GENERALIZED INTEGRAL TRANSFORMS OF DISTRIBUTIONS

Download or read book GENERALIZED INTEGRAL TRANSFORMS OF DISTRIBUTIONS written by Dr. B. B. Waphare and published by Lulu Publication. This book was released on 2021-02-03 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1.1 Introduction In recent years, integral transforms have become essential working tools of every engineer and applied scientist. The Laplace transform, which undoubtedly is the most familiar example, is being suited to solving boundary value problems. The classical methods of solution of initial and boundary value problems in physics and engineering sciences have their roots in Fourier’s pioneering work. An alternative approach through integral transforms methods emerged primarily through Heaviside’s efforts on operational techniques. In addition to being of great theoretical interest to mathematicians, integral transform methods have been found to provide easy and effective ways of solving a variety of problems arising in engineering and physical science. The use of integral transforms is somewhat analogous to that of logarithms. That is, a problem involving multiplication or division can be reduced to one involving simple processes addition or subtraction by taking logarithms. For almost two centuries the method of function transformations has been used successfully in solving many problems in engineering, mathematical physics and applied mathematics. Function transformations include, but are not limited to the well-known technique of linear integral transformations. A function transformation simply means a mathematical operation through which a real or complex valued function f is transformed into an other F, or into a sequence of numbers, or more generally into a set of data. Since its birth in the 1780’s in the work of the great mathematician Laplace, on probability theory, the theory of function transformations has flourished and continues to do so. In the last few years, in particular, it has received a great impetus from the advent of wavelets. Not only is the wavelet transform an example of how practical function transformations can be, but it is also an example of a transformation that has gone beyond what it was designed to do as a technique. It has contributed to the development of modern mathematical analysis just as the Fourier transformation contributed to the advancement of classical analysis in the earliest years of the nineteenth century.