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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 deriv...
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... In the Galerkin method, the two sets coincide; see [16][17][18]. In tau and collocation methods, we have more freedom to choose basis functions, for example; see [19][20][21][22][23]. The collocation method is advantageous due to its easy implementation; see, for example, [24][25][26][27]. ...
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.
... In [9], a numerical approach for a heat transfer model was proposed. In [10], the authors used a tau approach using Lucas polynomials for the time-fractional diffusion equation. ...
Citation: Abd-Elhameed, W.M.; Hafez, R.M.; Napoli, A.; Atta, A.G. A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations. Algorithms 2025, 18, 2. Abstract: This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms.
... The authors of [13,14] presented findings on the Fibonacci and Lucas polynomials and their relationships with various polynomials, such as various orthogonal polynomials. The authors of [15] derived some formulas for Lucas polynomials and employed them to find spectral solutions to the timefractional diffusion equation. Additional references for these polynomials are available in [16][17][18][19]. ...
This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas.
