In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated.

This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type ${}_5{F}_4\left(1\right)$. The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms.

This study presents Galerkin and collocation algorithms based on Telephone polynomials (TelPs) for effectively solving high-order linear and non-linear ordinary differential equations (ODEs) and ODE systems, including those with homogeneous and nonhomogeneous initial conditions (ICs). The suggested approach also handles partial differential equations (PDEs), emphasizing hyperbolic PDEs. The primary contribution is to use suitable combinations of the TelPs, which significantly streamlines the numerical implementation. A comprehensive study has been conducted on the convergence of the utilized telephone expansions. Compared to the current spectral approaches, the proposed algorithms exhibit greater accuracy and convergence, as demonstrated by several illustrative examples that prove their applicability and efficiency.

This article employs certain polynomials that generalize standard Fermat polynomials, called convolved Fermat polynomials, to numerically solve the fractional Burgers’ equation. New theoretical results of these polynomials are developed and utilized along with the collocation method to find approximate solutions of the fractional Burgers’ equation. The basic idea behind the proposed numerical algorithm is based on establishing the operational matrices of derivatives of both integer and fractional derivatives of the convolved Fermat polynomials that help to convert the equation governed by its underlying conditions into an algebraic system of equations that can be treated numerically. A comprehensive study is performed to analyze the error of the proposed convolved Fermat expansion. Some numerical examples are presented to test our proposed numerical algorithm, and some comparisons are made. The results indicate that the proposed algorithm is applicable and accurate.

This paper presents new results of Bernoulli polynomials. New derivative expressions of some celebrated orthogonal polynomials and other polynomials are given in terms of Bernoulli polynomials. Hence, some new connection formulas between these polynomials and Bernoulli polynomials are also deduced. The linking coefficients involve hypergeometric functions of different arguments that can be summed in some cases. Formulas that express some celebrated numbers in terms of Bernoulli numbers are displayed. Based on the new connection formulas between different polynomials and Bernoulli polynomials, along with some well-known integrals involving these polynomials, new closed forms for some definite integrals are given.

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.