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Abstracts

XX conference

The Problem of the Costructive Determination of Mathematical Physics Problems

Серовайский С.Я.

Faculty of Mathematics and Mechanics, 71 al Farabi av. Almaty 050078 Kazakhstan

The standard method of determination of mathematical physics problems implies the separation an elementary volume of the given domain. Then the balance relations of the change of the energy, pulse, mass, charge, etc. in this volume during a time interval are determined. Next step is the passing to the limit as the space and time set tends to a point. Mathematical physics equations are the result of these transformations. It is the basis of the mathematical model of the analyzed phenomenon.

This method is habitual and natural. So a circumstance is not taking into account frequently. There is the question before passing to the limit. Does this limit exist, in principle? This question is natural enough. So authors write: suppose the considered functions have the continuous derivatives. By this supposition the limit exists, and the final result is clear without doubt. But we receive now another question: why these functions are smooth enough?

The considered functions describe the state of the system. There are the temperature, the pressure, the velocity, the concentration, etc. It was the solutions of state equations. Why we know that it has the concrete properties, if we do not get so far the state equation? We could use the physical arguments. However it is not correct because we analyze the mathematical model but not the physical phenomenon. Besides the absence of the smoothness can have physical sense. It can be knock waves, an inhomogeneity of the environment, the phase change, etc.

We will use an analogue with limit definition in this situation. It is known that the definition of the limit is not constructive because it uses the preliminary information about of this limit. Therefore the convergence criterions are used for the proof of convergences. It is, for example, Caushy criterion. Unfortunately it is useful only for the complete spaces. However the completion technics can be used for the non-complete cases. This method was used for Cantor’s definition of the real numbers and for Huasdorf completion theorem for metric spaces. We will use this method for the determination of the mathematical physics problem without preliminary suppositions for properties of state functions.

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