Many boundary value problems which arise in applied mathematics are given in unbounded domains. Here we develop a theory for the imposition of boundary conditions at an artificial boundary which lead to finite domain problems that are equivalent to the unbounded domain problems from which they come. By considering the Cauchy problem with initial data in the appropriate space of functions on the artificial boundary, we show that satisfaction of the boundary conditions at infinity is equivalent to satisfaction of a certain projection condition, at the artificial boundary. This leads to an equivalent finite problem. The solvability of the finite problem is discussed and estimates of the solution in terms of the inhomogeneous data are given.
Applications of our reduction to problems whose coefficients are independent of the unbounded coordinate are considered first. For a class of problems we shall term 'separable', solutions in the tail can be developed in an eigenfunction expansion. These expansions are used to write down an explicit representation of the projection, which is useful in computations. Specific problems considered here include elliptic equations in cylindrical domains. Spatially unbounded parabolic and hyperbolic problems are also discussed. Here, the eigenfunction expansions must include continuous transform variables.
We use these 'constant tail' results to develop a perturbation theory for the case when the coefficients depend upon the unbounded coordinate. This theory is based on Duhamel's principle and is seen to be especially useful when the 'limiting' problem possesses an exponential dichotomy. We apply our results to the Helmholtz equation, perturbed hyperbolic systems and nonlinear problems. We present a numerical solution of the Bratu problem in a semi-infinite, two-dimensional, stepped channel to illustrate our method.
