The damped vibrating string equation on the positive half-line

J 2023

The damped vibrating string equation on the positive half-line

PAVLAČKOVÁ, Martina and Valentina TADDEI

Basic information

Original name

The damped vibrating string equation on the positive half-line

Authors

PAVLAČKOVÁ, Martina (203 Czech Republic, belonging to the institution) and Valentina TADDEI (380 Italy)

Edition

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2023, 1007-5704

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Netherlands

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Organization unit

Moravian Business College Olomouc

DOI

http://dx.doi.org/10.1016/j.cnsns.2023.107497

UT WoS

001079725700001

Keywords in English

Damped vibrating string equation; Second-order Cauchy problem; Banach spaces; Fundamental systém; Approximation solvability method; Mild solution

Abstract

V originále

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the righthand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Citovat

PAVLAČKOVÁ, Martina and Valentina TADDEI. The damped vibrating string equation on the positive half-line. COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION. 2023, Neuveden, No 126, p. 1-18. ISSN 1007-5704. Available from: https://dx.doi.org/10.1016/j.cnsns.2023.107497.

@article{349629,
   author = {Pavlačková, Martina and Taddei, Valentina},
   article_number = {126},
   doi = {http://dx.doi.org/10.1016/j.cnsns.2023.107497},
   keywords = {Damped vibrating string equation; Second-order Cauchy problem; Banach spaces; Fundamental systém; Approximation solvability method; Mild solution},
   language = {eng},
   issn = {1007-5704},
   journal = {COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION},
   title = {The damped vibrating string equation on the positive half-line},
   url = {https://www.sciencedirect.com/science/article/pii/S1007570423004185},
   volume = {Neuveden},
   year = {2023}
}

TY  - JOUR
ID  - 349629
AU  - Pavlačková, Martina - Taddei, Valentina
PY  - 2023
TI  - The damped vibrating string equation on the positive half-line
JF  - COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
VL  - Neuveden
IS  - 126
SP  - 1-18
EP  - 1-18
SN  - 10075704
KW  - Damped vibrating string equation
KW  - Second-order Cauchy problem
KW  - Banach spaces
KW  - Fundamental systém
KW  - Approximation solvability method
KW  - Mild solution
UR  - https://www.sciencedirect.com/science/article/pii/S1007570423004185
N2  - In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the righthand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.
ER  -

PAVLAČKOVÁ, Martina and Valentina TADDEI. The damped vibrating string equation on the positive half-line. \textit{COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION}. 2023, Neuveden, No~126, p.~1-18. ISSN~1007-5704. Available from: https://dx.doi.org/10.1016/j.cnsns.2023.107497.

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