Laguerre collocation solutions vs. analytic results for singular semilinear BVPs on the half line

Abstract

We make use of two Laguerre collocation techniques as a way to illustrate, deepen or extend some analytic results of existence, uniqueness and asymptotic behavior concerning the solutions to second and third orders nonlinear boundary value problems on the half line. We point out some particular features of solutions, e.g. boundary layers, which have not been noticed by theoretical studies and accurately resolve some analytic singularities. With respect to some challenging BVPs from engineering, we show that not absolutely all the analytic results are validated numerically. The free parameter, i.e., the scaling factor, involved in the Laguerre weight function is adjusted in order to ensure the most efficient convergence of the Newton type algorithms.

Authors

C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Keywords

Laguerre collocation, scaling, nonlinear equations, unbounded domain, analytic results, singularities, Kidder equation

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Paper coordinates

C.I. Gheorghiu, Laguerre collocation solutions vs. analytic results for singular semilinear BVPs on the half line, ROMAI J., 11 (2015), pp. 69-87

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ROMAI. J

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Editions de l’Academie Roumaine

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1841-5512

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2065-7714

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References

[1] R. P. Agarwal, D. O’Regan, Boundary Value Problems on the Half Line in the Theory of Colloids, Math. Probl. Eng., 8(2002), 143-150.

[2] R. P. Agarwal, D. O’Regan, Infinite Interval Problems Modeling the Flow of a Gas Through a Semi-Infinite Porous Medium, Stud. Appl. Math., 108(2002), 245-257.

[3] R. P. Agarwal, D. O’Regan, Infinite interval problems modelling phenomena which arise in the theory of plasma and electrical potential theory, Stud. Appl. Math., 111(2003), 339-358.

[4] R. P. Agarwal, Singular Boundary Value Problems With Real World Applications, (2008) http://www.dmmm.uniroma1.it/ agostino.prastaro/AGARWAL-LECTURE.pdf Accessed 22 Feb 2015

[5] H. Alici, H. Taseli,The Laguerre pseudospectral method for the radial SchrAsdinger equation , Appl. Numer. Math., doi:10.1016/j.apnum.2014.09.001

[6] O. Bayrak, I. Boztosun, Analytical solutions of the HulthASn and the Morse potentials by using  the Asymptotic Iteration Method, J. Mol. Structure: THEOCHEM, 802(2007), 17-21.

[7] J. V. Baxley, Existence and Uniqueness for Nonlinear Boundary Value Problems on Infinite Intervals, J. Math. Anal. Appl., 147(1990), 122-133.

[8] T. Benacchio, L. Bonaventura, Absorbing boundary conditions: a spectral collocation approach, Int. J. Numer. Meth. Fluids (2013), DOI: 10.1002/fld.3768

[9] C. Bernardy, Y. Maday, Spectral Methods, In Ciarlet, P, Lions, JL., (eds.) Handbook of Numerical Analysis, V.5 (Part 2), North-Holland, 1997.

[10] L. E. Bobisud, Existence of Positive Solutions to some Nonlinear Singular Boundary Value Problems on Finite and Infinite Intervals, J. Math. Anal. Appl., 173(1193), 69-83.

[11] J.P. Boyd, C. Rangan, P.H. Bucksbaum, Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom a comparison of the mapped Fourier sine method with Laguerre series and rational Chebyshev expansions, J. Comput. Phys. 188(2003), 56-74.

[12] C. V. Coffman, Uniqueness of the ground state solution for ∆u − u + u 3 = 0 and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46(1972), 12-95.

[13] M. Countryman, R. Kannan, Nonlinear Boundary Value Problems on Semi-Infinte Intervals, Comput. Math. Appl., 28(1994), 59-75.

[14] R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal., 33(1996), 1473-1483.

[15] C. I. Gheorghiu, Laguerre collocation solutions to boundary layer type problems, Numer. Algorithms 64(2013) 385-401.

[16] C. I. Gheorghiu, Pseudospectral solutions to some singular nonlinear BVPs. Applications in nonlinear mechanics, Numer. Algor., 68(2015), 1-14.

[17] C. I. Gheorghiu, Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond, Springer, Cham Heidelberg New York Dordrecht London, 2014.

[18] C. I. Gheorghiu, J. Rommes, Application of Jacobi-Davidson mathod to accurate analysis of singular hydrodynamic stability problems, Int. J. Numer. Meth. Fluids, 71(2013), 358-369.

[19] C. I. Gheorghiu, M. E. Hochstenbach, B. Plestenjak, J. Rommes, Spectral collocation solutions to multiparameter Mathieu’s system, Appl. Math. Comput., 218(2012), 11990-12000.

[20] R. Hammerling, O. Koch, C. Simon, E. B. Weinmuller, Numerical solution of singular eigenvalue problems for ODEs with a focus on problems posed on semi-infinite intervals, ASC Report No. 8/2010 Vienna University of Technology, Vienna, Austria

[21] J. Hoepffner, Implementation of boundary conditions, (2010) http://www.lmm.jussieu.fr/ hoepffner/research/realizing.pdf Accessed 2 Feb 2015

[22] R. Iacono, J. P. Boyd, The Kidder Equation: uxx + 2xux/ √ 1 − αu = 0., Stud. Appl. Math., 135(2014), 63-85.

[23] V. B. Mandelzweig, F. Tabakin, Quasilinearization approach to nonlinear problems in physics with applications to nonlinear ODEs, Comput. Phys. Commun., 141(2001), 268-281.

[24] K. McLeod, J. Serrin, Uniqueness of positive radial solutions of ∆u + f(u) = 0 in R n. Arch. Rational Mech. Anal., 99(1987), 115-145.

[25] K. McLeod, W. C. Troy, F. B. Weissler, Radial Solutions of ∆u+ f(u) = 0 with Prescribed Numbers of Zeros, J. Diff. Eq., 83(1987), 368-378.

[26] G. R. Ierley, O. G. Ruehr, Analytic and numerical solutions of a nonlinear boundary layer problem, Stud. Appl. Math., 75, (1986), 1-36
[27] G. R. Ierley, Boundary Layers in General Ocean Circulation, Annu. Rev. Fluid Mech., 22(1990), 111-142.

[28] B. Plestenjak, C. I. Gheorghiu, M. E. Hochstenbach, Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems, J. Comput. Phys., 298(2015), 585-601.

[29] R. O’Regan, Solvability of Some Singular Boundary Value Problems on the Semi-Infinite Interval, Can. J. Math., 48(1996), 143-158.

[30] A. Roy, The generalized pseudospectral approach to the bound states of the HulthASn and the  Yukawa potentials, PRAMANA–J. Phys., 65(2005), 1-15.

[31] J. Shen, T. Tang, L-L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer-Verlag Berlin Heidelberg, 2011.

[32] W. C. Troy, Solutions of Nonlinear Boundary Layer Problems Arising in Physical Oceanography, Rocky Mt. J. Math., 21(1991), 813-820.

[33] W. C. Troy, The existence and uniqueness of bound-state solutions of a semi-linear equation, Proc. R. Soc. A., 461(2005), 2941-2963.

[34] J. A. C. Weideman, S. C. Reddy, A MATLAB Differentiation Matrix Suite, ACM Trans. on Math. Software, 26(2000), 465-519.

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