This manuscript aims to provide numerical solutions for the FitzHugh–Nagumo (FH–N) problem. The suggested approximate solutions are spectral and may be achieved using the standard collocation technique. We introduce and utilize specific polynomials of the generalized Gegenbauer polynomials. These introduced polynomials have connections with Chebyshev polynomials. The polynomials' series representation, orthogonality property, and derivative expressions are among the new formulas developed for these polynomials. We transform these formulas to obtain their counterparts for the shifted polynomials, which serve as basis functions for the suggested approximate solutions. The convergence of the expansion is thoroughly examined. We provide several numerical tests and comparisons to confirm the applicability and accuracy of our proposed numerical algorithm.

This article introduces an efficient spectral collocation framework for numerically solving the time-fractional Huxley equation. New basis functions of shifted Dickson polynomials of the first kind are introduced and employed. To achieve this, new formulas for the shifted polynomials are derived, including a series representation, an inverse formula, and expressions for both integer and fractional derivatives, which together with the collocation method serve as the foundation of the proposed numerical algorithm for converting the equation with its governing conditions into a non-linear algebraic system. A convergence and error analysis of the proposed method is also provided. We present numerical results and compare them with existing methods to illustrate the high accuracy of the proposed algorithms and their applicability.

We establish a new sequence of polynomials that combines the Fibonacci and Lucas polynomials. We will refer to these polynomials as merged Fibonacci-Lucas polynomials (MFLPs). We will show that we can represent these polynomials by combining two certain Fibonacci polynomials. This formula will be essential for determining the power form representation of these polynomials. This representation and its inversion formula for these polynomials are crucial to derive new formulas about the MFLPs. New derivative expressions for these polynomials are given as combinations of several symmetric and non-symmetric polynomials. We also provide the inverse formulas for these formulas. Some new product formulas involving the MFLPs have also been derived. We also provide some definite integral formulas that apply to the derived formulas.

Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient.

The main aims of this study are the introduction of Pell coefficient polynomials and their numerical treatment of the first-order hyperbolic partial differential equations. Our suggested numerical algorithm will be derived from the utilization of some novel formulas of the Pell coefficient polynomials, along with the application of the spectral tau method. For the proposed expansion, we investigate the convergence and error estimations in detail. The presented numerical results indicate that the suggested numerical method is accurate, converges exponentially, and is computationally efficient.

In this work, a new spectral Galerkin approach to solving the time-fractional diffusion-wave equation (TFDWE) with non-homogeneous initial and boundary conditions is presented. A suitable transformation is used to convert the TFDWE governed by non-homogeneous conditions into a modified one governed by homogeneous conditions. New basis functions in terms of specific shifted Horadam polynomials are used. Some new definite integral formulas that are crucial to the numerical implementation are developed, and the Galerkin scheme is analyzed in detail to obtain the approximate solutions. A thorough convergence and error analysis of the proposed expansion is established. A number of numerical experiments are carried out to show the scheme's applicability and accuracy when compared to other methods.

This work introduces a computational method for solving the time-fractional cable equation (TFCE). We utilize the tau method for the numerical treatment of the TFCE, using generalized Chebyshev polynomials of the third kind (GCPs3) as basis functions. The integer and fractional derivatives of the GCPs3 are the essential formulas that serve to transform the TFCE with its underlying conditions into a matrix system. This system can be solved using a suitable algorithm to obtain the desired approximate solutions. The error bound resulting from the approximation by the proposed method is given. The numerical algorithm has been validated against existing methods by presenting numerical examples.