Ijraset Journal For Research in Applied Science and Engineering Technology

Abstract

This research presents a comprehensive study of generalized functions and distribution theory, exploring their fundamental concepts, properties, and applications in mathematics and mathematical physics. The research aims to provide a thorough understanding of this powerful mathematical framework, shedding light on its versatile applications in diverse areas of study.

Introduction

I. INTRODUCTION

Generalized functions and distributions, also known as generalized function theory or distribution theory, are mathematical tools used to extend the concept of functions beyond traditional functions that can be represented by ordinary mathematical expressions. They provide a framework for analysing and manipulating objects that are not necessarily well-defined functions in the classical sense.

Generalized functions and distribution theory emerged as a powerful mathematical framework in the mid-20th century to address the limitations of classical functions in describing certain phenomena and mathematical operations. Traditional functions often fail to capture singularities, impulses, and non-smooth phenomena encountered in various scientific and engineering disciplines. This necessitated the development of a more flexible and rigorous mathematical approach, leading to the formulation of generalized functions and distribution theory. In many scientific and engineering applications, traditional functions fail to capture certain phenomena or mathematical operations. For example, functions that describe point sources of energy or impulse-like events are difficult to represent using standard functions. Generalized functions offer a solution to this problem by providing a more flexible and powerful mathematical framework. The concept of generalized functions was first introduced by the French mathematician Laurent Schwartz in the 1940s. Schwartz developed the theory of distributions to provide a rigorous mathematical treatment of objects like the Dirac delta function, which represents an idealized point source. The theory of distributions was further developed and refined by many mathematicians, including Sergei Sobolev, Ivan Petrovsky, and Georges de Rham.

One key idea in distribution theory is that of a distribution as a linear functional acting on a space of test functions. Test functions are smooth and well-behaved functions with compact support, meaning they vanish outside a finite interval. Distributions can be thought of as generalized functionals that associate a value to each test function. This allows for the representation of objects like the Dirac delta function, which is not a conventional function but can be understood as a distribution.

Generalized functions and distributions find applications in various branches of mathematics, physics, and engineering. They are particularly useful in areas such as partial differential equations, Fourier analysis, signal processing, quantum mechanics, and general relativity. By employing the tools of distribution theory, researchers can handle singularities, impulses, and other non-smooth phenomena more rigorously and efficiently.

References

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Copyright

Copyright © 2023 Mariyam Kazmi, Chinta Mani Tiwari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.