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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 897–928 | Cite as

Collocation methods for general Riemann-Liouville two-point boundary value problems

  • Hui Liang
  • Martin StynesEmail author
Article

Abstract

General Riemann-Liouville linear two-point boundary value problems of order αp, where n − 1 < αp < n for some positive integer n, are investigated on the interval [0,b]. It is shown first that the natural degree of regularity to impose on the solution y of the problem is \(y\in C^{n-2}[0,b]\) and \(D^{\alpha _{p}-1}y\in C[0,b]\), with further restrictions on the behavior of the derivatives of y(n− 2) (these regularity conditions differ significantly from the natural regularity conditions in the corresponding Caputo problem). From this regularity, it is deduced that the most general choice of boundary conditions possible is \(y(0) = y^{\prime }(0) = {\dots } = y^{(n-2)}(0) = 0\) and \({\sum }_{j = 0}^{n_{1}}\beta _{j}y^{(j)}(b_{1}) =\gamma \) for some constants βj and γ, with b1 ∈ (0,b] and \(n_{1}\in \{0, 1, \dots , n-1\}\). A wide class of transformations of the problem into weakly singular Volterra integral equations (VIEs) is then investigated; the aim is to choose the transformation that will yield the most accurate results when the VIE is solved using a collocation method with piecewise polynomials. Error estimates are derived for this method and for its iterated variant. Numerical results are given to support the theoretical conclusions.

Keywords

Fractional derivative Riemann-Liouville derivative Two-point boundary value problem Volterra integral equation Collocation methods 

Mathematics Subject Classification (2010)

65L10 65R10 65R20 

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Notes

Acknowledgments

We thank the reviewers for their careful reading of this complicated paper and their helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsShenzhen UniversityShenzhenChina
  2. 2.Beijing Computational Science Research CenterBeijingChina

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