Identification Parameters of Mathematical Models

Another approach to identification parameters of the presented model is based on validation of the coefficient of diffusion D (jc). Both external and internal factors can influence this coefficient substantially. Therefore, its precise mathematical description cannot be obtained on the basis of a priori data, but only by using the special mathematic methods of data processing of the conditions of measuranent u (f, z) at a set temperature. These data can be determined by the limited number of measurements in points Xj,j= 1,. .., M of the spatial region [0, x ] and temporal region [0, T] (Xj e [0, X . ]), (( [0, T]), j = 1,, M, where M is the total number 

0] is the diagonal matrix scaled (M x M), characterizing the mode of measurement. Suppose that the solution of Equations (2.30)-(2.37) exists and it is the sole one. Then the identification of the coefficient of diffusion D (x) can be formulated in the following way. 

It is necessary to calculate the real coefficient of diffusion D (x) of Equations (2.30M2.37) on the basis of the discrete data in the space of measurement Zj (x) (/ = 1,. .., M), which causes the minimum to the quadratic function  

The formulated task is related to the class of problems on conditional extremes with due regard for Equations (2.30)-(2.37) as restrictions. It is necessary to import the Lagrange function L(D) for criterion (2.52) and Equation (2.30) in order to transfer the formulated task to the task solution on unconditional extreme [64]  

The correlation obtained for derivative dL(D) / dD can be used as an essential condition for minimization of the criterion (2.53) and, consequently, (2.54) by the coefficient of diffusion D employing one of the gradient methods. 

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