Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations

Sabermahani, Sedigheh; Ordokhani, Yadollah; Yousefi, S. A.

doi:10.1007/s40314-017-0547-5

Cited by 44 publications

(20 citation statements)

References 28 publications

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“…It is a fact that the analytical solution of a classical fractional differential equation is difficult to obtain, thus many numerical methods have been developed such as finite difference method, 13 finite element method, 14 spectral method, 15 nonpolynomial spline method, 16‐21 and other methods involving fractional‐order Lagrange polynomials, 22 Legendre‐Laguerre polynomials, 23 Genocchi hybrid functions, 24 Bernoulli wavelets, 25 to name a few. As one would expect, it is even more difficult to obtain analytical solutions of equations involving generalized fractional derivatives, and numerical treatment is more appropriate for such problems.…”

Section: Introductionmentioning

confidence: 99%

A higher order numerical scheme for generalized fractional diffusion equations

Ding

Wong

2020

Numerical Methods in Fluids

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In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.

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Section: Introductionmentioning

confidence: 99%

A higher order numerical scheme for generalized fractional diffusion equations

Ding

Wong

2020

Numerical Methods in Fluids

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“…Definition 1 Assume that f : [a, b] → R, ν ∈ R, ν > 0, n = ν , the following Riemann-Liouville fractional integral is defined as (Sabermahani et al 2018):…”

Section: Preliminariesmentioning

confidence: 99%

Two-dimensional Müntz–Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations

2020

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In this manuscript, we present a new numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Müntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works. Keywords Müntz-Legendre polynomial • Hybrid functions • Operational matrix • Two-dimensional hybrid functions Mathematics Subject Classification 65M70 • 65N35 Recently, fractional calculus is one of the worthy and essential tools for modeling many phenomena in physics (Hilfer 2000), engineering applications (Magin 2004), viscoelasticity Communicated by José Tenreiro Machado.

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“…In recent years, fractional equations appear in modeling of various real-life phenomena, for example dynamic viscoelasticity modeling (Larsson et al 2015), hydrologic (Benson et al 2013), economics (Baillie 1996), temperature and motor control (Bohannan 2008), solid mechanics (Rossikhin and Shitikova 1997), bioengineering (Magin 2004), medicine (Hall and Barrick 2008), porous media (He 1998), fluid-dynamic traffic model (He 1999), etc. Therefore, there is an increasing demand for numerical and analytical solutions of various types of fractional differential equations such as finite-difference/finite-element technique Abbaszadeh 2018a, b, 2019), compact difference schemes (Hu and Zhang 2012), homotopy analysis method (Dehghan et al 2010), homotopy perturbation method (Abdulaziz et al 2008), dual reciprocity boundary elements method (Dehghan and Safarpoor 2016), fifth-kind orthonormal Chebyshev polynomial method (Abd-Elhameed and Youssri 2018), B-spline functions (Lakestani et al 2012), fractional-order Bernoulli function method , hybrid method (Mashayekhi and Razzaghi 2015), Legendre wavelet method (Heydari et al 2014), fractional-order Lagrange polynomials (Sabermahani et al 2018), fractional-order Bernoulli wavelets method (Rahimkhani et al 2016), Müntz-Legendre wavelet method (Rahimkhani et al 2018), etc.…”

Section: Introductionmentioning

confidence: 99%

The bivariate Müntz wavelets composite collocation method for solving space-time-fractional partial differential equations

Rahimkhani

Ordokhani

2020

Comp. Appl. Math.

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Herein, we consider an effective numerical scheme for numerical evaluation of three classes of space-time-fractional partial differential equations (FPDEs). We are going to solve these problems via composite collocation method. The procedure is based upon the bivariate Müntz-Legendre wavelets (MLWs). The bivariate Müntz-Legendre wavelets are constructed for first time. The bivariate MLWs operational matrix of fractional-order integral is constructed. The proposed scheme transforms FPDEs to the solution of a system of algebraic equations which these systems will be solved using the Newton's iterative scheme. Also, the error analysis of the suggested procedure is determined. To test the applicability and validity of our technique, we have solved three classes of FPDEs.

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Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations

Cited by 44 publications

References 28 publications

A higher order numerical scheme for generalized fractional diffusion equations

A higher order numerical scheme for generalized fractional diffusion equations

Two-dimensional Müntz–Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations

The bivariate Müntz wavelets composite collocation method for solving space-time-fractional partial differential equations

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