Applying unrigorous mathematics: Heaviside's operational calculus

SummaryThe approximate supersonic flow past a slender ducted body of revolution having an annular intake is determined by using the Heaviside operational calculus applied to the linearised equation for the velocity potential. It is assumed that the external and internal flows are independent. The pressures on the body are integrated to find the drag, lift and moment coefficients of the external forces. The lift and moment coefficients have the same values as for a slender body of revolution without an intake, but the formula for the drag has extra terms given in equations (32) and (56). Under extra assumptions, the lift force due to the internal pressures is estimated. The results are applicable to propulsive ducts working under the specified condition of no “ spill-over “ at the intake.

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Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽

10.3390/sym13020354 ◽

2021 ◽

Vol 13 (2) ◽

pp. 354

Author(s):

Alexander Apelblat ◽

Francesco Mainardi

Keyword(s):

Laplace Transform ◽

Operational Calculus ◽

Inverse Laplace Transform ◽

Elementary Functions ◽

Hyperbolic Functions ◽

Error Functions ◽

Convolution Integrals ◽

Special Case ◽

Integral Identities ◽

Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

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