Fokas method
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems.
Traditionally, linear boundary value problems are analysed using either integral transforms and infinite series, or by employing appropriate fundamental solutions.
The analogous problem on a finite interval can be solved via an infinite series.
There exist traditional integral transforms and infinite series representations only for a very limited class of boundary value problems.
For example, there does not exist the analogue of the sine-transform for solving the following simple problem: supplemented with the initial and boundary conditions Eq.2.
satisfy the forced heat equation supplemental with the initial and boundary conditions Eq.2, where
The Fokas method is based on the fact that equation Eq.7 has a large domain of validity.
By using the inverse Fourier transform, the global relation yields an integral representation on the real line.
and substituting the resulting expression in Eq.9 we find If is important to note that the unknown term
yields a zero contribution.Equation Eq.11 can be rewritten in a form which is consistent with the Ehrenpreis fundamental principle: if the boundary condition is specified for
is a given positive constant, then using Cauchy's integral theorem, it follows that Eq.11 is equivalent with the following equation: where Uniform convergence The unified transform constructs representations which are always uniformly convergent at the boundaries.
For the details of effective numerical quadrature using the unified transform, we refer the reader to,[2] which solves the advection-dispersion equation on the half-line.
Recent work has extended the method and demonstrated a number of its advantages; it avoids the computation of singular integrals encountered in more traditional boundary based approaches, it is fast and easy to code up, it can be used for separable PDEs where no Green's function is known analytically and it can be made to converge exponentially with the correct choice of basis functions.
both satisfy Laplace's equation in the interior of a convex bounded polygon
It follows that Then Green's theorem implies the relation In order to express the integrand of the above equation in terms of just the Dirichlet and Neumann boundary values, we parameterize
, leading to the global relation for Laplace's equation: A similar argument can also be used in the presence of a forcing term (giving a non-zero right-hand side).
lead to respective global relations and These three cases deal with more general second order elliptic constant coefficient PDEs through a suitable linear change of variables.
can be approximated in terms of Legendre polynomials: where for the cases of the Dirichlet, Neumann or Robin boundary value problems either
, the relevant system is diagonally dominant, thus its condition number is very small.
In the case of straight edges, Green's representation theorem leads to Due to the orthogonality of the Legendre polynomials, for a given
, the integrals in the above representation are Legendre expansion coefficients of certain analytic functions (written in terms of
The method can be extended to variable coefficient PDEs and curved boundaries in the following manner (see [6]).
Eq.25 can be written in the form Integrating across the domain and applying the divergence theorem we recover the global relation (
, we define the following important transform: Using Eq.28 , the global relation Eq.27 becomes For separable PDEs, a suitable one-parameter family of solutions
To compute the integrals that form the approximate global relation, we can use the same trick as before - expanding the function integrated against Legendre polynomials in a Chebyshev series and then converting to a Legendre series.
A major advantage of the above collocation method is that the basis choice (Legendre polynomials in the above discussion) can be flexibly chosen to capture local properties of the solution along each boundary.
The complex collocation points are allowed precisely because of the radiation condition.
in terms of weighted Chebyshev polynomials of the second kind: These have the following Fourier transform: where
, a suitable basis choice are Bessel functions of fractional order (to capture the singularity and algebraic decay at infinity).
and shows the quadratic-exponential convergence of the method, namely the error decreases like
