Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials

Abstract

a b s t r a c tThis paper presents a computational technique based on the collocation method and Müntz polynomials for the solution of fractional differential equations. An appropriate representation of the solution via the Müntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations. The main advantage of the present method is its superior accuracy and exponential convergence. Consequently, one can obtain good results even by using a small number of collocation points. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

“…Although there is comprehensive literature on the numerical methods for solving equations involving fractional derivatives and integrals (cf. [8,9,11,3,10,35]), there seems to exist a few literature on automatic quadrature for the fractional derivatives, see e.g. [20,24,39].…”

“…Problems with algebraic or/and logarithmic singularities can be solved using Müntz systems and quadratures of this type (cf. [30,32,11]) or using a procedure proposed in [20].…”

Numerical solution of fractional differential equations via a Volterra integral equation approach

“…Ford and Connolly [13], Diethelm et al [9,14] and Magin et al [11] have reviewed some of the existing methods and demonstrated their respective strengths and weaknesses. There are some further methods, such as fractional Adams method [15], operational matrix [16], product integration rules [17], spectral collocation method [18], a modified variational iteration method [19] and other methods [20][21][22][23][24][25][26][27]. All these methods essentially deal with the non-locality of the fractional operator in the same way, namely, by sampling the function on a more or less regular grid in [0 T ].…”

“…Furthermore, the fractional integrals and derivatives of a classical polynomial are not polynomials, so we may not be able to obtain a good approximation for the fractional integrals and derivatives via the classical polynomials. However, the fractional order polynomials give more accurate results than the classical polynomials [18]. In the current article, a different approach for the numerical treatment of fractional differential equations is proposed.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…(2) Use some nonpolynomial (or singular) basis functions or collocation spectral methods to capture the singularity of the solutions of (1.1), see [1], [5], [13], [14], [34], [27], [59], [63], [67]. (3) Separate the solution into two parts: smooth and nonsmooth parts.…”