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

Abstract

General Riemann-Liouville linear two-point boundary value problems of order α_p, where n − 1 < α_p < 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 β_j 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.