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2018 | Buch

Complements of Higher Mathematics

verfasst von: Prof. Dr. Marin Marin, Dr. Andreas Öchsner

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book highlights the remarkable importance of special functions, operational calculus, and variational methods. A considerable portion of the book is dedicated to second-order partial differential equations, as they offer mathematical models of various phenomena in physics and engineering.

The book provides students and researchers with essential help on key mathematical topics, which are applied to a range of practical problems. These topics were chosen because, after teaching university courses for many years, the authors have found them to be essential, especially in the contexts of technology, engineering and economics. Given the diversity topics included in the book, the presentation of each is limited to the basic notions and results of the respective mathematical domain. Chapter 1 is devoted to complex functions. Here, much emphasis is placed on the theory of holomorphic functions, which facilitate the understanding of the role that the theory of functions of a complex variable plays in mathematical physics, especially in the modeling of plane problems.

In addition, the book demonstrates the importance of the theories of special functions, operational calculus, and variational calculus. In the last chapter, the authors discuss the basic elements of one of the most modern areas of mathematics, namely the theory of optimal control.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Complex Functions

Abstract

This chapter contains the basic results on complex functions of real as well as of complex variables. For more details and more results, the readers are referred to the books in the bibliography section.

Marin Marin, Andreas Öchsner

Chapter 2. Special Functions

Abstract

Consider the semi-plane $\Delta _0=\{z\in C,\;z=x+iy:x>0\}$. The complex function $\Gamma :\Delta _0\rightarrow C$ defined by

$$ \Gamma (z)=\int \limits _{0}^{\infty }t^{z-1}e^{-t}\mathrm {d}t, $$

is called the Euler’s function of first species.

Marin Marin, Andreas Öchsner

Chapter 3. Operational Calculus

Abstract

An useful instrument in tackling differential equations and partial differential equations is proved to be the Laplace’s transform which we study in this paragraph. The Laplace’s transform makes the correspondence between two functions set, one having difficult operations, and second, more accessible. For instance, a differential equation in the first functions set is transformed in an algebrical equation in the second functions set. This correspondence is made by means of a transformation. We will deal only with the Laplace’s transform and Fourier’s transform.

Marin Marin, Andreas Öchsner

Chapter 4. Fourier’s Transform

Abstract

Consider the trigonometrical series of the following form

$$ \frac{a_0}{2}+\sum \limits _{n=1}^{\infty }\left( a_n\cos n\omega x+ b_n\sin n\omega x\right) . $$

Since the functions $\cos n\omega x$ and $\sin n\omega x$ are periodical functions having the period $T=2\pi /\omega $ we say that the series (4.1.1) is a periodical series.

Marin Marin, Andreas Öchsner

Chapter 5. Calculus of Variations

Abstract

The modern engineer often has to deal with problems that require a sound mathematical background and set skills in the use of various mathematical methods. Expanding the mathematical outlook of engineers contributes appreciably to new advances in technology. The calculus of variations is one of the most important divisions of classical mathematical analysis in regards to applications.

Marin Marin, Andreas Öchsner

Chapter 6. Quasi-linear Equations

Abstract

Let $\Omega $ be a bounded domain in the $n-$dimensional Euclidean space $\mathrm{I}\!\mathrm{R}^n$. The general form of a partial differential equation is: $F(x,u, u_{x_1},\ldots ,u_{x_n}, u_{x_1x_2},\ldots , u_{x_ix_j}, u_{x_1x_2\ldots x_i},\ldots , u_{x_1x_2\ldots x_n})=0,$ where by $u_{x_i},u_{x_ix_j}, u_{x_ix_jx_k},\ldots ,$ we have denoted partial derivatives $\frac{\partial u}{\partial x_i}, \frac{\partial ^2 u}{\partial x_i\partial x_j}, \frac{\partial ^3 u}{\partial x_i\partial x_j\partial x_k},\ldots $.

Marin Marin, Andreas Öchsner

Chapter 7. Hyperbolical Equations

Abstract

The main representative of hyperbolical equations is considered to be the equation of the vibrating chord, also called the equation of waves.

Marin Marin, Andreas Öchsner

Chapter 8. Parabolical Equations

Abstract

The main exponent of the parabolical equations is the equation of heat conduction in a body. The general aim of this paragraph is to study the following initial-boundary values problem, attached to the equation (homogeneous, in first instance) of the heat conduction in a rod. This is a bar with its cross section small in comparison with the length.

Marin Marin, Andreas Öchsner

Chapter 9. Elliptic Partial Differential Equations

Abstract

Let us consider the three-dimensional regular domain $D\subset R^3$ bounded by the Liapunov surface $S=\partial D$. In the classical mathematical analysis the following formula is proved which is called the Gauss-Ostrogradski-Green’s formula.

Marin Marin, Andreas Öchsner

Chapter 10. Optimal Control

Abstract

In this section we will introduce some notions and results which are specific to the functional analysis and are necessary in the whole present chapter. Since we consider these notions and results being subordinate to the main objectives of this chapter, we shall renounce to prove the results. Our readers are invited, for more details, to consult the titles cited in the bibliography dedicated to functional analysis and convex analysis.

Marin Marin, Andreas Öchsner

Backmatter

Titel: Complements of Higher Mathematics
verfasst von: Prof. Dr. Marin Marin
Dr. Andreas Öchsner
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-74684-5
Print ISBN: 978-3-319-74683-8
DOI: https://doi.org/10.1007/978-3-319-74684-5

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