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.
Highlights
The fractional-order differential equations have been widely used for describing a variety of phenomena in physics, astrophysics, medicine, chemistry, optimal control, engineering, biology, fluid dynamics, etc
The ordinary/partial differential equations that contain fractional-order derivatives provide more flexible models compared with the classical ones that are characterized by integer-orders [2, 7, 10]
7 Conclusions In the present article, we have introduced a trustworthy method for solving a mathematical physics model of fractional-order, advection–dispersion equation, numerically
Summary
The fractional-order differential equations have been widely used for describing a variety of phenomena in physics, astrophysics, medicine, chemistry, optimal control, engineering, biology, fluid dynamics, etc. (see, for instance, [9, 18, 26,27,28]). Agarwal and El-Sayed Advances in Difference Equations (2020) 2020:626 lutions depend on several techniques such as finite difference, finite volume, variational iteration, Legendre polynomials, Chebyshev collocation, homotopy perturbation, operational matrix, variational iteration, Adams–Bashforth, nonstandard finite difference, sinccollocation, compact finite difference, tau method, block pulse, decomposition, radial basis, Taylor collocation, and wavelets spectral (see, for example, [5, 11, 24, 25, 35, 38]) All these methods introduce numerical solutions for many types of fractional-order differential equations. We strive to solve the fractional-order advection–dispersion equation numerically via a nonstandard finite difference method besides a collocation method that depends on a new class of orthogonal polynomials (Vieta–Lucas). √1 4–x2 is the weight function corresponding to VLn(x)
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