3D simulation on the unit duct in the shell side of the ROD by Wu J., Dong Q., Liu M.

By Wu J., Dong Q., Liu M.

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1 are irregular and a derivative cannot be defined according to the mean value theorem. This is because of the fact that the function changes erratically within small intervals, however small that interval may be. Therefore we have to devise new mathematical tools that would be useful in dealing with these irregular, non-differentiable functions. 1) i 1 where  n  max(ti  ti  1 ) . 1 i  n If Vf([a,b]) is finite such as in continuous differentiable functions then f is called a function of finite variation on [a,b].

Obviously a function can only have countable number of jumps in a given range. From the mean value theorem in calculus, it can be shown that we can differentiate a function in a given interval only if the function is either continuous or has a discontinuity of the second kind during the interval. Stochastic calculus is the calculus dealing with often non-differentiable functions having jumps without discontinuities of the second kind. One such example of a function is the Wiener process (Brownian motion).

In this section we will focus only on strong solutions. In many situations, finding analytical solutions to SDEs is impossible and therefore we will review a minimum number of SDEs and their solutions in order to facilitate the discussion in the subsequent chapters. 9) with Y (0)  1 . Thus Y(t ) is called the stochastic exponential of X (t). 10) and, for any process X(t), Y (t )  e ( t ) satisfies the stochastic differential given above when 1 2  (t )  X(t )  X(0)  [ X , X ](t ) . 11) [X, X](t) is quadratic variation of X (t) and for a continuous function with finite variation [X,X](t) = 0.

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