Nonpolynomial collocation approximation of solutions to fractional differential equations

By Neville J. Ford, Maria Luísa Morgado, Magda Rebelo. 2013

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We propose a non-polynomial collocation method for solving fractional differential equations. The construction of such a scheme is based on the classical equivalence between certain fractional differential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional differential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/dc771adc9d7c8b36c6dd4429ecb866f70f062fc8

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Citations in the corpus

Higher order numerical methods for solving fractional differential equations (2014, methodology): “Ford, Morgado and Rebelo recently (see [16]) used a nonpolynomial collocation method to achieve good convergence properties without”
An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017, unknown): “1) are sufficiently smooth, for example u ∈ C2[0, T ], there are many time discretisation schemes in the literature, for example, Lubich’s fractional multistep methods [7], [30], [55], [65], [61], [62], [64], the L1 scheme and its modification [28], [26], [18], [29], [47], the spectral method [58] [”