Abstract
Using the functional discrete approach and Adomian polynomials, we propose a numerical algorithm for an eigenvalue problem with a potential that consists of a nonlinear autonomous part and a linear part depending on an independent variable. We prove that the rate of convergence of the algorithm is exponential and improves as the order number of an eigenvalue increases. We investigate the mutual influence of the piecewise-constant approximation of the linear part of the potential and the nonlinearity on the rate of convergence of the method. Theoretical results are confirmed by numerical data.
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References
V. L. Makarov, “About functional-discrete method of arbitrary order of accuracy for solving Sturm-Liouville problem with piecewise smooth coefficients,” Soviet DAN SSSR, 320, No. 1, 34–39 (1991).
V. L. Makarov, “FD-method: the exponential rate of convergence,” J. Math. Sci., 104, No. 6, 1648–1653 (1997).
V. L. Makarov and O. L. Ukhanev, “FD-method for Sturm-Liouville problems. Exponential rate of convergence,” Appl. Math. Inform., 2, 1–19 (1997).
B. I. Bandyrskii, V. L. Makarov, and O. L. Ukhanev, “Sufficient conditions for the convergence of non-classical asymptotic expansions for Sturm-Liouville problems with periodic conditions,” Different. Equat., 35, No. 3, 369–381 (1999).
B. I. Bandyrskii and V. L. Makarov, “Sufficient conditions for eigenvalues of the operator with Ionkin-Samarskii conditions to be real-valued,” Comput. Math. Math. Phys., 40, No. 12, 1715–1728 (2000).
B. I. Bandyrskii, I. I. Lazurchak, and V. L. Makarov, “A functional-discrete method for solving Sturm-Liouville problems with an eigenvalue parameter in the boundary conditions,” Comput. Math. Math. Phys., 42, No. 5, 646–659 (2002).
B. I. Bandyrskii, I. P. Gavrilyuk, I. I. Lazurchak, and V. L. Makarov, “Functional-discrete method (FD-method) for matrix Sturm-Liouville problems,” Comput. Meth. Appl. Math., 5, No. 4, 1–25 (2005).
B. I. Bandyrskii, V. L. Makarov, and N. O. Rossokhata, “Functional-discrete method with a high order of accuracy for the eigenvalue transmission problems,” Comput. Meth. Appl. Math., 4, No. 3, 324–349 (2004).
B. I. Bandyrskii, V. L. Makarov, and N. O. Rossokhata, “Functional-discrete method for an eigenvalue transmission problem with periodic boundary conditions,” Comput. Meth. Appl. Math., 5, No. 2, 201–220 (2005).
V. L. Makarov and N. O. Rossokhata, “Estimates for the rate of convergence of the FD-method for the Sturm-Liouville problem with potential from the space L1,” Zb. Prats’ Inst. Mat. Nat. Akad. Nauk Ukr., 1, No. 3, 1–16 (2005).
I. P. Gavrilyuk, A. V. Klimenko, V. L. Makarov, and N. O. Rossokhata, “Exponentially convergent parallel algorithm for nonlinear eigenvalue problems,” IMA J. Numer. Anal. (2007).
P. E. Zhidkov, “Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems,” El. J. Different. Equat., No. 28, 1–13 (2000).
T. Shibata, “Precise spectral asymptotics for nonlinear Sturm-Liouville problems,” J. Different. Equat., 180, No. 3, 1–16 (2005).
K. Abbaoui, M. J. Pujol, Y. Cherruault, N. Himoun, and P. Grimalat, “A new formulation of Adomian method. Convergence result,” Kybernetes, 30, No. 39/10, 1183–1191 (2001).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 14–28, January, 2007.
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Havrylyuk, I.P., Klymenko, A.V., Makarov, V.L. et al. FD-method for an eigenvalue problem with nonlinear potential. Ukr Math J 59, 12–27 (2007). https://doi.org/10.1007/s11253-007-0002-7
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DOI: https://doi.org/10.1007/s11253-007-0002-7