A numerical implementation of the unified Fokas transform for evolution problems on a finite interval

TL;DR

This paper develops a numerical method based on the Fokas transform to solve third order linear PDE boundary value problems on finite intervals, effectively handling boundary conditions via contour deformation for accurate solutions.

Contribution

It introduces a novel numerical implementation of the Fokas transform for third order PDEs, addressing boundary condition imposition through contour deformation techniques.

Findings

Accurate numerical solutions for third order PDE boundary value problems.

Effective contour deformation strategy for integral evaluation.

First known numerical implementation of Fokas transform for such problems.

Abstract

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.