Spectral parameter power series for fourth-order Sturm–Liouville problems

• Published in: Applied mathematics and computation Vol. 219; no. 8; pp. 3610 - 3624
• Main Authors: Khmelnytskaya, Kira V., Kravchenko, Vladislav V., Baldenebro-Obeso, Jesús A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.12.2012
• Subjects: Complex eigenvalue; Computation; Convergence; Fourth-order Sturm–Liouville equation; Fourth-order Sturm–Liouville problem; Mathematical analysis; Mathematical models; Numerical solution of eigenvalue problems; Power series; Representations; Spectra; Spectral parameter power series; Taylor series; Spectral parameter power series; Fourth-order Sturm–Liouville equation; Complex eigenvalue; Fourth-order Sturm–Liouville problem; Numerical solution of eigenvalue problems
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2012.09.055

A general solution of the fourth-order Sturm–Liouville equation is presented in the form of a spectral parameter power series (SPPS). The uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure. Based on the SPPS representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their Taylor series expansions. This leads to a simple and efficient numerical method for solving the spectral problems for fourth-order Sturm–Liouville equations. Several examples of application are discussed.