scholarly journals On the nonhomogeneous second-order Euler operator differential equation: Explicit solutions

1992 ◽  
Vol 177 ◽  
pp. 145-156 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Second Order ◽  
Explicit Solutions ◽  
Operator Differential Equation ◽  
Euler Operator

scholarly journals On the Stepanov-almost periodic solution of a second-order operator differential equation

1975 ◽  
Vol 19 (3) ◽  
pp. 261-263 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Almost Periodic Solution ◽  
Second Order ◽  
Almost Periodicity ◽  
Operator Differential Equation ◽  
Almost Periodic ◽  
Order Operator ◽  
Image Position

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).

On Heisenberg's elementary theory of turbulence

1949 ◽  
Vol 200 (1060) ◽  
pp. 20-33 ◽  
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Explicit Solution ◽  
Order Differential Equation ◽  
Elementary Theory ◽  
Second Order ◽  
Parametric Family ◽  
Explicit Solutions ◽  
Second Order Differential Equation ◽  
Equilibrium Spectrum

In this paper the spectrum of turbulence is considered on the basis of an elementary theory recently developed by Heisenberg. Explicit solutions for the spectrum have been obtained both when the conditions are stationary and an equilibrium spectrum obtains and when the conditions are non-stationary and the turbulence is decaying. In the former case the problem admits of an explicit solution. In the latter case the problem reduces to determining a one-parametric family of solutions of a certain second-order differential equation. The decay spectra for various values of the Reynolds number (which remains constant during the decay) are illustrated.

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