fa.functional analysis – Fractional Laplacian problem on half-line

Is it possible to obtain an explicit solution for the following fractional problem on the half-line?
$$(-Delta)^alpha u(x) + M u'(x) + K u(x) + C = 0 quad text{ in } (0,infty)$$
$$u(x) = a, quad u'(x) = b text{ in } (-infty, 0)$$
where $M,K,C,a,b$ are constants and $(-Delta)^alpha$ is the Fractional Laplacian.