Fokas-type closed-form solution formulae for Sobolev-type equations with time-dependent coefficients

TL;DR

This paper develops explicit integral solution formulas for Sobolev-type evolution PDEs with time-dependent coefficients using the Fokas method, enabling analysis of complex mixed-derivative equations on the half-line.

Contribution

It extends the Fokas unified transform method to handle Sobolev-type equations with time-dependent coefficients and mixed derivatives, providing closed-form solutions for a broad class of equations.

Findings

Derived explicit integral representations for solutions

Successfully applied to various Sobolev-type equations with mixed derivatives

Resolved complex-analytic and algebraic challenges in the process

Abstract

We analytically derive novel explicit integral representations for the solution of nonhomogeneous initial-boundary-value problems for a large category of evolution partial differential equations of Sobolev-Galpern type with generic temporally variable coefficients, satisfying suitable mild conditions, and with arbitrary data in classical function spaces. This work is based on the careful implementation of the pioneering Fokas unified transform methodology alongside its recently-proposed extension for solving a class of linear evolution equations with dispersion relation of specific polynomial type and time-dependent coefficients. We herein effectively extend those techniques to a special collection of evolution equations with time-dependent coefficients and mixed spatiotemporal derivatives, which induce rational dispersion relations. The new approach is exhibited in detail through illustrative generation of closed-form solutions for a multitude of such equations (such as Milne-Taylor-Barenblatt-Coleman-Ting-Chen-type, Benjamin-Bona-Mahony-type, as well as numerous higher-order variants) posed on the half-line. Challenging technical difficulties of complex-analytic and of algebraic flavour naturally emerge due the presence of mixed-derivative terms, and these are appropriately resolved in each case. The new formulas are of utility in subsequent investigation of qualitative properties and analysis of nonlinear counterparts too. Further extensions, generalizations, rigorous aspects and implementations to other types of problems as well will soon be reported in forthcoming publications.