Abstract
This paper concerns accurate spectral collocation solutions, more precisely Chebyshev collocation (ChC), to some third-order nonlinear and singular boundary value problems on unbounded domains. The problems model some draining or coating fluid flows. We use exclusively ChC, in the form of Chebfun, avoid any obsolete shooting-type method, and provide reliable information about the convergence and accuracy of the method, including the order of Newton’s method involved in solving the nonlinear algebraic systems. As a complete novelty, we combine a graphical representation of the convergence of the Newton method with a numerical estimate of its order of convergence for a more realistic value. We treat five challenging examples, some of which have only been solved by approximate methods. The found numerical results are judged in the context of existing ones; at least from a qualitative point of view, they look reasonable.
Authors
C.I. Gheorghiu
Romanian Academy, Tiberiu Popoviciu Institute of Numerical Analysis, Cluj-Napoca, Romania
Keywords
Paper coordinates
C.I. Gheorghiu, Chebyshev collocation solutions to some nonlinear and singular third-order problems relevant to thin-film flows, Journals Modern Mathematical Physics, 1 (2025) no. 1, art. no. 5, 15 pp., https://doi.org/10.3390/mmphys1010005
https://www.mdpi.com/3042-5034/1/1/5
About this paper
Print ISSN
3042-5034
Online ISSN
google scholar link
[1] Chifu, E.; Gheorghiu, C.I.; Stan, I. Surface Mobility of Surfactant Solutions XI. Numerical Analysis for the Marangoni and Gravity Flow in a Thin Liquid Layer of Triangular Section. Rev. Roum. Chim. 1984, 29, 31–42. [Google Scholar]
[2] Gheorghiu, C.I.; Pop, I.S. A modified Chebyshev-tau method for a hydrodynamic stability problem. In Proceedings of the ICAOR (International Conference on Approximation, Optimisation), Cluj-Napoca, Romania, 27 July–1 August 1996; Volume II, pp. 119–126. Available online: https://www.ictp.acad.ro/gheorghiu/ (accessed on 15 April 2025).
[3] Gheorghiu, C.-I. Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations. Computation 2021, 9, 2. [Google Scholar] [CrossRef]
[4] Gheorghiu, C.-I. Accurate Chebfun solutions to third-order nonlinear BVPs on the half-line. Applications in boundary layer theory. Phys. Scr. 2025, 100, 055204. [Google Scholar] [CrossRef]
[5] Gheorghiu, C.-I. Spectral collocation solutions to systems of boundary layer type. Numer. Algor. 2016, 73, 1–14. [Google Scholar] [CrossRef]
[6] Fornberg, B. A Practical Guide to Pseudospectral Methods; CUP: Cambridge, UK, 1998. [Google Scholar]
[7] Trefethen, L.N. Approximation Theory and Approximation Practice; SIAM: Philadelphia, PA, USA, 2013. [Google Scholar]
[8] Tuck, E.O.; Schwartz, L.W. A Numerical and Asymptotic Study of some Third-Order Ordinary Differential Equations Relevant to Draining and Coating Flows. SIAM Rev. 1990, 32, 453–469. [Google Scholar] [CrossRef]
[9] Acheson, D.J. Elementary Fluid Dynamics; Oxford Academic: Oxford, UK, 2023. [Google Scholar] [CrossRef]
[10] Bertozzi, A.L.; Pugh, M. The Lubrication Approximation for Thin Viscous Films: Regularity and Long-Time Behaviour of Weak Solutions. Nonlinearity 1994, 7, 1535–1564. [Google Scholar] [CrossRef]
[11] Moriarty, J.A.; Schwartz, L.W.; Tuck, E.O. Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids 1991, 3, 733–742. [Google Scholar] [CrossRef]
[12] Agarwal, R.P.; O’Regan, D. Singular Problems on Infinite Interval Modelling Fenomena in Draining Flows. IMA J. Appl. Math. 2001, 66, 621–635. [Google Scholar] [CrossRef]
[13] Trefethen, L.N. Numerical computation with functions instead of numbers. Commun. ACM. 2015, 58, 91–97. [Google Scholar] [CrossRef]
[14] Trefethen, L.N.; Birkisson, A.; Driscoll, T.A. Exploring ODE; SIAM: Philadelphia, PA, USA, 2018. [Google Scholar]
[15] Boyd, J.P. Chebyshev and Fourier Spectral Methods, 2nd ed.; Dover: Mineola, NY, USA, 2001. [Google Scholar]
[16] Boyd, J.P. Exponentially convergent Fourier-Chebyshev quadrature schemes on bounded and infinite intervals. J. Sci. Comput. 1987, 2, 99–109. [Google Scholar] [CrossRef]
[17] Catinas, A.E. How many steps still left to x*? SIAM Rev. 2021, 63, 585–624. [Google Scholar] [CrossRef]
[18] Troy, W.C. Solutions of Third-Order Differential Equations Relevant to Draining and Coating Flows. SIAM J. Math. Anal. 1993, 24, 155–171. [Google Scholar] [CrossRef]
[19] Snoeijer, J.H.; Ziegler, J.; Andreotti, B.; Fermigier, M.; Eggers, J. Thick films coating a plate withdrawn from a bath. PRL 2008, 100, 244502. [Google Scholar] [CrossRef] [PubMed]
[20] Bernis, F. On some Nonlinear Singular Boundary Value Problems of Higher Order. Nonlinear Anal. 1996, 26, 1061–1078. [Google Scholar] [CrossRef]
[21] Bernis, F.; Peletier, L.A. Two Problems from Draining Flows Involving Third-Order Ordinary Differential Equations. SIAM J. Math. Anal. 1996, 27, 515–527. [Google Scholar] [CrossRef]
[22] Howes, F.A. Differential Inequalities of Higher Order and the Asymptotic Solution of Nonlinear Boundary Value Problems. SIAM J. Math. Anal. 1982, 13, 61–80. [Google Scholar] [CrossRef]
[23] Howes, F.A. The Asymptotic Solution of a Class of Third-Order Boundary Value Problems Arising in the Theory of Thin Film Flows. SIAM J. Appl. Math. 1983, 43, 993–1004. [Google Scholar] [CrossRef]
[24] King, A.C.; Tuck, E.O. Thin liquid layers supported by steady air-flow surface traction. J. Fluid Mech. 1993, 251, 709–718. [Google Scholar] [CrossRef]
[25] Gheorghiu, C.I. Accurate Laguerre collocation solutions to a class of Emden–Fowler type BVP. Phys. A Math. Theor. 2023, 56, 17LT01. [Google Scholar] [CrossRef]