A Study on Fourth-Order Coupled Boundary Value Problems: Existence, Uniqueness and Approximations

Year 2025, , 19 - 30, 31.03.2025

Emirhan Hacıoğlu

https://doi.org/10.33401/fujma.1613301

Abstract

This study examines the existence and approximation of solutions for a coupled system of fourth-order boundary value problems (4th-BVPs), which model the interactions between two distinct but interrelated physical systems. These coupled boundary value problems arise in various applications in engineering and physics, including the analysis of bending behaviors in beams and vibrations in interconnected structural components. By leveraging Green`s functions and building upon prior research in fourth-order differential equations, we derive sufficient conditions for the existence and uniqueness of solutions to the system. Additionally, we provide a numerical framework for approximating these solutions, offering practical insights for real-world applications.

Keywords

Fourth-order boundary value problem, Existence, Convergence, Approximation

References

• [1] R.P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differ. Integral Equ., 2(1) (1989), 91-110. $ \href{http://dx.doi.org/10.57262/die/1372191617}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33947175349&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+fourth+order+boundary+value+problems+arising+in+beam+analysis%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} $
• [2] E.R. Kaufmann and K. Nickolai, Elastic beam problem with higher order derivatives, Nonlinear Anal. Real World Appl., 8(3) (2007), 811- 821. $ \href{https://doi.org/10.1016/j.nonrwa.2006.03.006}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33846231387&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Elastic+beam+problem+with+higher+order+derivatives%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000245444200007}{\mbox{[Web of Science]}} $
• [3] D. Habib, S. Benaicha and N. Bouteraa, Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem, Fundam. J. Math. Appl., 1(2) (2018), 205-211. $\href{https://doi.org/10.33401/fujma.418934}{\mbox{[CrossRef]}} $
• [4] S.S. Almuthaybiri and C.C. Tisdell, Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis, Open Math., 18(1) (2020), 1006-1024. $ \href{https://doi.org/10.1515/math-2020-0056}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85093673803&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Sharper+existence+and+uniqueness+results+for+solutions+to+fourth-order+boundary+value+problems+and+elastic+beam+analysis%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000577014400001}{\mbox{[Web of Science]}} $
• [5] H. Chen and Y. Cui,Existence and uniqueness of solutions to the nonlinear boundary value problem for fourth-order differential equations with all derivatives, J. Inequal. Appl., 2023(1) (2023), 23. $ \href{https://doi.org/10.1186/s13660-022-02907-9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85147750270&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+and+uniqueness+of+solutions+to+the+nonlinear+boundary+value+problem+for+fourth-order+differential+equations+with+all+derivatives%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000934971500002}{\mbox{[Web of Science]}} $
• [6] C. Zhai and D.R. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl., 375(2) (2011), 388-400. $ \href{https://doi.org/10.1016/j.jmaa.2010.09.017}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78149412425&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+sum+operator+equation+and+applications+to+nonlinear+elastic+beam+equations+and+Lane-Emden-Fowler+equations%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000284343200002}{\mbox{[Web of Science]}} $
• [7] A. Granas, R.B. Guenther and J.W. Lee, Nonlinear Boundary Value Problems for Ordinary Differential Equations, Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), (1985). $ \href{https://eudml.org/doc/268365}{\mbox{[Web]}} $
• [8] R. Rao and J.M. Jonnalagadda Existence of a unique solution to a fourth-order boundary value problem and elastic beam analysis Math. Model. Control., 4(3) (2024), 297-306. $ \href{https://doi.org/10.3934/mmc.2024024}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85204229290&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+of+a+unique+solution+to+a+fourth-order+boundary+value+problem+and+elastic+beam+analysis%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001314913000003}{\mbox{[Web of Science]}} $