Existence of a unique solution to a fourth-order boundary value problem and elastic beam analysis

Ravindra Rao; Jagan Mohan Jonnalagadda; Ravindra Rao; Jagan Mohan Jonnalagadda

doi:10.3934/mmc.2024024

Mathematical Modelling and Control

2024, Volume 4, Issue 3: 297-306. doi: 10.3934/mmc.2024024

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Research article

Existence of a unique solution to a fourth-order boundary value problem and elastic beam analysis

Ravindra Rao ,
Jagan Mohan Jonnalagadda ^,

Department of Mathematics, Birla Institute of Technology & Science Pilani, Hyderabad, Telangana 500078, India

Received: 09 February 2024 Revised: 27 June 2024 Accepted: 18 August 2024 Published: 04 September 2024

We study the existence and uniqueness of solutions to a particular class of two-point boundary value problems involving fourth-order ordinary differential equations. Such problems have exciting applications for modeling the deflections of beams. The primary tools employed in this study include the application of Banach's and Rus's fixed point theorems. Our theoretical results are applied to elastic beam deflections when the beam is subjected to a loading force and both ends are clamped. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.
- fourth-order boundary value problem,
- Green's function,
- fixed point,
- existence,
- uniqueness,
- elastic beam analysis
Citation: Ravindra Rao, Jagan Mohan Jonnalagadda. Existence of a unique solution to a fourth-order boundary value problem and elastic beam analysis[J]. Mathematical Modelling and Control, 2024, 4(3): 297-306. doi: 10.3934/mmc.2024024

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Abstract

We study the existence and uniqueness of solutions to a particular class of two-point boundary value problems involving fourth-order ordinary differential equations. Such problems have exciting applications for modeling the deflections of beams. The primary tools employed in this study include the application of Banach's and Rus's fixed point theorems. Our theoretical results are applied to elastic beam deflections when the beam is subjected to a loading force and both ends are clamped. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.

References

[1]	R. P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differ. Integral Equations, 2 (1989), 91–110. https://doi.org/10.57262/die/1372191617 doi: 10.57262/die/1372191617
[2]	S. S. Almuthaybiri, C. C. Tisdell, Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis, Open Math., 18 (2020), 1006–1024. https://doi.org/10.1515/math-2020-0056 doi: 10.1515/math-2020-0056
[3]	A. Cabada, L. Saavedra, Existence of solutions for $n^th$-order nonlinear differential boundary value problems by means of fixed point theorems, Nonlinear Anal., 42 (2018), 180–206. https://doi.org/10.1016/j.nonrwa.2017.12.008 doi: 10.1016/j.nonrwa.2017.12.008
[4]	H. Chen, 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 (2023), 23. https://doi.org/10.1186/s13660-022-02907-9 doi: 10.1186/s13660-022-02907-9
[5]	D. Franco, D. O'Regan, J. Perán, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174 (2005), 315–327. https://doi.org/10.1016/j.cam.2004.04.013 doi: 10.1016/j.cam.2004.04.013
[6]	A. Granas, R. Guenther, J. Lee, Nonlinear boundary value problems for ordinary differential equations, Diss. Math., 244 (1985), 1–128.
[7]	C. P. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Anal. Appl., 135 (1988), 208–225. https://doi.org/10.1016/0022-247X(88)90149-7 doi: 10.1016/0022-247X(88)90149-7
[8]	D. Jiang, H. Liu, X. Xu, Nonresonant singular fourth-order boundary value problems, Appl. Math. Lett., 18 (2005), 69–75. https://doi.org/10.1016/j.aml.2003.05.016 doi: 10.1016/j.aml.2003.05.016
[9]	E. R. Kaufmann, N. Kosmatov, Elastic beam problem with higher order derivatives, Nonlinear Anal., 8 (2007), 811–821. https://doi.org/10.1016/j.nonrwa.2006.03.006 doi: 10.1016/j.nonrwa.2006.03.006
[10]	Y. Li, Y. Gao, Existence and uniqueness results for the bending elastic beam equations, Appl. Math. Lett., 95 (2019), 72–77. https://doi.org/10.1016/j.aml.2019.03.025 doi: 10.1016/j.aml.2019.03.025
[11]	F. Minhós, I. Coxe, Systems of coupled clamped beams equations with full nonlinear terms: existence and location results, Nonlinear Anal., 35 (2017), 45–60. https://doi.org/10.1016/j.nonrwa.2016.10.006 doi: 10.1016/j.nonrwa.2016.10.006
[12]	Y. Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc., 104 (1988), 175–180. https://doi.org/10.2307/2047481 doi: 10.2307/2047481
[13]	C. Zhai, D. R. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl., 375 (2011), 388–400. https://doi.org/10.1016/j.jmaa.2010.09.017 doi: 10.1016/j.jmaa.2010.09.017
[14]	E. Zeidler, Nonlinear functional analysis and its applications. Ⅰ: fixed-point theorems, Springer-Verlag, 1986.
[15]	I. A. Rus, On a fixed point theorem of Maia, Stud. Univ. Babes Bolyai Math., 22 (1977), 40–42.
[16]	M. Arda, J. Majak, M. Mehrparvar, Longitudinal wave propagation in axially graded Raylegh-Bishop nanorods, Mech. Compos. Mater., 59 (2024), 1109–1128. https://doi.org/10.1007/s11029-023-10160-4 doi: 10.1007/s11029-023-10160-4
[17]	J. Majak, M. Pohlak, M. Eerme, B. Shvartsman, Solving ordinary differential equations with higher-order Haar wavelet method, AIP Conf. Proc., 2116 (2019), 330002. https://doi.org/10.1063/1.5114340 doi: 10.1063/1.5114340

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