Fractional derivative of Abel type on a half-line

Abstract:

We consider the initial-boundary value problem on a half-line for an evolution equation \[ \left ( \partial _{t}+\left \vert \partial _{x}\right \vert ^{\alpha }\right ) u(x,t)=f(x,t),t>0,\ x>0, \] with a fractional derivative of Abel type \begin{equation*} \text {(0.1)\qquad \qquad \qquad \qquad \qquad }\left \vert \partial _{x}\right \vert ^{\alpha }u=\mathcal {R}^{1-\alpha +\left [ \alpha \right ] }\partial _{x}^{\left [ \alpha \right ] +1}u, \text {\qquad \qquad \qquad \qquad \quad }\end{equation*} where $\left [ \alpha \right ]$ denotes the integer part of number $\alpha >0,\alpha$ is not equal to an integer, and\[ \mathcal {R}^{\alpha }u=\frac {1}{2\Gamma (\alpha )\sin (\frac {\pi }{2}\alpha )}\int _{0}^{+\infty }\frac {\mathrm {sign}(x-y)u(y)}{\left \vert x-y\right \vert ^{1-\alpha }}dy \] is the modified Riesz potential. We study traditionally important problems of a theory of partial differential equations, such as existence and uniqueness of solution. We propose a new method of solution. Also we get a closed form of the solution.