-n t2n2-2n(67)With = 1/2 and = /2, the L i mirnov distributions emerge.

-n t2n2-2n(67)With = 1/2 and = /2, the L i mirnov distributions emerge. Remark six. The entropy from the = 1, = two case was computed in (39) 4.4. Space-Dominant Case When , the method we followed above is just not appropriate, since the series becomes divergent. Hence, we’ve to seek out a way where the two orders play reverse roles. Theorem three. Let and 1. The Fourier inverse of G (, t), (47), is now given by g( x, t) =1 +(-1) sin (2n + 2) 2 n =nn =(-1)n cos(2n + 1)((2n + 1) /) sin((2n + 1)/) 2n -(2n+1) / x t (2n)! sin((2n + 1)/) ((2n + two) /) sin((2n + 2)/) 2n+1 -(2n+2) / x t (2n + 1)! sin((2n + two)/)(68)Remark 7. We have to note that (68) is often a generalization for any significantly less than with the benefits known for the steady distributions corresponding to = 1 that emerges as a certain case. Proof. We create the inverse of (21) as g( x, t) = 1 (two )2 i 1 s est eix dsd, s s + (69)Rwhere is really a vertical straight line inside the ideal half complicated plane. From it, define a brand new integration path u that results from by the transformation u = s which will be made use of within the integrand. This path consists of two half straight lines producing angles of with all the 2 actual axis. Then, we acquire: g( x, t) = 1 (2 )two i 1 eu t eix dud, u +(70)uRFractal Fract. 2021, five,16 ofHowever,1 = u +e- e-u d,that enables us to write g( x, t) = 1 (two )2 ie- ev t eix dvde-v d =R1 2iI ( x,)evte-v dv d.(71)We’re going to consider initially the inverse FT I ( x,) =1e- eix dRIf we expand e- in Taylor series,.