Partial Differential Equations on Bounded Domains

Abstract

The standard technique for solving PDEs on bounded domains is called separation of variables(or the method of eigenfunction eχpansions or the Fourier method). As we have already noted in Section 3.4, the idea is to assume that the unknown function u = u(χ, t) (we assume that the independent variables are χ and t) in an initial boundary value problem can be written as a product of a function of χ and a function of t, that is, u(χ, t) = y(χ)g(t). Thus, the variables separate. If the method is to be successful, when this product is substituted into the PDE, the PDE separates into two ODEs, one for y(χ) and one for g(t). Substitution of the product into the boundary conditions leads to boundary conditions on y(χ). Therefore, we are left with an ODE boundary value problem for y(χ) and an ODE for g(t). When we solve for y(χ) and g(t), we will have a product solution u(χ, t) of the PDE that satisfies the boundary conditions. As it turns out, the boundary value problem for y(χ) is a Sturm-Liouville problem and will, in fact, have infinitely many solutions; consequently, we will have infinitely many product solutions u₁(χ, t), u₂(χ, t), … that satisfy the boundary conditions. One can then superimpose these solutions, or add them up in a special way, to determine a solution of the PDE and boundary conditions that also satisfies the initial condition(s).