Boundary value problems for a loaded wave equation

Abstract

We consider Cauchy, Goursat and Darboux problems for a loaded wave equation uxx uyy = λu ( x 0 , y ). We construct an explicit representation for solution of Cauchy problem, which coincides with the classical representation for solution of the Cauchy problem for the wave equation as λ = 0. We describe the domains of dependence, influence and definition of Cauchy data. It is shown their essential difference from corresponding domains in the case of the Cauchy problem for the wave equation. We formulate Darboux and Goursat problems in nonlocal setting and give an algorithm for construction of their solutions.