High precision methods in eigenvalue problems and their applications. (English) Zbl 1086.34001

Akulenko, Leonid D.; Nesterov, S. V.

Differential and Integral Equations and Their Applications 6. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-415-30993-X/hbk). xix, 239 p. (2005).

The book is devoted to the development and applications of a method for solving eigenvalue problems for second-order differential equations of Sturm-Liouville type (including generalized Sturm-Liouville equations in which the spectral parameter is nonlinearly involved in the equation or in the boundary conditions, and vector-valued Sturm-Liouville equations) as well as for certain fourth-order selfadjoint equations. The method (called by the author the method of accelerated convergence) provides the refinement of eigenvalues and eigenfunctions given by some rough approximation (e.g., of the Rayleigh-Ritz type) by solving a sequence of Cauchy problems and finding the roots of the respective solutions. The method introduces a sequence of small parameters, in terms of which the refined eigenvalues are expanded. The method is tested on a number of model equations admitting analytic solutions and is shown to give solutions with good accuracy while sparing computational resources, which allows on-the-fly calculations of eigenvalues and eigenfunctions by using personal computers. The method is also applied to systems described by partial differential equations, where the problems of finding frequencies and shapes of free vibrations of membranes and plates are addressed. A distinctive feature of such problems is that the independent variables cannot be separated completely and one has to solve two or more coupled boundary value problems of Sturm-Liouville type.

Reviewer: Dmitry Shepelsky (Kharkov)

MSC:

34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34L16	Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L15	Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B07	Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34B24	Sturm-Liouville theory
65L10	Numerical solution of boundary value problems involving ordinary differential equations
65L15	Numerical solution of eigenvalue problems involving ordinary differential equations

Keywords:

eigenvalue Sturm-Liouville problems; variational formulation of boundary value problems; regular perturbations method; accelerated convergence method