doi:10.1515/cmam-2014-0024engStynes,M.Gracia Lozano, José LuisBoundary layers in a two-point boundary value problem with a caputo fractional derivativeART-2015-89299A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Caputo fractional derivative of order ¿ with 1 < ¿ < 2. Writing ¿ for the solution of the problem, it is known that typically ¿¿¿(¿) blows up as ¿ ¿ 0. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: ¿ may exhibit a boundary layer at ¿ = 1 when ¿ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for ¿). This analysis proves that usually no boundary layer can occur in the solution ¿ at ¿ = 0, and that the quantity ¿ = max¿¿[0,1] ¿(¿), where ¿ is the coefficient of the first-order term in the differential operator, is critical: when¿ < 1,noboundarylayerispresentwhen¿isnear1,butwhen¿ = 1thenaboundarylayerat¿ = 1 is possible. Numerical results illustrate the sharpness of most of our results.2015http://zaguan.unizar.es/record/5608710.1515/cmam-2014-0024http://zaguan.unizar.es/record/56087oai:zaguan.unizar.es:56087info:eu-repo/grantAgreement/ES/MEC/MTM2010-16917Computational Methods in Applied Mathematics 15, 1 (2015), 79-95byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess