Klein-Gordon equation as a bi-dimensional moment problem

Abstract

We consider the solution of one-dimensional linear and nonlinear Klein-Gordon equations by first transforming them into bi-dimensional integral equations which are then handled as bi-dimensional moment problems. The integral equations are obtained by either Laplace transforming the linear PDE or by using Green identity for the linear as well as the nonlinear cases. The discretization of the so obtained integral equations results, for the linear and nonlinear problems, respectively, into a bi-dimensional Hausdorff problem and into a generalized moment problem (in which the kernel set { }nm n m x y has been replaced by sets { ( )} m m g x, y of more general linearly independent functions). In both cases, the corresponding inverse problem is numerically solved by approximating the associated finite moment problem by a truncated expansion.