Sequential Models of Mathematical Physics

Simon Serovajsky

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9780429893650
• Edition: 1
• Topic: Matematica
• Subtopic: Matematica generale

The equations of mathematical physics are the mathematical models of the large class of phenomenon of physics, chemistry, biology, economics, etc. In Sequential Models of Mathematical Physics, the author considers the justification of the process of constructing mathematical models. The book seeks to determine the classic, generalized and sequential solutions, the relationship between these solutions, its direct physical sense, the methods of its practical finding, and its existence.

Features

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• Describes a sequential method based on the construction of space completion, as well as its applications in number theory, the theory of distributions, the theory of extremum, and mathematical physics

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• Presentation of the material is carried out on the simplest example of a one-dimensional stationary heat transfer process; all necessary concepts and constructions are introduced and illustrated with elementary examples, which makes the material accessible to a wide area of readers

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• The solution of a specific mathematical problem is obtained as a result of the joint application of methods and concepts from completely different mathematical directions

Part III

Sequential objects

The necessity of justifying the procedure for constructing mathematical models led us to the study of the problem of convergence of the sequences. The existence of the limit can be proven by the Cauchy criterion. Its direct application is possible only in the case of completeness of the space. We can use the completion procedure for the general situation, which is the basis of the sequential method. The most natural realization of this idea is the definition of real numbers based on the completion of the set of rational numbers.

Before applying this apparatus to construct mathematical models, we will consider other forms of sequential objects. Particularly, the set of real numbers is not the unique extension of the class of rational numbers by the sequential method. The extremely interesting and important class of p-adic numbers is another sequential extension of the set of rational numbers. A qualitatively different application of this method is the optimal control theory, where the sequential method is a means of analyzing extremum problems that do not have any solutions in the natural sense. In the concluding chapter of this part, we consider a sequential interpretation of the set of distributions associated with the extension of the solution of mathematical physics problems. These results are already close to solving the problem of justifying mathematical models.

Chapter 6

p-adic numbers

6.1Comparisons of integers modulo

6.2Integer p-adic numbers

6.3General p-adic numbers

6.4p-adic metrics

6.5Sequential definition of p-adic numbers

6.6Conclusions

Notes

The justification of mathemat...