On the Construction of Cosine Operator Functions and Semigroups on Function Spaces with Generator a(x)(d 2 dx 2 )+b(x)(d/dx)+c(x); Theory

Abstract

In this paper we develop a method to solve exactly partial differential equations of the type (∂n/∂tn)f(x,t)=(a(x)(∂n/∂xn)+b(x) (∂/∂x+c(x))f(x,t); n=1,2, with several boundary conditions, where f·,t) lies in a function space. The most powerful tool here is the theory of cosine operator functions and their connection to (holomorphic) semigroups. The method is that generally we are able to unify and generalize many theorems concerning problems in the theories of holomorphic semigroups, cosine operator functions, and approximation theory, especially these dealing with approximation by projections. These applications will be found in [14].