients in the eigenfunction
expansion are given in terms of the initial PDF with an inner product defined with respect
to the steady-state PDF as a weight functionak (i=o, ijk)
f Po (X) Peq (X)-#4(x)dx,
indicating that the FP operator is self-adjoint on the space spanned by the basis functions
pk-71
such that
(167)
(168)
04(x, t)
at(169)
(170)
as follows
(171)
(172)
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Beig, Mirza Tanweer Ahmad. Fractional Calculus and Dynamic Approach to Complexity, dissertation, December 2015; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc822832/m1/80/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .