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.
 A: This is just an extended comment, where I list what I know about the question, but it does not actually answer the question. However, it suggests that the general answer might be complicated.
Suppose that we consider zero "exterior" condition: $u(x) = 0$ when $x \leqslant 0$ (I think I have very little to say in the more general case).

*

*The equation $(-\Delta)^\alpha u = 0$ (i.e. $M = K = C = 0$) admits two linearly independent positive solutions: $u(x) = x^\alpha$ and $x^{\alpha - 1}$. This is a rather old result: more general case has been studied by Silverstein in 1980, https://www.jstor.org/stable/2243167.


*The equation $(-\Delta)^\alpha u + M u' = 0$ (i.e. $K = C = 0$, $M \ne 0$) is also covered by Silverstein. There should be an "explicit" expression (involving some integrals): roughly speaking, one has to identify the appropriate Wiener–Hopf factor and invert the Laplace transform.


*The equation $(-\Delta)^\alpha u + C = 0$ (i.e. $M = K = 0$, $C \ne 0$) has no solutions. Off the top of my head, I do not have a reference, but the argument is based on the following fact: the same equation in $(0, R)$ has a solution $c_\alpha C (x - R x)_+^\alpha$, which goes to infinity as $R \to \infty$ — and there is no way to compensate that by imposing appropriate behaviour for $x > R$.


*The eigenvalue problem $(-\Delta)^\alpha u + K u = 0$ (i.e. $M = C = 0$, $K \ne 0$) has an "explicit" solution if $K < 0$, which behaves as $\sin(|K|^{1/\alpha} x + \tfrac{(1 - \alpha) \pi}{4})$ away from the boundary, but the formula is somewhat complicated. See Example 6.1 in my paper from 2011, https://doi.org/10.4064/sm206-3-2. Another formula involves the double sine function, see my paper with Alexey Kuznetsov from 2018, https://doi.org/10.1214/18-EJP134. The method


*The same equation for $K > 0$ (or, more generally, complex $K \notin (-\infty, 0]$) is somewhat simpler and, if I remember correctly, it is also addressed in the above paper with Alexey Kuznetsov.


*The equation $(-\Delta)^\alpha u + M u' + K u = 0$ can be dealt with by similar methods. This is not (yet) written anywhere, I suppose, but some preliminary work is in my preprint from 2018, https://arxiv.org/abs/1811.06617.


*My (vague) impression is that if $C \ne 0$ and $M < 0$, there will be no solution, while if $C \ne 0$ and $M > 0$, there should be a solution that could possibly be found again by applying similar methods.
(I am sorry if the above looks like I was just advertising my own work. This is not what I intended; it just so happened that I had been working quite a while on particular variants of your equation.)
