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.
Highlights
The variable-order fractional (VO-F) operators [8,17], which are generalizations of constant-order fractional operators [23], open up new possibilities for robust mathematical modeling and simulation of diverse physical problems in science and engineering, such as modeling of diffusive-convective effects on the oscillatory flows [15], linear and nonlinear oscillators with viscoelastic damping [8], processing of geographical data using VO-F derivatives [9], constitutive laws in viscoelastic continuum mechanics [16], signature verification through variable/adaptive fractional order differentiators [21], anomalous diffusion problems [10, 24] and chloride ions sub-diffusion in concrete structures [22]
Fu et al [10] applied the method of approximate particular solutions to VO-F diffusion models
Tayebi et al [20] proposed an accurate and robust meshless method based on the moving least squares approximation and the finite difference scheme for the numerical solution of VO-F advection-diffusion equation on two-dimensional arbitrary domains
Summary
The variable-order fractional (VO-F) operators [8,17], which are generalizations of constant-order fractional operators [23], open up new possibilities for robust mathematical modeling and simulation of diverse physical problems in science and engineering, such as modeling of diffusive-convective effects on the oscillatory flows [15], linear and nonlinear oscillators with viscoelastic damping [8], processing of geographical data using VO-F derivatives [9], constitutive laws in viscoelastic continuum mechanics [16], signature verification through variable/adaptive fractional order differentiators [21], anomalous diffusion problems [10, 24] and chloride ions sub-diffusion in concrete structures [22]. Tayebi et al [20] proposed an accurate and robust meshless method based on the moving least squares approximation and the finite difference scheme for the numerical solution of VO-F advection-diffusion equation on two-dimensional arbitrary domains. We focus on the computation of the VO-F integrals and derivatives of the modified generalized Laguerre polynomials. Using the modified generalized Laguerre polynomials as the basis functions, we develop Laguerre-Gauss collocation methods to solve fractional differential equations of variable and constant orders on the half line. Numerical algorithms for calculating the VO-F integral and the Caputo derivative are presented in Sections 3 and 4, respectively.
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