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## Main Question or Discussion Point

Hi, in the link https://www.researchgate.net/profil...ods-for-deterministic-parabolic-equations.pdf and equation 3 you will see the exponential form of the solution, also operator "D" includes non-linear operator structure written as ##\partial_x x##. Besides, I think in previous link x is dependent on t. I can understand that when operator "D" only consists of linear operators like ##\partial_x## then exponential form is consistent but I can not understand this situation is also possible when non-linear operators are included in "D". Could you provide me with mathematical demonstration to show that exponential form can be written also for non-linear operator structures??

When I expand exponential form of operator "D", I can see it is totally consistent with first order but when I come to second order expansion of operator "D" then it is not going well and not consistent contrary to link I shared. If "D" only included linear operators, it would be ok but here we have nonlinear operator, x depends on t so at the second order it can not be written as 1/2*D^2*##\Delta_t##

When I expand exponential form of operator "D", I can see it is totally consistent with first order but when I come to second order expansion of operator "D" then it is not going well and not consistent contrary to link I shared. If "D" only included linear operators, it would be ok but here we have nonlinear operator, x depends on t so at the second order it can not be written as 1/2*D^2*##\Delta_t##