The heat equation is one of the most important partial differential equations in mathematics and natural sciences. Its solution can rarely be expressed in closed, explicit form. It can however be expressed in integral form with the so-called fundamental solution or as a series of solutions to the eigenvalue problem, if the domain is bounded. In this master's thesis a third method is explored, where the solution to the heat equation is expressed as conditional expectation of appropriate functionals of Brownian motion. In computational sense it is similar to the expression in terms of the fundamental solution, but conceptually unrelated. In addition, the Brownian approach gives rise to new numerical algorithms for solving the heat equation.
|
