Operational Calculus

krabbe , gregers

ISBN 10: 3540048960 / ISBN 13: 9783540048961

Editore: Springer Verlag, 1970

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Riassunto:

Since the publication of an article by G. DOETSCH in 1927 it has been known that the Laplace transform procedure is a reliable sub- stitute for HEAVISIDE'S operational calculus*. However, the Laplace transform procedure is unsatisfactory from several viewpoints (some of these will be mentioned in this preface); the most obvious defect: the procedure cannot be applied to functions of rapid growth (such as the 2 function t ~ exp (t )). In 1949 JAN MIKUSINSKI indicated how the un- necessary restrictions required by the Laplace transform can be avoided by a direct approach, thereby gaining in notational as well as conceptual simplicity; this approach is carefully described in MIKUSINSKI'S textbook "Operational Calculus" [M 1J. . The aims of the present book are the same as MIKUSINSKI'S [M 1J: a direct approach requiring no un-necessary restrictions. The present operational calculus is essentially equivalent to the "calcul symbolique" of distributions having left-bounded support (see 6.52 below and pp. 171 to 180 of the textbook "Theorie des distributions" by LAURENT SCHWARTZ).

Contenuti: 1.- §0. Operators.- Test-functions and Operators.- Convolution.- Entering Functions.- § 1. Perfect Operators.- Algebra of Perfect Operators.- The Operator of an Entering Function.- Translation Property.- K-functions.- Numbers and Operators.- Ratios of Perfect Operators.- 2.- §2. The Basic Facts.- Translation Property.- Numbers and Operators.- Functions Having no Jumps on [0, ?).- Various Consequences.- Three Integral Equations.- Differential Equations: Procedure.- Higher Derivatives.- Continuity of the Convolution Integral.- § 3. Elementary Applications.- Table of Formulas.- Illustrative Examples.- § 4. Partial Fraction Decomposition.- Decompositions Obtained by Division.- Heaviside’s Expansion.- Decomposition into Two Parts.- Repeated Linear Factors.- Heaviside Procedure.- Repeated Quadratic and Linear Factors.- Illustrative Examples.- 3.- §5. Further Applications.- Series of Translates.- Periodic Functions.- Advantages of Operational Calculus.- Use of Laplace Transform Tables.- Difference Equations.- Bibliographical Comments.- Notes and Further Comments.- § 6. Calculus of Operators.- Convergence.- Value of an Operator at a Point.- Integral of an Operator.- Function-operators.- Impulse and Dipole.- Distributions and Operators.- Two Examples.- Multiplication by t.- Static Deflection of Beams.- §7. Vectors.- Electric Circuits.- The General Case.- §8 Non-integrable Functions.- Convolution of Operators.- Finite Part of a Divergent Integral.- Indexed Functions.- Derivatives on the Open Interval (0, ?).- Duhamel’s Formula.- 4.- § 9. Partial Differential Equations.- Differentiation with Respect to x.- Vibrating String.- A Diffusion Problem.- Wave Problems.- The Infinite Interval.- Classical Solutions.- The Finite Interval.- § 10. Diffusion Problems.- The Infinite Interval.- Insulated Rod.- The Finite Interval.- Comparison with the Laplace Transformation.- 5.- § 11. Series of Operators.- Series of Impulses.- Analytic Functions of h/D.- A Functional Calculus for h/D.- Applications.- Difference-differential Equations.- § 12. A Functional Calculus for D.- Applications.- § 13. Non-linear Equations.- Applications.- § 14. Differential Equations with Polynomial Coefficients.- Table of Formulas.- Illustrative Examples.- § 15. Theorems.- Three Basic Theorems.- A Theorem for § 6.- A Theorem for § 9.- A Theorem for § 11.- Glossary of Terminology and Notations.- Terminology.- Notations.- Summary of Results and Table of Formulas.- Elementary Formulas.- Periodic Functions.- Bibliographical Comments.- Subject and Author Index.

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Titolo: Operational Calculus
Casa editrice: Springer Verlag
Data di pubblicazione: 1970
Legatura: hardcover
Condizione: Acceptable