Numerical approximation of heat equation in bounded domains in Mathematica

Oana Rachieru

Transilvania University

Abstract

We consider numerical approximation of heat equation with Neumann condition and Dirichlet condition on the boundary of domain Dc(R^d) , that is we will consider the problem of finding a (C^(1,2)) 
function  u=u(t,z):(0,∞)×(OverBar[D])->R satisfying the Neumann condition on the  boundary domain, respectively Dirichlet condition on the boundary domain . Using probabilistic methods we can give explicit representation of solutions of heat equation on both problems by: 1.Neumann condition: u(t,z)=(E^z)(Subscript[u, 0])((Subscript[B, t])), where (Subscript[B, t]) is a reflecting Brownian motion in D starting at (Subscript[B, 0])=z 
, and (E^z)
 denotes the expectation of function (Subscript[u, 0]) at (Subscript[B, t]), and  2. Dirichlet condition: u(t,z)=(E^z)((Subscript[u, 0])((Subscript[B, t]))(Subscript[1, t<(Subscript[τ, D])])), where  (Subscript[B, t]) is a Brownian motion killed on the boundary of D starting at (Subscript[B, 0])=z 
, and (E^z)
 denotes the expectation of function (Subscript[u, 0]) at (Subscript[B, t]) with {t<(Subscript[τ, D])}. We present a Mathematica implementation of function u(t,z) for different choices of (Subscript[u, 0]) and domain D (half-plane, unit disc, rectangle, triangle). We will describe the effectiveness of this probabilistic method and how Mathematica was helpful in verifying the right solutions for equations.