A unified spectral strategy for 1D and 2D nonlinear hyperbolic telegraph equations using shifted Vieta–Lucas operational matrices

This work introduces an innovative spectral collocation framework for addressing one- and two-dimensional nonlinear hyperbolic telegraph-type (NHTT) equations. The methodology employs specially designed shifted Vieta–Lucas polynomials (SVLPs) that naturally conform to prescribed initial-boundary conditions (IBCs). Through the development of operational matrices (OMs) for polynomial derivatives, the approach transforms differential equations into algebraic systems. The technique effectively handles both homogeneous and nonhomogeneous boundary constraints via appropriate transformations. Validation through diverse test problems with constant and variable coefficients confirms the method’s computational strength, achieving near-machine precision with absolute errors reaching [Formula: see text]. Theoretical examinations of convergence and stability, framed within weighted Sobolev spaces, substantiate the robustness of the proposed framework for complex telegraph-type models in computational physics and engineering domains.