![]()
[ 1 ]
Introduction
1.1 Preliminary Remarks
In Heavisideās operational calculus, in particular in the application of this operational calculus to partial differential equations, difficulties arise as a result of the occurrence of certain operators whose meaning is not at all obvious. The interpretation of such operators as given by Heaviside and his successors is difficult to justify, and the range of validity of the calculus so developed remains unclear. A similar lack of clarity with regard to the range of validity also arises in connection with the use of the delta function and other impulse functions both in operational calculus and in other branches of applied mathematics.
In view of this situation one can either use operational calculus and impulse functions as a kind of shorthand or heuristic means for obtaining tentative solutions to be verified, if necessary, by the techniques of classical analysis (such an attitude seems to have been envisaged originally by Dirac when he introduced the delta function); or else it becomes necessary to develop a mathematical theory that will justify the process.
In this book such a theory will be developedānamely, the theory of
convolution quotients due to the Polish mathematician Jan Mikusi
ski. This theory is based on an extension of the concept āfunction,ā an extension somewhat akin to the extension of the concept ānumberā from integers to rational numbers (fractions). The resulting abstract entities of Mikusi
skiās theory may be interpreted either as operators or as generalized functions, and they include the operators of differentiation, integration, and related operators, and also the delta function and other impulse functions. Functions (in the ordinary sense of the word) and numbers also find their places in the system of convolution quotients.
Mikusi
skiās theory provides a satisfactory basis for operational calculus, and it can be applied successfully to ordinary and partial differential equations with constant coefficients, difference equations, integral equations, and also in some other fields.
In
sections 1.2 to
1.4 some problems are reviewed that arise in connection with operational calculus and impulse functions and some of the solutions that have been proposed are briefly indicated, including a preview of Mikusi
skiās theory. These sections are not required for the understanding of what follows and may be omitted.
Section 1.5 includes comments on the notion of integral to be used in this book: readers possessing an adequate knowledge of integration
theory may omit this discussion. Some of the notations and conventions that are used in the sequel are explained in
section 1.6.
Chapters 2 and 3 contain the elementary theory of convolution quotients and its application to ordinary linear differential equations with constant coefficients and to certain integral equations. These two chapters form a self-contained whole, and a short course may be based on them. In chapter 4 the convergence theory of convolution quotients is developed and operator functions are introduced; and in chapter 5 differential equations involving operator functions are discussed and exponential functions of operators are introduced. The tools developed in these chapters are then applied in chapters 6 and 7 to problems in partial differential equations.
1.2 Heaviside Calculus; Laplace Transforms
In some contextsāfor instance, for the solution of ordinary linear differential equations with constant coefficientsāit is usual to treat the operator of differentiation as an algebraic entity. The differential equation
is written in the form P(D) z = f, where D = d/dt, and
is a polynomial with constant coefficients. The solution of (1) appears as
and this solution i...
