The Cauchy problem for second order differential equations in Banach spaces

1967 ◽  
Vol 8 (1) ◽  
pp. 52-68 ◽  
Author(s):  
V. G. Raskin ◽  
P. E. Sobolevskii
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Second Order ◽  
Second Order Differential Equations ◽  
The Cauchy Problem

Solving a second-order algebro-differential equation with respect to the derivative

2021 ◽  
pp. 414-420
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Rigid Body ◽  
Fredholm Operator ◽  
Arbitrary Dimension ◽  
Second Order ◽  
Index Zero ◽  
Second Order Differential Equations ◽  
The Cauchy Problem ◽  
One Step

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

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