An efficient collocation method for a Caputo two-point boundary value problem

Abstract

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Caputo fractional-order derivative of order 2 − δ with 0 < δ < 1. The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity u (x) − u (0), where u is the solution of the original problem. A collocation method that uses piecewise polynomials of arbitrary order is developed and analysed for this Volterra problem; then by postprocessing an approximate solution u h of u is computed. Error bounds in the maximum norm are proved for u − u h and u − u h . Numerical results are presented to demonstrate the sharpness of these bounds.

Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

“…One can easily deduce from the proof of that equivalence an existence result for the Caputo problem that is analogous to Theorem 2.5 and a uniqueness result analogous to Theorem 2.8. Furthermore, both these results require (in the notation of [6]) only α 0 ≥ 0 instead of the more restrictive hypothesis α 0 ≥ 1/(1 − δ) that was used throughout that paper.…”

“…Our aim here is to reformulate (1.5) in terms of Volterra integral equations in order to show existence, uniqueness and regularity of a solution to (1.5), and furthermore to facilitate its efficient numerical solution. A related reformulation was used in [6], where a Caputo boundary value problem was rewritten in terms of the continuous variable u , but in (1.5)-as we saw in Example 1.1-one may have u / ∈ C[0, 1], which would not fit with the standard Volterra theory in [1] so a different reformulation will be necessary here. Thus we shall rewrite (1.5) in terms of a lower-order fractional derivative of u that lies in C[0, 1].…”

“…Thus we shall rewrite (1.5) in terms of a lower-order fractional derivative of u that lies in C[0, 1]. Fundamentally, the Riemann-Liouville boundary value problem (1.5) is less well behaved than the analogous Caputo problem of [6] and requires more work for its satisfactory analysis and accurate numerical solution.…”

Boundary Layers in a Two-Point Boundary Value Problem with a Caputo Fractional Derivative

“…Set ( ) = ὔ ( ) − ὔ (0) for 0 ≤ ≤ 1. On multiplying (1.2a) by ( − ) −2 /Γ( − 1) then integrating from = 0 to = , after some manipulation of the fractional derivative term one obtains [7] a weakly singular Volterra integral equation of the second kind in the unknown : for 0 < ≤ 1,…”

“…It is also discussed in [1]. Numerical methods for its solution are presented in [5,7,15] and their references.…”