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

Abstract

Abstract A two-point boundary value problem is considered on the interval [ 0 , 1 ] $[0,1]$ , where the leading term in the differential operator is a Caputo fractional derivative of order δ with 1 < δ < 2 $1<\delta <2$ . Writing u for the solution of the problem, it is known that typically u ' ' ( x ) $u^{\prime \prime }(x)$ blows up as x → 0 $x\rightarrow 0$ . A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: u may exhibit a boundary layer at x = 1 when δ 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 u). This analysis proves that usually no boundary layer can occur in the solution u at x = 0, and that the quantity M = max x ∈ [ 0 , 1 ] b ( x ) $M = \max _{x\in [0,1]}b(x)$ , where b is the coefficient of the first-order term in the differential operator, is critical: when M < 1 $M<1$ , no boundary layer is present when δ is near 1, but when M ≥ 1 then a boundary layer at x = 1 is possible. Numerical results illustrate the sharpness of most of our results.