Numerical exploration of a class of fractional order partial differential equations with generalised Narayana polynomials

Abstract This research proposes a novel approach to construct operational matrices for Liouville–Caputo fractional derivatives in the framework of generalised Narayana polynomials, ensuring exact representation rather than mere approximation. By virtue of operational matrices, the differential problem is transformed into a system of equations, where the unknowns are the coefficients used to approximate the solution. This system is numerically solved by the software “Mathematica” by employing the collocation technique in conjunction with operational matrices. These matrices derive approximate solutions with negligible error and reduce computational time for a class of fractional partial differential equations of orders within (0, 2] that subsume a range of well-known differential equations, including but not limited to the Klein–Gordon, Burger, and advection-reaction-diffusion equations. Moreover, the methodology permits unrestricted selection of domain size, allowing complete flexibility in defining the range of variables in contrast to available results for Legendre and Genocchi polynomials, etc. Further, bounds for the approximation of the solution are derived for the reliability of the approximation. The efficacy of this method is demonstrated through extensive numerical examples, with the results showing a strong correlation with existing solutions, particularly in terms of accuracy and reduced computational time.