Computability aspects for 1st-order partial differential equations via characteristics
In this paper, it is shown that the method of characteristics can be used to compute local solutions of the boundary value problems for the first-order partial differential equations at feasible instances, under the natural definition of computability from the view point of application; but the maximal region of existence of a computable local solution may not be computable. It is also shown that the problem whether a boundary value problem has a global solution is not algorithmically decidable. The negative results retain even within the class of quasilinear equations defined by analytic computable functions over particularly simple domains (quasilinear equations are among the simplest first-order nonlinear partial differential equations). This fact shows that the algorithmic unsolvability is intrinsic.
method of characteristics computability analysis 1st-order pdes