Fourth-order modified method for the Cauchy problem for the Laplace equation

We consider the Cauchy problem for the Laplace equation in the half plane x > 0 , y ∈ R where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x ⩽ 1 . The problem is ill-posed: the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically, it is necessary to modify the equation so that a bound on the solution is imposed. We study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the solution of the Cauchy problem for the Laplace equation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. Numerical implementation is considered and a simple example is given.