New Recursive Approximations for Variable-Order Fractional Operators With Applications

Abstract

To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.

Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

“…Now, we apply directly the collocation method to solve (27)- (28). Using the nodes x i (0 ≤ i ≤ M) which are the shifted Legendre-Gauss-Lobatto roots of L L M (x) and t j (0 ≤ j ≤ N − 1) is the shifted Legendre roots of L τ N (x).…”

“…A generalization to the classical fractional calculus is given in their work by introducing the study of fractional integration and differentiation when the order is a function instead of a constant of arbitrary order [24,25]. As a result, a new generation of mathematicians and physicist is concerned with studying physical problems involving the variable order derivatives due to the property of memory incorporation for changes with time or spatial location (see, for example, [26][27][28]). Lorenzo and Hartley [29] gave the idea where the variable order operator is a varying function of the independent variables of differentiation or other unrelated variables which lead to the introduction of distributed order fractional operators.…”

A Simultaneous Inversion Problem for the Variable‐Order Time Fractional Differential Equation with Variable Coefficient

“…We refer to some work given by F. Liu and his group, see [6][7][8][9] for instance, also see the work given by Sun et al [10], and the master degree thesis given by Zhang [11] for numerical solutions to the variable-order time fractional diffusion models with variable coefficients. Recently, we refer to the work by Bhrawy and Zaky et al [12][13][14][15][16] for numerical methods for multidimensional space/time variable-order fractional differential equations, where approximate solutions of the forward variable-order fractional differential equations and spectral techniques are discussed based on orthogonal polynomials. On concrete applications of the fractional calculus models, we always encounter with inverse problems which need to determine some unknowns in the model using suitable additional data.…”

The minimax approach for a class of variable order fractional differential equation

“…Coimbra et al used this definition in the modeling of viscous‐viscoelastic oscillator. Recently, the Liouville‐Caputo variable‐order fractional derivative has been introduced in several physical fields . For the power function, the Liouville‐Caputo operator is given by This paper introduces a family of fractional‐order Chebyshev polynomials to solve a class of differential equations with Liouville‐Caputo variable‐order fractional derivative.…”

“…Recently, the Liouville-Caputo variable-order fractional derivative has been introduced in several physical fields. [45][46][47][48][49] For the power function, the Liouville-Caputo operator D α t ð Þ c is given by 33…”