We solve by Laguerre collocation method two second order genuinely nonlinear singularly perturbed boundary value problems and a fourth order nonlinear one formulated on the half-line. First, we are concerned with solutions to a model equation for low Reynolds number flow, the so-called Cohen, Fokas, Lagerstrom model. Then we compute monotonously increasing solutions, the so-called “bubble type solutions”, to density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. We compare the results obtained in the latter case with those carried out by shooting or polynomial collocation both coupled with domain truncation. A fourth order nonlinear boundary value problem supplied with mixed boundary conditions in origin as well as at infinity is also solved. The solution and its first three derivatives are obtained and compared with their counterparts obtained by Keller’s box scheme. The collocation based on Laguerre functions avoids the domain truncation, exactly impose any kind of boundary conditions at infinity and resolve with reliable accuracy the sharp interior or exterior layer of the solutions. The method is fairly easy implementable and efficient in treating various types of nonlinearities. The accuracy of the method is also tested on the classical Blasius problem (a third order one). We solve with the same method the eigenvalue problem obtained by linearization around the constant solution, which corresponds to the case of a homogeneous fluid (without bubbles), and observe that this solution is stable.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Singular nonlinear BVPs; collocation Laguerre functions; semi-infinite domain; Blasius problem; low Reynolds number flows; density profile equation; foundation engineering problem.
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C.I. Gheorghiu, Pseudospectral solutions to some singular nonlinear BVPs. Applications in nonlinear mechanics. Numer. Algor., 68 (2015) 1-14.
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1. Boyd, J.P.: Chebyshev and fourier spectral methods, 2nd edn. Dover Publications, New York (2000)
2. Boyd, J.P., Rangan, C., Bucksbaum, P.H.: Pseudospectral methods on semi-infinite interval with application to the hydrogen atom; a comparison of mapped Fourier sinc method with Laguerre series and rational Chebyshev expansion. J. Comput. Phys. 188, 56–74 (2003)
3. Chaitin-Chatelin, F., Fraisse, V.: Lectures on finite precision computation. SIAM, Philadelphia (1996)
4. Cohen, D.S., Fokas, A., Lagerstrom, P.A.: Proof of some asymptotic results for a model equation for low Reynolds number flow. SIAM J. Appl. Math. 35, 187–207 (1978)
5. Fazio, R.: A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals. J. Comput. Appl. Math. 140, 331–344 (2002)
6. Fazio, R.: A free boundary value approach and Keller’s box scheme for BVPs on infinite intervals. Int. J. Comput. Math. 80, 1549–1560 (2003)
7. Gheorghiu, C.I.: On the spectral characterization of some Chebyshev type methods; dimension vs. structure. Studia Univ. “Babes-Bolyai” L, 61–66 (2005)
8. Gheorghiu, C.I.: Spectral methods for differential problems. Casa Cartii de Stiinta Publishing House, Cluj-Napoca (2007)
9. Gheorghiu, C.I., Rommes, J.: Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems. Int. J. Numer. Meth. Fl. 71, 358–369 (2012)
10. Gheorghiu, C.I.: Laguerre collocation solutions to boundary layer type problems. Numer. Algor. 64, 358–401 (2012)
11. Henrici, P.: Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices. Numer. Math. 4, 24–40 (1962)
12. Hsiao, G.C.: Singular perturbations for a nonlinear differential equation with a small parameter. SIAM J. Math. Anal. 4, 283–301 (1973)
13. Kitzhofer, G., Koch, O., Lima, P.M., Weinmuller, E.: Efficient numerical solution of the density ̈ profile equation in hydrodynamics. J. Sci. Comput. 32, 411–424 (2007)
14. Konyukhova, N.B., Lima, P.M., Morgado, M.L., Soloviev, M.B.: Bubbles and droplets in nonlinear physics models: analysis and numerical simulation of singular nonlinear boundary value problems Comput. Math. Math. Phys. 48, 2018–2058 (2008)
15. Kulikov, G.Y., Lima, P.M., Morgado, M.L.: Analysis and numerical approximation of singular boundary value problems with p-Laplacians in fluid mechanics. J. Comput. Appl. Math. doi:10.1016/j.cam.2013.09.071
16. Lentini, M., Keller, H.B.: Boundary value problems on semi-infinite intervals and their numerical solution. SIAM J. Numer. Anal. 17, 577–604 (1980)
17. Lima, P.M., Konyukhova, N.B., Chemetov, N.V., Sukov, A.I.: Analytical- numerical investigation of bubble-type solutions of nonlinear singular problems. J. Comput. Appl. Math. 189, 260–273 (2006)
18. O’Regan, D.: Solvability of some singular boundary value problems on the semi-infinite interval. Can. J. Math. 48, 143–158 (1996)
19. Markowich, P.A.: Analysis of boundary value problems on infinite intervals. SIAM J. Math. Anal. 14, 11–37 (1983)
20. Pruess, S., Fulton, C.T.: Mathematical software for Sturm-Liouville problems. ACM T. Math. Softw. 19, 360–376 (1993)
21. Rubel, L.A.: An estimation of the error due to the truncated boundary in the numerical solution of the Blasius equation. Q. Appl. Math. 13, 203–206 (1955)
22. Tang, T., Trummer, M.R.: Boundary layer resolving pseudospectral methods for singular perturbation problem. SIAM J. Sci. Comput. 17, 430–438 (1996)
23. Trefethen, L.N.: Computation of pseudospectra. Acta Numer. 8, 247–295 (1999)
24. Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM T. Math. Softw. 26, 465–519 (2000)