A Boundary-Adapted Crank–Nicolson-Induced Mamadu–Njoseh Basis Functions for the Solution of Parabolic Partial Differential Equations

Abstract

A boundary-adapted Crank–Nicolson-induced Mamadu–Njoseh basis-functions (CN-MNBF) framework is developed for the numerical solution of one-dimensional parabolic partial differential equations. The method combines a weighted Mamadu–Njoseh polynomial approximation in space with Crank–Nicolson time discretization. To enforce homogeneous Dirichlet conditions exactly, the numerical implementation uses boundary-adapted trial functions obtained by multiplying each Mamadu–Njoseh polynomial (MNP) by the boundary factor (1 - x2) on the transformed interval [-1,1]. The resulting mass and spatial-operator matrices are assembled directly in this constrained space. For the diffusion problem, a weighted energy argument establishes dissipativity and the associated Crank–Nicolson update is contractive in the mass-matrix norm. The fully discrete error splits into spatial approximation and temporal discretization components, with the temporal component second-order under standard smoothness assumptions. Numerical illustrations show decreasing spatial error and, for a sufficiently rich spatial approximation, approximately second-order temporal convergence. We interpret the induced basis as an operator-adapted representation of the Crank–Nicolson evolution rather than a new finite-dimensional spatial span.