A finite difference method for a two-point boundary value problem with a Caputo fractional derivative

Abstract

A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order δ ∈ (1, 2) is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution u of the boundary value problem are established, showing that u ′′ (x) may be unbounded at the interval endpoint x = 0. These bounds and a discrete comparison principle are used to prove pointwise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield stability on coarse meshes for all values of δ. Numerical results are presented to illustrate the performance of the method. Fractional differential equation; Caputo fractional derivative; boundary value problem; derivative bounds; finite difference method; convergence proof.

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

“…All these results show that our main convergence result (Theorem 4.2) is sharp. We see also that our collocation method is much more accurate (and less expensive) than the finite difference methods for (1.5) that are discussed in [6,12].…”

“…In the current paper, we contribute to these developments by describing and analysing a numerical method for a two-point boundary value problem whose leading term is a Caputo fractional derivative. The conditions c ≥ 0 and (1.6) guarantee that (1.5) satisfies a suitable comparison/maximum principle, from which existence and uniqueness of the solution u of (1.5) follows; see [12] and Theorem 2.1 below.…”

“…It is also discussed in [1]. Numerical methods for its solution are presented in [6,12] and their references. We assume that b, c, f ∈ C[0, 1]; further hypotheses will be placed later on the regularity of these functions.…”

Solving fuzzy fractional differential equations using fuzzy Sumudu transform

“…Later, it was studied in [31], where the authors proposed some applications. When dealing with fractional differential equations, the terms such as Riemann-Liouville, Grünwald-Letnikov and Caputo fractional derivative are considered by many authors [21,25,30,39,42]. Of the three definitions for derivative stated, Riemann-Liouville and Caputo fractional derivatives appeared to be more popular.…”

Estimates of higher order fractional derivatives at extreme points

“…The estimates were derived for the Caputo and RiemannLiouville fractional derivatives with fractional derivatives 0 < δ 1 < 1 and 1 < δ 2 < 2. These results were used to develop new maximum principles and establish existence and uniqueness results for several types of fractional equations [1, 3, 5-7, 11, 17], and to develop numerical schemes for certain fractional differential equations [16,18,22]. Also, the results concerning the Riemann-Liouville fractional derivative, were used and generalized for weaker conditions on [6,18].…”