Mathematical Description of Elastic Phenomena 2 which Uses Caputo or Riemann-Liouville Fractional 3 Order Partial Derivatives is Nonobjective 4

In this paper it is shown that mathematical description of strain, constitutive law and 12 dynamics obtained by direct replacement of integer order derivatives with Caputo or Riemann13 Liouville fractional order partial derivatives, having integral representation on finite interval, in 14 case of a guitar string, is nonobjective. The basic idea is that different observers, using this type of 15 descriptions, obtain different results which cannot be reconciled, i.e. transformed into each other 16 using only formulas that link the coordinates of the same point in two fixed orthogonal reference 17 frames and formulas that link the numbers representing the same moment of time in two different 18 choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: 19 which one of the obtained results is correct? 20


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The mathematical description of a real world phenomenon is objective if it is independent on 26 the observer. That is, it is possible to reconcile observation of the phenomenon into a single coherent

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In classical theory of elasticity [2] a material particle Q of a material body  76  77  3  ,  2  ,  1  ) , , , ( X X X are the coordinates, with respect to O R , of the point P ( which represents 80 the material particle

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Because (1) and (2) describe the movement of the same material particle Q the following 114 relations hold:  (1) and (2) and make possible the description by 141 one of them. This means that the description (1) of the material particles movement in elasticity is 142 objective. Two observers who describe the material particles movement of an elastic body with (1),

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Observer O describes the displacement of the particle Q of B at the moment of time M 145 by the vector valued function :

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Relations which reconcile the displacement description made by (6) with that made by (7) and 155 make possible the description of the displacement by one of them, are the following:

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Observer O describes the principal directions of strains and the principal strains with the 182 solution of the equations

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The objectivity of the above presented descriptions implies that, different observers describing 209 the same phenomenon using integer order partial derivatives, obtain results which can be reconciled.

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That is why in our work we consider that this statement is only a conjecture or a belief based on

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Remember that for a continuously differentiable function Remark that the derivative defined with (15) was considered by other people before Caputo, like 262 Gherasimov (see [14]). So, the name of Caputo, used in this paper, may be is not appropriate.

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For a continuously differentiable function

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We chose for the special issue Mathematical Modelling in Applied Sciences the very simple case that 280 of the guitar string. Thus was "born" sections 2 , 3 and 4 of the manuscript in which we analyzed the 281 objectivity of the description of guitar string strain defined instead of integer order partial derivative 282 (formula (10) ) with spatial Caputo fractional order partial derivative having integral representation 283 on finite interval . That is: The strain considered by us is not the strain considered in [3] which is:

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The similarity consists only in the fact that in both cases Caputo fractional partial derivatives,

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The functions  and  * appearing in (17)

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Observer O describes a movement of the point P from Using Caputo fractional spatial partial derivative of order    If the considered description is objective, then for the following equalities hold:

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In particular if the description is objective, then It follows that : if the strain description is objective, then the next identity holds:

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If the considered description is obiective, then: for 2 , 1 ,  j i the following equalities hold: (31)

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In particular, if the description is objective, then

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On the other hand, equality It follows that : if the considered description is objective, then the next identity holds:

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It is easy to see that For 455 E 0 =0 and E 1 =1 this law become: Observer O describes a movement of a point P from given by: For observer O the components 1 U , 2 U of the displacement vector are :

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It follows that if the description is objective, then the next identity holds: For objectivity the additional terms which appear in case of Caputo or Riemann-Liouville 738 fractional order spatial derivative has to be equal to zero. At the end of section 2.and 3. there is a short 739 discusion about the situation when the additional terms are equal to zero.But even if the additional 740 term is equal to zero, the objectivity of the description does not result because the condition is only 741 necessary. This means that following this way we cannot find an answer to the question: which is the 742 suitable choices of fractional-order assuring the objectivity?