Distributions In The Physical And Engineering Sciences Volume: Exploring the Mathematical Foundation of Physical Phenomena

Distributions play a crucial role in the physical and engineering sciences, providing a mathematical framework to represent physical phenomena with varying degrees of complexity. They provide a powerful tool that allows scientists and engineers to understand and model a wide range of natural and artificial systems. In this article, we will explore the concept of distributions and their applications in the physical and engineering sciences.

Understanding Distributions

Distributions, also known as generalized functions, are mathematical objects that extend the concept of functions to include non-smooth and discontinuous functions. Unlike traditional functions, distributions can be defined on a larger class of objects, including discontinuous and piecewise-defined functions.

Distributions in the Physical and Engineering Sciences, Volume 3: Random and Anomalous Fractional Dynamics in Continuous Media (Applied and Numerical Harmonic Analysis)

by Alexander I. Saichev(1st ed. 2018 Edition, Kindle Edition)

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Language	:	English
File size	:	9396 KB
Print length	:	423 pages

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One of the key features of distributions is their ability to represent physical quantities that contain impulses or singularities, such as the delta function. These quantities are commonly encountered when modeling real-world physical phenomena, and distributions provide an elegant way to mathematically represent and work with them.

The Role of Distributions in Physical Phenomena

Distributions have a wide range of applications in the physical and engineering sciences. They allow scientists and engineers to describe and analyze physical phenomena that involve singularities, such as point sources of energy or mass. By using distributions, researchers can mathematically model and simulate the behavior of these phenomena, leading to a better understanding of the underlying principles at work.

For example, in electromagnetism, distributions are used to represent point charges or currents. These distributions can be used to calculate the electric and magnetic fields generated by these sources. Without the mathematical framework provided by distributions, accurately modeling and analyzing electromagnetic systems would be significantly more challenging.

Distributions are also widely used in fluid dynamics to represent flows around obstacles or vortices. By defining appropriate distributions, researchers can accurately describe the behavior of fluids in a variety of scenarios, including the interaction between fluids and solid objects.

Applications in Engineering

In engineering, distributions are crucial for understanding and designing systems that involve energy or signal transmission. For instance, in signal processing, distributions can be used to model and analyze the behavior of signals that contain impulses or highly concentrated energy. This is particularly relevant in communication systems, where the accurate representation of signals is essential for the design and optimization of transmission protocols.

Distributions are also used in control theory, a branch of engineering that deals with the design and analysis of systems with desired behaviors. By using distributions, engineers can mathematically describe the behavior of control systems and design feedback mechanisms to achieve desired performance.

Distributions are a fundamental concept in the physical and engineering sciences, providing a mathematical framework to represent and analyze physical phenomena with varying degrees of complexity. The ability to work with non-smooth and discontinuous functions allows scientists and engineers to accurately model and simulate a wide range of natural and artificial systems. From electromagnetism to fluid dynamics and engineering applications, distributions play a crucial role in understanding the mathematical foundation of physical phenomena. By further exploring the applications and properties of distributions, scientists and engineers can continue to advance our understanding of the physical world and develop innovative technologies that shape our future.

Distributions in the Physical and Engineering Sciences, Volume 3: Random and Anomalous Fractional Dynamics in Continuous Media (Applied and Numerical Harmonic Analysis)

by Alexander I. Saichev(1st ed. 2018 Edition, Kindle Edition)

5 out of 5

Language	:	English
File size	:	9396 KB
Print length	:	423 pages

FREE

Continuing the authors’ multivolume project, this text considers the theory of distributions from an applied perspective, demonstrating how effective a combination of analytic and probabilistic methods can be for solving problems in the physical and engineering sciences.  Volume 1 covered foundational topics such as distributional and fractional calculus, the integral transform, and wavelets, and Volume 2 explored linear and nonlinear dynamics in continuous media.  With this volume, the scope is extended to the use of distributional tools in the theory of generalized stochastic processes and fields, and in anomalous fractional random dynamics.  
Chapters cover topics such as probability distributions; generalized stochastic processes, Brownian motion, and the white noise; stochastic differential equations and generalized random fields; Burgers turbulence and passive tracer transport in Burgers flows; and linear, nonlinear, and multiscale anomalous fractional dynamics in continuous media.  The needs of the applied-sciences audience are addressed by a careful and rich selection of examples arising in real-life industrial and scientific labs and a thorough discussion of their physical significance. Numerous illustrations generate a better understanding of the core concepts discussed in the text, and a large number of exercises at the end of each chapter expand on these concepts.
Distributions in the Physical and Engineering Sciences is intended to fill a gap in the typical undergraduate engineering/physical sciences curricula, and as such it will be a valuable resource for researchers and graduate students working in these areas.  The only prerequisites are a three-four semester calculus sequence (including ordinary differential equations, Fourier series, complex variables, and linear algebra),and some probability theory, but basic definitions and facts are covered as needed.  An appendix also provides background material concerning the Dirac-delta and other distributions.