A Boundary-Adapted Crank–Nicolson-Induced Mamadu–Njoseh Basis Functions for the Solution of Parabolic Partial Differential Equations
Ebimene James Mamadu *
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
Jude Chukwuyem Nwankwo
Department of Mathematics, University of Delta, Agbor, Nigeria.
Newton Ighomaro Okposo
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
Ebikonbo-owei Anthony Mamadu
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
Nkonyeasua Ignatius Njoseh
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
Henrietta Ify Ojarikre
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
Jonathan Tsetimi
Department of Mathematics, Faculty of Science, Delta State University, Abraka, Nigeria.
*Author to whom correspondence should be addressed.
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.
Keywords: Crank–Nicolson method, Mamadu–Njoseh polynomials, boundary-adapted basis functions, weighted spectral method, parabolic partial differential equations