Herrera, Ismael Boundary methods. An algebraic theory. (English) Zbl 0549.35004 Applicable Mathematics Series. Boston-London-Melbourne: Pitman Advanced Publishing Program. VIII, 136 p. £16.50 (1984). This book provides a presentation of an abstract theory of boundary value problems developed by the author over several years of work. This theory is based on an algebraic structure which systematically occurs in boundary value problems which are linear and it is formulated in the setting of general functional-valued operators defined on arbitrary linear spaces which, generally, do not possess an inner product or metric. The theory is developed only for (linear) formally symmetric operators; inside this class it is applicable irrespective of the type of equation, or system of equations; in particular it can be applied to both time-independent and time-dependent equations. An application of the theory is shown in problems formulated by variational principles, also in the case of problems in discontinuous fields, with prescribed jump conditions, as the problem of matching or connecting. Other types of application are the numerical solution of boundary value problems and the development of biorthogonal systems of functions, to obtain generalized Fourier series developments. Reviewer: P.Secchi Cited in 50 Documents MSC: 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35A35 Theoretical approximation in context of PDEs 35C10 Series solutions to PDEs 35G15 Boundary value problems for linear higher-order PDEs 35A25 Other special methods applied to PDEs 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 65N99 Numerical methods for partial differential equations, boundary value problems 35G05 Linear higher-order PDEs 35G10 Initial value problems for linear higher-order PDEs Keywords:abstract theory; boundary value problems; algebraic structure; functional-valued operators; (linear) formally symmetric operators PDFBibTeX XML