Summary: We consider a Cauchy problem for the heat equation in a quarter plane with data given along the line \(x=1\). The solution is sought for in the interval \(0\leq x<1\). This problem is ill-posed in the sense that a solution does not depend continuously on the data. Hölder type stability estimates have been obtained for the heat equation, and a stabilised problem can be formulated, where a bound on the solution is imposed. We study an approximation of the problem, where the heat equation is modified so that a hyperbolic equation is obtained, for which the Cauchy problem is well posed. The problem of choosing the coefficients in the hyperbolic equation is discussed, and an error estimate is given, which shows that as the errors become small we only have a logarithmic type error estimate.