The Fokas unified transform method

Abstract: The method was developed to solve initial-boundary value problems for integrable nonlinear partial differential equations (PDEs) having a Lax pair. It turned out that the method also provides a unified algorithmic approach to solving initial-boundary and boundary value problems for linear PDEs. At the same time, the Fokas method allows solving problems that cannot be solved by the Fourier method.

In the talk, we describe the Fokas method as applied to solving initial-boundary value problems for linear evolutionary PDEs. We show how the Fokas method is used to solve initial-boundary value problems for a pseudoparabolic PDE and systems of linear PDEs, see [1-4]. In the final part of the talk, the Fokas method is demonstrated in its most general form (using the Lax pair, reducing to the Riemann-Hilbert problem), which is applicable to both linear and nonlinear PDEs.

References:

1. S. A. Rukolaine, Integral Representation of Solutions to Initial-Boundary Value Problems on a Finite Interval in the Framework of the Hyperbolic Heat Equation, Math. Meth. Appl. Sci., 48:13760-13772, 2025. doi.org/10.1002/mma.11140

2. S. A. Rukolaine, Analytical representation of heat waves on a finite interval in the framework of the hyperbolic heat equation. Proceedings of the International Conference ``Days on Diffraction 2025'', St. Petersburg, Russia, June 16-20, 2025, pp. 181-187. doi.org/10.1109/DD66835.2025.11263496

3. A. Chatziafratis, S. A. Rukolaine, E. C. Aifantis, Integral representations for the Barenblatt-Sobolev-Galpern pseudo-parabolic equation: boundary-value and interface problems on unbounded and finite intervals, Russ. J. Math. Phys., 2026. (Accepted)

4. S. A. Rukolaine, The Guyer-Krumhansl model of heat conduction on a finite interval. (Submitted) arxiv.org/abs/2502.20057v2

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods