A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation
Coupled systems of differential-algebraic equations (DAEs) and partial differential equations (PDEs) appear in various fields of applications such as electrical engineering, bio-mathematics, or multi-physics. They are of particular interest for the modeling and simulation of flow networks, for insta...
Saved in:
| Main Author: | |
|---|---|
| Format: | Online |
| Language: | English |
| Published: |
Logos Verlag Berlin
2024
|
| Subjects: | |
| Online Access: | ONIX_20241119_9783832557737_17 |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1869516205153320960 |
|---|---|
| author | Groh, Dennis |
| author_browse | Groh, Dennis |
| author_facet | Groh, Dennis |
| author_sort | Groh, Dennis |
| collection | Directory of Open Access Books |
| description | Coupled systems of differential-algebraic equations (DAEs) and partial differential equations (PDEs) appear in various fields of applications such as electrical engineering, bio-mathematics, or multi-physics. They are of particular interest for the modeling and simulation of flow networks, for instance energy transport networks. In this thesis, we discuss a system in which an abstract DAE and a second order hyperbolic PDE are coupled through nonlinear coupling functions. The analysis presented is split into two parts: In the first part, we introduce the concept of matrix-induced linear operators which arise naturally in the context of abstract DAEs but have surprisingly not been discussed in literature on abstract DAEs so far. We also present a novel index-1-like criterion that allows to separate dynamical and non-dynamical parts of the abstract DAE while allowing for a considerable reduction of required assumptions, compared to existing theoretical results for abstract DAEs. In the second part, we build upon the developed techniques. We show how to combine the theoretical frameworks for abstract DAEs and second order hyperbolic PDEs in a way such that both parts of the solution are of similar regularity. We then use a fixed-point approach to prove existence and uniqueness of local as well as global solutions to the coupled system. In the last part of this thesis, we throw a glance at a related optimal control problem and prove existence of a global minimizer. |
| format | Online |
| id | doab-20.500.12854ir-148063 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Logos Verlag Berlin |
| publisherStr | Logos Verlag Berlin |
| record_format | ojs |
| spelling | doab-20.500.12854ir-1480632024-11-19T16:32:27Z A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation Groh, Dennis PDAE Abstract DAE Wave Equation Existence and Uniqueness Optimal Control thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKA Calculus thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKF Functional analysis and transforms thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations thema EDItEUR::P Mathematics and Science::PB Mathematics::PBU Optimization thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists thema EDItEUR::T Technology, Engineering, Agriculture, Industrial processes::TJ Electronics and communications engineering::TJF Electronics engineering::TJFC Electronics: circuits and components Coupled systems of differential-algebraic equations (DAEs) and partial differential equations (PDEs) appear in various fields of applications such as electrical engineering, bio-mathematics, or multi-physics. They are of particular interest for the modeling and simulation of flow networks, for instance energy transport networks. In this thesis, we discuss a system in which an abstract DAE and a second order hyperbolic PDE are coupled through nonlinear coupling functions. The analysis presented is split into two parts: In the first part, we introduce the concept of matrix-induced linear operators which arise naturally in the context of abstract DAEs but have surprisingly not been discussed in literature on abstract DAEs so far. We also present a novel index-1-like criterion that allows to separate dynamical and non-dynamical parts of the abstract DAE while allowing for a considerable reduction of required assumptions, compared to existing theoretical results for abstract DAEs. In the second part, we build upon the developed techniques. We show how to combine the theoretical frameworks for abstract DAEs and second order hyperbolic PDEs in a way such that both parts of the solution are of similar regularity. We then use a fixed-point approach to prove existence and uniqueness of local as well as global solutions to the coupled system. In the last part of this thesis, we throw a glance at a related optimal control problem and prove existence of a global minimizer. 2024-11-19T16:32:26Z 2024-11-19T16:32:26Z 2024 book ONIX_20241119_9783832557737_17 9783832557737 https://directory.doabooks.org/handle/20.500.12854/148063 eng image/jpeg Attribution 4.0 International https://www.logos-verlag.de/cgi-bin/engbuchmid?isbn=5773&lng=eng&id= https://www.logos-verlag.de/ebooks/OA/978-3-8325-5773-7.pdf Logos Verlag Berlin Logos Verlag Berlin 10.30819/5773 10.30819/5773 04b263a1-7fba-4491-9eae-1c394ac42fc3 9783832557737 Logos Verlag Berlin 150 Berlin open access |
| spellingShingle | PDAE Abstract DAE Wave Equation Existence and Uniqueness Optimal Control thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKA Calculus thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKF Functional analysis and transforms thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations thema EDItEUR::P Mathematics and Science::PB Mathematics::PBU Optimization thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists thema EDItEUR::T Technology, Engineering, Agriculture, Industrial processes::TJ Electronics and communications engineering::TJF Electronics engineering::TJFC Electronics: circuits and components Groh, Dennis A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title | A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title_full | A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title_fullStr | A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title_full_unstemmed | A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title_short | A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation |
| title_sort | coupled system of differential algebraic equation and hyperbolic partial differential equation |
| topic | PDAE Abstract DAE Wave Equation Existence and Uniqueness Optimal Control thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKA Calculus thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKF Functional analysis and transforms thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations thema EDItEUR::P Mathematics and Science::PB Mathematics::PBU Optimization thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists thema EDItEUR::T Technology, Engineering, Agriculture, Industrial processes::TJ Electronics and communications engineering::TJF Electronics engineering::TJFC Electronics: circuits and components |
| topic_facet | PDAE Abstract DAE Wave Equation Existence and Uniqueness Optimal Control thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKA Calculus thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKF Functional analysis and transforms thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations thema EDItEUR::P Mathematics and Science::PB Mathematics::PBU Optimization thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists thema EDItEUR::T Technology, Engineering, Agriculture, Industrial processes::TJ Electronics and communications engineering::TJF Electronics engineering::TJFC Electronics: circuits and components |
| url | ONIX_20241119_9783832557737_17 |
| work_keys_str_mv | AT grohdennis acoupledsystemofdifferentialalgebraicequationandhyperbolicpartialdifferentialequation AT grohdennis coupledsystemofdifferentialalgebraicequationandhyperbolicpartialdifferentialequation |