Solving boundary value problems with complex parameters by the Monte Carlo method
The authors consider the three-dimensional Dirichlet problem for the Helmholtz equation \[ \Delta u + cu = -g,\qquad u|_{\Gamma} = \psi \] in a domain \(D\subset\mathbb{R}^3\) with boundary \(\Gamma\), where \(c=a+bi\) is a complex parameter. The estimates under consideration are constructed on the base of the process of a ``random walk by spheres'' in the domain \(D\). New in this paper is the fact, that it extends the estimators of the Monte Carlo type, constructed earlier by the same authors for a real parameter, to the case of a complex parameter. The obtained results enable in the second part of the paper to implement the Fourier transform while solving the mixed problem for a parabolic equation \[ u_t-\Delta u = f(r,t),\qquad u|_{t=0} = \phi(r),\qquad u|_{\Gamma}=v(r,t)\qquad\text{with}\qquad v(r,0)|_{\Gamma}=\psi(r) \] by the Monte Carlo method.
scientific article; zbMATH DE number 1023929