A numerical implementation of the unified Fokas transform for evolution problems on a finite interval
This work provides a numerical method for a class of boundary value problems that previously lacked computational solutions, but it is incremental as it applies an existing transform with contour deformation.
The authors present a numerical implementation of the unified Fokas transform to solve two-point boundary value problems for a third-order linear PDE on a finite interval, achieving accurate results by deforming integration contours. No existing computations for this problem were known.
We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary conditions, and to our knowledge, no such computations exist. Instead of computing the evolution numerically, we evaluate the solution representation formula obtained by the unified transform of Fokas. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results.