The aim of the reviewed monograph is to present the constructive and practical aspects of spectral methods forwarded to solve nonstandard eigenvalue problems (EP) arising in physical applications such as stability of elastic systems and hydrodynamics. By nonstandard EPs the author means singular Sturm-Liouville problems, singular and nonsingular generalized EPs of larger than order two, EPs containing the spectral parameter in boundary conditions and multipalameter EPs.
The introductory Chapter 1 contains the general information on the spectral approximation as a weighted residual method, provides the tau and Galerkin methods based on Chebyshev polynomials. The Chebyshev collocation (ChC) is introduced together with the collocation differentiation matrices with applications to a regular second-order Sturm-Liouville problem. In Chapter 2, for some linear fourth-order EP at various boundary conditions, the based on Chebyshev polynomials tau variant of spectral methods is described with some applications. These are ChTau approximation for transverse vibrations of a uniform elastic bar and buckling column with mixed boundary conditions. Then, as Galerkin method application for the fourth-order EPs, spectral schemes are built which conditioned discretization matrices. They are illustrated on the example of Orr-Sommerfeld EPs. In conclusion of Chapter 2, ChTau methods are applied to EPs with dependent on spectral parameter boundary conditions, to examples of second-order Sturm-Liouville problems, some fourth-order stability problems for elastic systems and in their modification to a particular case of the Orr-Sommerfeld problem.
Chapter 3 is devoted to various realizations of the Chebyshev collocation. At first, they are presented as comparison with ChGalerkin and ChT methods for solving the fourth-order boundary EP on column buckling with various boundary conditions. Then they are applied to Viola’s EP described in the monograph of B. Straughan [The energy method, stability, and nonlinear convection. Applied Mathematical Sciences. 91. New York etc.: Springer-Verlag (1992; Zbl 0743.76006)]: \[ D^2[(1-\theta x)^3 D^2u]- \lambda(1-\theta x)u=0,\; x\in (0,1);\; u= D^2u=0,\; x=0,1;\; 0\leq \theta <1, \] and three interesting EPs for ordinary differential equations of the fourth and sixth and even eighth orders arising correspondingly in problems of linear hydrodynamic stability of thermal convection with variable gravity field and electro-hydrodynamic convection between two parallel walls. Here, to reduce the high-order problems to systems of second-order equations with Dirichlet boundary conditions, the so-named D-strategy or factorization technique is suggested and developed.
In Chapter 3, multiparameter Mathieu’s problems are considered, for their solving the ChC discretization techniques is developed with the usage of Jacobi-Davidson methods in solving of large algebraic multiparameter EPs.
In Chapter 4, the spectral collation method is developed, based on the Laguerre functions as smoothly decreasing to zero at infinity together with derivatives. It allows to solve high-order generalized EPs arising in fluid mechanics such as Blasius, Falkner-Skan, density profile equation and Ekmann boundary layer.
Chapter 5 is devoted to conclusions and further developments. The Appendix A contains necessary material on algebraic two-parameter EPs, particularly, the tensor product of matrix techniques.