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Sequential models of mathematical physics
The equations of mathematical physics are the mathematical models of the large class of phenomenon of physics, chemistry, biology, economics, et cetera 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 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 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 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
Cover; Half Title; Title Page; Copyright Page; Dedication; Contents; Preface; Author; Part I: Mathematical physics problems; Chapter 1: Classic models; 1.1 Mathematical analysis of a physical phenomenon; 1.2 Definition of a mathematical model; 1.3 Classic solution of the system; 1.4 Approximate solution of the system; 1.5 Validity of the classic method; 1.6 Conclusions; Chapter 2: Generalized models; 2.1 Generalized solution of the problem; 2.2 Determination of the generalized model; 2.3 Generalized derivatives; 2.4 Approximation of the generalized model
2.5 Validity of the generalized method2.6 Conclusions; Part II: Sequential method; Chapter 3: Convergence and Cauchy principle; 3.1 Definitions of the convergence; 3.2 Non-constructiveness of the limit; 3.3 Cauchy criterion of the convergence; 3.4 Picard's method for differential equations; 3.5 Banach fixed point theorem; 3.6 Conclusions; Chapter 4: Completeness and real numbers; 4.1 Inapplicability of the Cauchy criterion; 4.2 Complete metric spaces; 4.3 Completion problem; 4.4 Real numbers by Cantor; 4.5 Conclusions; Chapter 5: Real numbers and completion
5.1 Axiomatic definition of real numbers5.2 Weierstrass real numbers; 5.3 Properties of Weierstrass real numbers; 5.4 Properties of Cantor real numbers; 5.5 Completion of metric spaces; 5.6 Conclusions; Part III: Sequential objects; Chapter 6: p-adic numbers; 6.1 Comparisons of integers modulo; 6.2 Integer p-adic numbers; 6.3 General p-adic numbers; 6.4 p-adic metrics; 6.5 Sequential definition of p-adic numbers; 6.6 Conclusions; Chapter 7: Sequential controls; 7.1 Optimal control problems; 7.2 Insolvable optimal control problems; 7.3 Sequential controls
7.4 Extension of the easiest extremum problem7.5 Extension of the optimal control problem; 7.6 Non-uniqueness of the optimal control; 7.7 Tihonov well-posedness of the optimal control problems; 7.8 Conclusions; Chapter 8: Distributions; 8.1 Test functions; 8.2 Schwartz distributions; 8.3 Sequential distributions; 8.4 Sobolev spaces; 8.5 Conclusions; Part IV: Sequential models; Chapter 9: Sequential models of mathematical physics phenomena; 9.1 Sequential model of the heat transfer phenomenon; 9.2 Justification of sequential modeling; 9.3 Generalized model of the heat transfer phenomenon
9.4 Classic model of the heat transfer phenomenon9.5 Models of mathematical physics problems; 9.6 Conclusions; Bibliography; Index