A Multi‐Domain Space‐Time Spectral Collocation Method for the Huxley Equation

ABSTRACT A multi‐domain space‐time spectral collocation scheme is proposed for the mixed problem of the Huxley equation, utilizing Legendre–Gauss–Lobatto nodes as interpolation points. The Huxley equation is transformed into a system of nonlinear matrix equations, with the numerical solution obtained through fixed‐point iteration. A novel composite quasi‐orthogonal approximation is introduced. Unlike existing methods that only ensure accurate fitting of function values at the interval endpoints, this new approximation precisely fits not only the function values at the endpoints but also at the common points of adjacent subdomains. Using the Petrov–Galerkin spectral method with numerical integration, the convergence and stability of the proposed scheme for solving the Huxley equation are established. Numerical experiments confirm the high accuracy of the proposed algorithm.