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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

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