Vieta–Lucas polynomials for solving a fractional-order mathematical physics model

Abstract

In this article, a fractional-order mathematical physics model, advection–dispersion equation (FADE), will be solved numerically through a new approximative technique. Shifted Vieta–Lucas orthogonal polynomials will be considered as the main base for the desired numerical solution. These polynomials are used for transforming the FADE into an ordinary differential equations system (ODES). The nonstandard finite difference method coincidence with the spectral collocation method will be used for converting the ODES into an equivalence system of algebraic equations that can be solved numerically. The Caputo fractional derivative will be used. Moreover, the error analysis and the upper bound of the derived formula error will be investigated. Lastly, the accuracy and efficiency of the proposed method will be demonstrated through some numerical applications.

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

“…Vieta-Lucas polynomials belong to the class of orthogonal polynomials and can be created by using the recurrence relation [35]. Consider |z| ≤ 2; then, the Vieta-Lucas polynomials of degree n ∈ N 0 in the variable z can be defined as…”

Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

“…Let j P N 0 " N Y t0u. The VL polynomial of degree j P N 0 is given by [36,37] VL j pξq :" 2 cospjϕq, ξ " 2 cos ϕ, ϕ P r0, πs.…”

Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

“…For some recent results for boundary value problems involving left or/and right fractional derivatives, we refer to the papers [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and references therein.…”