Collocation Method to Solve Analytically Nonseparable Boundary Problems

Boundary problems in which the boundary does not coincide with coordinate lines or surfaces of coordinate systems in which the pertinent partial differential equation is not separable into ordinary differential equations are usually called nonseparable [1]. In these days such problems are mainly solved numerically by the finite elements method. However, the general solution of a partial differential equation contains one or more arbitrary functions. Since an arbitrary function is equivalent to an infinite set of coefficients (partial amplitudes of modes) one has an infinite number of constants available to satisfy arbitrary boundary conditions on arbitrary boundaries. In practice, a finite number of these coefficients is used to satisfy the boundary condition exactly in a finite number of given collocation points chosen on the boundary. A plot of the boundary given by the collocation points check the satisfaction of the boundary condition not only in the collocation points but also in other points. A theorem of Courant [2] guarantees smoothness and analyticity of the solution between the collocation points.