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.
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We thank the reviewers for their careful reading of this complicated paper and their helpful comments.
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Communicated by: Jan Hesthaven
The research of the first author is supported in part by the National Natural Science Foundation of China under grants 11771128 and 11101130, the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province (No. UNPYSCT-2016020), and the special fund of Heilongjiang University of the Fundamental Research Funds for Universities in Heilongjiang Province (No. HDJCCX-2016211). The research of the second author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF U1530401.
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Liang, H., Stynes, M. Collocation methods for general Riemann-Liouville two-point boundary value problems. Adv Comput Math 45, 897–928 (2019). https://doi.org/10.1007/s10444-018-9645-1
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DOI: https://doi.org/10.1007/s10444-018-9645-1
Keywords
- Fractional derivative
- Riemann-Liouville derivative
- Two-point boundary value problem
- Volterra integral equation
- Collocation methods